master thesis - uni-tuebingen.de · in the sub-riemannian geometry of the heisenberg group as the...

Università degli Studi di TrentoFacoltà di Scienze

Matematiche, Fisiche e Naturali

Corso di Laurea Magistralein Matematica

Eberhard KarlsUniversität Tübingen

Studiengang Mathematik

Master Thesis

The Isoperimetric Problemin the Sub-Riemannian Heisenberg Group

Supervisors:Prof. Frank LooseProf. Francesco Serra Cassano

Student:Elisa Paoli

Academic Year 2010-2011

Contents

Introduction 5

1 Recalls on Riemannian geometry and Geometric MeasureTheory 91.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Vector fields and Frobenius’ Theorem . . . . . . . . . . 91.1.2 Lie groups and Lie algebras . . . . . . . . . . . . . . . 13

1.2 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 Affine connection and Riemannian connection . . . . . 151.2.2 The exponential map . . . . . . . . . . . . . . . . . . . 171.2.3 Curvature of a Riemannian manifold . . . . . . . . . . 191.2.4 Second fundamental form and shape operator . . . . . 21

1.3 Recalls on geometric measure theory . . . . . . . . . . . . . . 251.3.1 The divergence theorem for manifolds embedded in Rn 251.3.2 Hausdorff measures . . . . . . . . . . . . . . . . . . . . 281.3.3 Rademacher’s Theorem and Area and Coarea formulas 291.3.4 First variation formula . . . . . . . . . . . . . . . . . . 321.3.5 Doubling measure and the Hardy-Littlewood

maximal operator . . . . . . . . . . . . . . . . . . . . . 33

2 The Isoperimetric problem in Euclidean Space 372.1 The isoperimetric inequality in the plane . . . . . . . . . . . . 372.2 The classical isoperimetric problem . . . . . . . . . . . . . . . 41

2.2.1 BV functions and perimeter . . . . . . . . . . . . . . . 422.2.2 Sobolev’s and Poincaré’s Inequalities for C1

c functions . 462.2.3 Isoperimetric Inequalities . . . . . . . . . . . . . . . . . 51

3 The Heisenberg Group and Sub-Riemannian Geometry 553.1 The first Heisenberg group H . . . . . . . . . . . . . . . . . . 55

3.1.1 The horizontal distribution and the horizontal linearmaps on H . . . . . . . . . . . . . . . . . . . . . . . . . 60

1

3.2 Carnot-Carathéodory distance . . . . . . . . . . . . . . . . . . 623.2.1 Horizontal paths . . . . . . . . . . . . . . . . . . . . . 623.2.2 The Korányi gauge and metric . . . . . . . . . . . . . . 653.2.3 Sub-Riemannian structure . . . . . . . . . . . . . . . . 67

3.3 Geodesic and bubble sets . . . . . . . . . . . . . . . . . . . . . 713.4 Riemannian approximants to H . . . . . . . . . . . . . . . . . 74

3.4.1 The gL metrics . . . . . . . . . . . . . . . . . . . . . . 743.4.2 Levi-Civita connection and curvature in the

Riemannian approximants . . . . . . . . . . . . . . . . 763.4.3 Gromov-Hausdorff convergence . . . . . . . . . . . . . 783.4.4 Carnot-Carathéodory geodesics and

Gromov-Hausdorff convergence . . . . . . . . . . . . . 81

4 Horizontal geometry of hypersurfaces in H 854.1 The second fundamental form in (R3, gL) . . . . . . . . . . . . 854.2 Horizontal geometry of hypersurfaces . . . . . . . . . . . . . . 86

4.2.1 The Legendrian foliation . . . . . . . . . . . . . . . . . 89

5 Sobolev and BV Spaces and first variation of the perimeterin H 915.1 Sobolev spaces, perimeter measure and total variation . . . . . 91

5.1.1 Riemannian perimeter approximation . . . . . . . . . . 955.2 Embedding theorems for the Sobolev and BV spaces . . . . . 97

5.2.1 The geometric case (Sobolev-Gagliardo-Nirenberg in-equality) . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.2 Compactness of the embedding BV ⊂ L1 on John do-mains . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 First variation of the perimeter . . . . . . . . . . . . . . . . . 1095.3.1 Parametric surface and noncharacteristic variations . . 1095.3.2 General variations . . . . . . . . . . . . . . . . . . . . 116

6 The Isoperimetric problem in the Sub-Riemannian Heisen-berg group 1236.1 The Isoperimetric Inequality in H . . . . . . . . . . . . . . . . 1236.2 The Isoperimetric Profile of H and Pansu’s conjecture . . . . . 1256.3 Existence of minimizers . . . . . . . . . . . . . . . . . . . . . . 1266.4 A C2 isoperimetric profile has constant horizontal mean cur-

vature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5 Recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.5.1 Existence and characterization of minimizers with ad-ditional symmetries . . . . . . . . . . . . . . . . . . . . 133

3

6.5.2 The C2 isoperimetric profile in H . . . . . . . . . . . . 1346.5.3 The convex isoperimetric profile of H . . . . . . . . . . 1346.5.4 The isoperimetric profile in a reduced symmetry setting 134

Conclusions 135

Bibliography 137

Introduction

In this survey we will discuss the isoperimetric problem in the sub-Rieman-nian Heisenberg group. This problem has its origin in the more ancientisoperimetric problem in the Euclidean plane, which has absorbed mathe-maticians of any time. There the question is to find the figure with maximalarea, bounded by a fixed perimeter.

This is a practical problem known for so long time that it is documentedeven in the legend of the foundation of Carthage. According to the story,the Phoenician princess Dido, because of dynastic fight, was forced to takerefuge on the coast of North Africa. There she managed to obtain from thelocal potentate a small bit of land for temporary refuge, as wide as could beencompassed by the hide of a bull. That’s not a lot, but Dido found a veryingenious solution: she instructed to cut the hide into a series of long thinstrips, to obtain a long string, by which she could mark out a vast region,forming the eventual line of the walls of ancient Carthage.

The problem that Dido had to solve was a variant of the classical isoperi-metric problem: she had to find the maximal land that can be enclosed witha fixed perimeter. An equivalent formulation is: given a fixed area, find theregion with this area and minimal perimeter. The solution is very intuitiveand, as Dido correctly guessed, is the disk with length of the circumferenceequal to the given perimeter.

The formal solution of this problem has requested over the centuries theefforts of many scientists, starting from the great Greek mathematicians.A first partial solution was found by the Swiss Jacob Steiner (1796-1863),who introduced its celebrated method of simmetrization, even if he didn’tproved that an isoperimetric profile exists. The question in the Euclideanspace was solved in 1958 by Ennio De Giorgi (1928-1996), when he presentedthe formalization of the problem and proved the isoperimetric property ofthe circle in the plane, the sphere in the space (and more generally of thehypersphere in a space of arbitrary dimension) among a very broad class ofsets called sets of finite perimeter.

Here we are interested in the isoperimetric problem in spaces with a sub-

5

6 Introduzione

Riemannian metric structure, also known as Carnot-Carathéodory spaces.This metric structure may be viewed as a constrained geometry, where motionis only possible along a given set of directions, changing from point to point.They play a central role in the general problem of analysis and geometry inmetric spaces, with applications ranging from differential geometry to nonholomorphic mechanics, getting through partial differential equations andcontrol theory. One of the most significant and simplest example of such aspace is the so-called (first) Heisenberg group.

In 1982 the French mathematician Pierre Pansu studied the isoperimetricproblem in the Heisenberg group and proposed a conjecture for its solution.From then on several developments have been made and lots of instancessupport Pansu’s conjecture, but the problem is still an open question. Thegoal of this survey is to introduce the notions for the comprehension of theconjecture and the presentation of the recent results in this topic.

After a first chapter, in which we recall the main concepts of analysis andgeometry necessary for the comprehension of this work, Chapter 2 is devotedto the study of the isoperimetric problem in the Euclidean space, with thepurpose to give an idea of the notions that are involved.

The next chapters treat the generalization of the problem to the Heisen-berg groups, in particular to the first Heisenberg group. These are special Liegroups, whose Lie algebra is nilpotent and can be expressed as a direct sumV = V1 ⊕ V2 of two vector spaces such that the commutator [V1, V1] = V2.We will see that this property allows us to introduce a distance, providedwe give a Riemannian metric on V1 and we extend it to a sub-Riemannianmetric on the whole space. This construction will be done in Chapter 3, inwhich we introduce the Heisenberg group, its structure and its Riemannianapproximants.

In Chapter 4 and Chapter 5 we deal with the main geometric quantitiesin the sub-Riemannian geometry of the Heisenberg group as the perimetermeasure and its first variation.

Finally, the last chapter is the most interesting from the point of viewof the research. We will describe how Pansu generalized the isoperimetricproblem to the Heisenberg group and his conjecture. In particular we willpresent the proof of the existence of an isoperimetric set among sets of finiteperimeter, which is inspired by the Euclidean technique due to De Giorgi.

At this point I would like to thank professor Francesco Serra Cassano,whose experience and competence have been an example for me and a greathelp for my study in these months of work. I’d would like to thank himalso for the time he has given me, but even more for the humane words ofsupport.

7

Finally I would like to thank professor Frank Loose for his helpfulnessand kindness. I have appreciated his enthusiasm for mathematics which hecommunicates to the students and the readiness with which he accepted tosupervise me.

Chapter 1

Recalls on Riemannian geometryand Geometric Measure Theory

In this section we recall that definitions and theorems on Riemannian geom-etry and geometric measure theory, which will be used in the survey. We willsometimes state these classical results without proof.

1.1 Manifolds

We assume here that the concepts of C∞ manifold, differentiable map, tan-gent space, differential of a function and tangent bundle are known (for is-tance see [20] or [5]). We restrict ourself to recall those notions on manifoldand Riemannian geometry which will be heavily used in this survey.

1.1.1 Vector fields and Frobenius’ Theorem

Let M be a m-dimensional manifold.

Definition 1.1. A vector field X on an open set U in M is a map X : U →TM such that X(p) (often denoted Xp) is a vector in TpM , the tangent spaceto M at p. The vector field is smooth if it is smooth as a map from U toTM , the tangent bundle of M .

We denote by X(M) the vector space of all smooth vector fields on M .

If (U,ϕ) is a neighborhood of p ∈M , the vectors(∂

∂x1

)p

:= (dϕp)−1

(∂

∂u1

), . . . ,

(∂

∂xm

)p

:= (dϕp)−1

(∂

∂um

)9

10 Recalls on Riemannian geometry and Geometric Measure Theory

form a basis for the tangent space TpM . Here u1, . . . , um denote the canonicalcoordinates in Rm. Then the vector field X is smooth if and only if its localexpression

X =m∑i=1

ai∂

∂xi

has smooth coordinates functions a1, . . . , am : U → R.

Definition 1.2. If X and Y are smooth vector fields on M , we define avector field [X, Y ] called the Lie bracket of X and Y by setting

[X, Y ]p(f) = Xp(Y f)− Yp(Xf).

The following proposition underlines some properties of the Lie bracket,whose proofs are an easy computation.

Proposition 1.3. For each X, Y, Z ∈ X(M) hold:

1. [X, Y ] is indeed a smooth vector field on M .

2. If f, g ∈ C∞(M), then [fX, gY ] = fg[X, Y ] + f(Xg)Y − g(Y f)X.

3. [X, Y ] = −[Y,X].

4. (Jacobi identity)

[[X, Y ], Z] + [[Y, Z], X] + [[Z,X], Y ] = 0.

A vector space with a bilinear operation satisfying (3) and (4) is called aLie algebra.

Definition 1.4. Let X be a smooth vector field on M . A C∞ curve γ in Mis an integral curve of X if γ(t) = X(γ(t)) for each t in the domain of γ.

Since a differentiable manifold is locally diffeomorphic to Rn, the fun-damental theorem on existence, uniqueness and dependence on initial con-ditions of ordinary differential equations extends naturally to differentiablemanifolds. By this theorem, to each p ∈ M there exists a neighborhoodU ⊂ M of p, an intervall (−δ, δ), with δ > 0, and a differentiable mappingϕ : (−δ, δ) × U → M such that the curve t → ϕ(t, q), with t ∈ (−δ, δ)and q ∈ U is the unique integral curve of X with ϕ(0, q) = q. The mapϕt : U →M such that ϕt(q) = ϕ(t, q) is called the local flow of X.

Let us generalize these concepts to greater dimensional objects.

1.1 Manifolds 11

Definition 1.5. Let M be a m-dimensional manifold and let n ≤ m be aninteger. A n-dimensional distribution ∆ on M is a choice for each pointp ∈ M of an n-dimensional subspace ∆p of the tangent space TpM . ∆ issmooth if for each p ∈ M there is a neighborhood U of p and there are nlinearly independent C∞-vector fields X1, . . . , Xn on U which form a basis of∆q for every q ∈ U .

We say that a vector field X on M lies in the distribution ∆, X ∈ ∆, ifXp ∈ ∆p for each p ∈ M . A distribution ∆ is called involutive if it is closedunder the operation of Lie brackets, that is if [X, Y ] ∈ ∆ for each X, Y ∈ ∆.

Let ∆ be a C∞ distribution on M and N a connected C∞ manifold.Consider a function F : N → M . If F is a one-to-one immersion such thatfor each q ∈ N we have dFq(TqN) ⊂ ∆F (q), then we say that the immersedsubmanifold F (N) ⊂M is an integral manifold of ∆. Notice that an integralmanifold may be of lower dimension than ∆.

A more restrictive definition characterizes a completely integrable distri-bution. Let ∆ be a distribution on M . If each point p ∈ M admits acoordinate neighborhood (U,ϕ) such that the first n vectors

∂

∂x1

, . . . ,∂

∂xn

of the local basis of TpM are a local basis for ∆ on U , we say that thedistribution ∆ is completely integrable. In other words, for a completelyintegrable distribution there exists an n-dimensional integral manifold Nwith ∆ as tangent space. In fact, if (q1, . . . , qm) denote the local coordinatesof q, then the integral manifold through q is the set of all points whosecoordinates satisfy un+1 = qn+1, . . . , um = qm, that is N = ϕ−1(u ∈ ϕ(U) :uj = qj, j = n+ 1, . . . ,m).

Notice that if ∆ is completely integrable, then it is also involutive, in fact[(∂

∂xi

)p

,

(∂

∂xj

)p

]= dϕ−1

[(∂

∂ui

)p

,

(∂

∂uj

)p

]= 0,

lies in ∆p for each i, j = 1, . . . , n.Frobenius’ theorem states that also the inverse is true.

Theorem 1.6 (Frobenius). A distribution ∆ on a manifold M is completelyintegrable if and only if it is involutive.

To conclude this section we give a classical geometric interpretation ofthe Lie bracket.


Proposition 1.7. Let X, Y be differentiable vector fields on a differentiablemanifold M , let p ∈ M , and let ϕt be the local flow of X in a neighborhoodU of p. Then

[X, Y ](p) = limt→0

1

t[Y − dϕtY ](ϕt(p)).

For the proof, we need the following lemma from calculus.

Lemma 1.8. Let δ > 0 and h : (−δ, δ)× U → R be a differentiable functionwith h(0, q) = 0 for all q ∈ U . Then there exists a differentiable mappingg : (−δ, δ)× U → R with h(t, q) = tg(t, q), in particular,

g(0, q) =∂h(t, q)

∂t

∣∣∣∣t=0

.

Proof of Lemma 1.8. It sufficies to define for fixed t

g(t, q) =

ˆ 1

0

∂h(ts, q)

∂tsds,

indeed, after a change of variable, we have

tg(t, q) =

ˆ t

0

∂h(r, q)

∂rdr = h(t, q).

Proof of Proposition 1.7. Let f be a differentiable function in a neighborhoodof p. Define

h(t, q) = f(ϕt(q))− f(q)

and apply the lemma to h to obtain a differentiable function g(t, q) such that

f ϕt(q) = f(q) + tg(t, q)

and g(0, q) = Xf(q). Then we obtain

((dϕtY )f)(ϕt(p)) = (Y (f ϕt))(p) = Y f(p) + t(Y g(t, p))

and we can conclude that

limt→0

1

t[Y − dϕtY ]f(ϕt(p)) = lim

t→0

(Y f)(ϕt(p))− Y f(p)

t− (Y g(0, p))

= (X(Y f))(p)− (Y (Xf))(p)

= ((XY − Y X)f)(p) = [X, Y ]f(p).

1.1 Manifolds 13

1.1.2 Lie groups and Lie algebras

Definition 1.9. A Lie group, (G, ·), is a group with a structure of differentialmanifold with respect to which the operation of the group and the inverseoperation are smooth functions, that is

(x, y) 7−→ x−1y ∀x, y ∈ G

is a smooth function between G×G and G.

An example of Lie group that will be usefull later is the general lineargroup, GL(n,R), the set of all n×n invertible matrices. The operation is theusual product between matrices and the differentiable structure is inducedby Rn2 .

On a Lie group G are naturally defined the left translation, Lx, and theright translation, Rx, that are given respectively by

Lx : G −→ G, Lx(y) = xy (1.1)Rx : G −→ G, Rx(y) = yx. (1.2)

They are smooth functions, so we may consider their differential dLx and dRx.A smooth vector field, X, on the Lie group G, is said to be left invariant ifdLxX = X for all x ∈ G, in other words, for each x, y ∈ G it holds

dLx(y)(X(y)) = X(xy).

In the same way, we say that X is right invariant if dRxX = X for eachx ∈ G.

The vector space of all left invariant vector fields on G is a Lie algebra.In fact, let X, Y be two left invariant smooth vector fields on G, then

dLx[X, Y ]f = [X, Y ](f Lx)= X(dLxY )f − Y (dLxX)f

= (XY − Y X)f

= [X, Y ]f,

for each element x ∈ G and each differentiable function f on G. Then theLie product [X, Y ] of two left invariant vector fields is still left invariant.

Notice that a left invariant smooth vector field is completely determinedby its value at a unique point, for example at the neutral element e of thegroup. In fact X(x) = dLx(X(e)) for each x ∈ G. This observation allowsus to give to TeG a structure of Lie algebra induced from that of the left


invariant vector fields: let us identify each vector Xe ∈ TeG with the leftinvariant vector field defined by

Xx = dLxXe ∀x ∈ G

and let us define the Lie product of two vectors Xe, Ye ∈ TeG as

[Xe, Ye] = [X, Y ]e.

Thus TeG equipped with this operation is a Lie algebra.The Lie algebra of G is denoted by g and may be equivalently defined

as the tangent space at the identity, TeG, or as the set of all left invarianttangent vector fields.

1.2 Riemannian geometryIn this section we recall the notion of curvature and exponential map ona Riemannian surface. We start by the notions of curvature for a spacecurve and for a plane curve. Then we introduce the affine connection andin particular the Riemannian connection, in order to give at the end thedefinitions of exponential map, curvature and second fundamental form of aRiemannian manifold.

Let I denote an open interval in R. A C2 space curve γ : I → R3 is calledregular if γ′(t) 6= 0 for each t. Given an interval [a, b] ⊂ I, we recall that thelength of γ between a and b is defined as

Length(γ) := supn∑i=1

|γ(ti)− γ(ti−1)| : a = t0 < t1 < · · · < tn = b

where the supremum is taken over all finite parametrizations of [a, b]. Aclassical result of geometry states that for a regular space curve,

Length(γ) =

ˆ b

a

|γ′(t)|dt.

This computation for the length of a curve, suggests us to take a preferredparameter for γ. Given a point γ(t0) the arclength parameter is defined by

s(t) :=

ˆ t

t0

|γ′(h)|dh.

In particular, if γ is parametrized by arclength, the vector γ′ has constantunitary norm.

1.2 Riemannian geometry 15

Let γ : I → R3 be parametrized by arclength. The Frenet orthonormalframe of γ is given by

~t(s) := γ′(s) ~n(s) :=~t ′(s)

|~t ′(s)|and ~b(s) := ~t(s) ∧ ~n(s).

Notice that, since ~t has constant unitary norm, ddt

⟨~t,~t⟩

= 2⟨~t ′,~t

⟩= 0 and

~n is orthogonal to ~t. Hence ~t, ~n,~b form indeed an orthonormal frame. Thevector

~k := ~t ′ = γ′′

is called curvature vector of γ and represents the acceleration of the curve.Therefore it is a normal vector to γ that points towards the inside of thecurve. The quantity k(s) such that

~t ′(s) = k(s)~n(s)

is called curvature of γ. In particular k = |γ′′| is always positive.Some special comment is required for the case of curves lying on a plane.Let γ(t) = (γ1(t), γ2(t)) define a regular C2 plane curve in R2. The length

of the curve and the arclength parameter are defined as for a space curve.Let γ be parametrized by arclength. As before, the tangent vector to γ isdefined by

~t(s) := γ′(s) = (γ′1, γ′2)(s).

The normal unit vector is now choosen so that ~t and ~n form an orthonormalframe. So ~n := (−γ′2, γ′1). The curvature vector still represents the accelera-tion of γ and is given by ~k := γ′′; so it is the vector normal to γ that pointstowards the inside of the curve. Therefore it can go in the same direction of ~nor in the opposite direction and the curvature k(s) such that ~t ′(s) = k(s)~n(s)may be positive, zero or negative.

Let us explicite the last formulas in the identification of R2 with thecomplex plane C. Let γ : I → C be a regular C2 curve, parametrized byarclength. Then

~t = (γ′1, γ′2) ~n = i(γ′1, γ

′2) ~k = (γ′′1 , γ

′′2 ) = k(s)~n = k(s)i(γ′1, γ

′2). (1.3)

1.2.1 Affine connection and Riemannian connection

We now introduce the concept of affine connection on a smooth manifold.The underlying idea is the following: given a smooth manifold M immersedin Rn and a tangent vector field X on M , the Euclidean derivative of Xalong a curve lying in M will not in general be tangent to the manifold.


Hence, the concept of differentiating a vector field is not an intrinsic geometricnotion on M . We will consider therefore the orthogonal projection of thederivative of X on the tangent space toM and we will call this projection thecovariant derivative of X along the curve. The idea of the affine connectionis a generalization of the covariant derivative.

Let us indicate with X(M) the set of all smooth vector fields on M .

Definition 1.10. An affine connection D on a differentiable manifold M isa mapping

D : X(M)× X(M)→ X(M) : (X, Y ) 7→ DXY

which satisfies the following properties:

(i) DfX+gYZ = fDXZ + gDYZ;

(ii) DX(Y + Z) = DXY +DXZ;

(iii) DX(fY ) = fDXY +X(f)Y

for each X, Y, Z ∈ X(M) and f, g ∈ C∞(M).

We remark that an affine connection is frequently denoted by ∇. Here wehave adopted this convention for coherence with the notation used in nextchapters and to avoid confusion with the gradient operator.

The following proposition will clarify the relation between affine connec-tion and covariant derivative.

Proposition 1.11. Let M be a differentiable manifold with an affine con-nection D. There exists a unique correspondence which associates to a vectorfield X along a differentiable curve γ : (−ε, ε)→ M another vector field DX

dt

along γ, called the covariant derivative of X along γ, such that:

1. Ddt

(X + Y ) = DdtX + D

dtY ;

2. Ddt

(fX) = dfdtX + f DX

dtfor each smooth vector field Y along γ and

differentiable function f on (−ε, ε);

3. if X is the restriction to γ of a vector field X ∈ X(M), then DXdt

=

Ddγ/dtX.

Proof. See for example [10].

Let M be endowed with a Riemannian metric 〈·, ·〉M . We say that theconnection D is compatible with the metric 〈·, ·〉M , if for each vector fieldX, Y, Z ∈ X(M) we have

X〈Y, Z〉M = 〈DXY, Z〉M + 〈Y,DXZ〉M . (1.4)


We say that D is symmetric if

DXY −DYX = [X, Y ]

for each X, Y ∈ X(M).Finally we state the main result of this section.

Theorem 1.12 (Levi-Civita). Let (M, 〈·, ·〉M) be a Riemannian manifold.There exists a unique affine connection D on M that is symmetric and com-patible with the Riemannian metric.

Proof. We first prove the uniqueness. Let us suppose that such a connectionexists. Since D is compatible with the metric we have

X〈Y, Z〉M = 〈DXY, Z〉M + 〈Y,DXZ〉M (1.5)Y 〈Z,X〉M = 〈DYZ,X〉M + 〈Z,DYX〉M (1.6)Z〈X, Y 〉M = 〈DZX, Y 〉M + 〈X,DZY 〉M . (1.7)

Adding (1.5) and (1.6) and subtracting (1.7), and using the symmetry of D,we obtain

X〈Y, Z〉M + Y 〈Z,X〉M − Z〈X, Y 〉M =

= 〈[X,Z], Y 〉M + 〈[Y, Z], X〉M + 〈[X, Y ], Z〉M + 2〈Z,DYX〉M .

Therefore

〈Z,DYX〉M =1

2X〈Y, Z〉M + Y 〈Z,X〉M − Z〈X, Y 〉M

− 〈[X,Z], Y 〉M − 〈[Y, Z], X〉M − 〈[X, Y ], Z〉M .(1.8)

This expression is usually referred to as the Kozul identity. It shows inparticular that D is uniquely determined by the metric 〈·, ·〉M , hence if itexists, it will be unique.

To prove existence, let us define D as in (1.8). It is easy to verify that Dis well-defined and that it satisfies the desired conditions.

Definition 1.13. Given a Riemannian manifold M , the unique symmetricconnection D compatible with the metric is called the Levi-Civita (or Rie-mannian) connection on M .

1.2.2 The exponential map

The definition of affine connection allows us to give a notion of accelerationof a smooth vector field on a manifold. Consequently we may define the


geodesic curves, that are smooth curves which have constant velocity withrespect to the manifold.

Let M be a Riemannian manifold endowed with the respective Rieman-nian connection.

Definition 1.14. A parametrized curve γ : (−ε, ε) → M is called geodesicat t0 ∈ (−ε, ε) if D

dt(γ′) = 0 at t0. If γ is a geodesic at all points t ∈ (−ε, ε)

we say that γ is a geodesic.

Notice that, if γ : (−ε, ε)→M is a geodesic, then using the fact that theRiemannian connection is compatible with the metric, we have

d

dt

⟨dγ

dt,dγ

dt

⟩= 2

⟨D

dt

dγ

dt,dγ

dt

⟩= 0

and the velocity vector dγdt

is constant as desired.With respect to the local coordinates, it may be seen that the equation

describing a geodesic satisfyies a second order differential equation.It may be stated a theorem of existence and uniqueness of the solution of

differential equation on a manifold M analogous to the classical one in Rn.For the proof and more details, see [10].

Proposition 1.15. Let M be a Riemannian manifold, endowed with its Rie-mannian connection. Given a point p ∈M , there exist a neighborhood V of pin M and a number ε > 0, such that if we denote by U = (q, w) ∈ TM ; q ∈V,w ∈ TqM, |w| < ε, there exists a C∞ mapping γ : (−2, 2)× U →M suchthat t → γ(t, q, w), t ∈ (−2, 2), is the unique geodesic of M which, at theinstant t = 0, passes through q with velocity w, for every q ∈ V and for everyw ∈ TqM , with |w| < ε.

Definition 1.16. Let p ∈ M and let U ⊂ TM be an open set as in Propo-sition 1.15. The exponential map on U is defined as the map exp : U → Msuch that

exp(q, v) = γ(1, q, v) = γ(|v|, q, v|v|

), (q, v) ∈ U .

We will frequently utilize the restriction of exp to an open subset of thetangent space TqM , that is, we define

expq : Bε(0) ⊂ TqM →M

by expq(v) = exp(q, v). Here Bε(0) denotes the open ball with center at theorigin of TqM and of radius ε.

Geometrically, expq(v) is a point of M obtained by going out the lengthequal to |v|, starting from q, along a geodesic which passes through q withvelocity equal to v

|v| .


1.2.3 Curvature of a Riemannian manifold

Let M denote a Riemannian manifold with Riemannian connection D.

Definition 1.17. The curvature R of a Riemannian manifold M is a corre-spondence that associates to every pair X, Y ∈ X(M) a mapping R(X, Y ) :X(M)→ X(M) given by

R(X, Y )Z = DYDXZ −DXDYZ +D[X,Y ]Z

for each Z ∈ X(M).

If for example M = Rn, then R(X, Y )Z = 0 for all X, Y, Z ∈ X(Rn). Forthis reason we can think of R as a way of measuring how much M deviatesfrom being Euclidean.

Another way of viewing the curvature is to consider a local system ofcoordinates (x1, . . . , xn). Since

[∂∂xi, ∂∂xj

]= 0, we obtain

R

(∂

∂xi,∂

∂xj

)∂

∂xk=

(D ∂

∂xj

D ∂∂xi

−D ∂∂xi

D ∂∂xj

)∂

∂xk,

that is, the curvature measures the non-commutativity of the covariant de-rivative.

The following proposition is a consequence of the definition and can beproved by direct computations.

Proposition 1.18. The curvature R has the following properties:

1. R is bilinear in X(M)× X(M), that is,

R(fX1 + gX2, Y1) = fR(X1, Y1) + gR(X2, Y1)

R(X1, fY1 + gY2) = fR(X1, Y1) + gR(X1, Y2)

for each f, g ∈ C∞(M) and X1, X2, Y1, Y2 ∈ X(M).

2. for any X, Y ∈ X(M), the curvature operator is linear R(X, Y ), thatis,

R(X, Y )(fZ +W ) = fR(X, Y )Z +R(X, Y )W,

for each f ∈ C∞(M), Z,W ∈ X(M).

We recall also the celebrated Bianchi identity :

Proposition 1.19 (Bianchi identity).

R(X, Y )Z +R(Y, Z)X +R(Z,X)Y = 0,

for all X, Y, Z ∈ X(M).


Moreover for the curvature tensor the following properties hold:

Proposition 1.20. For each X, Y, Z, T ∈ X(M) we have

1. 〈R(X, Y )Z, T 〉+ 〈R(Y, Z)X,T 〉+ 〈R(Z,X)Y, T 〉 = 0;

2. 〈R(X, Y )Z, T 〉 = −〈R(Y,X)Z, T 〉;

3. 〈R(X, Y )Z, T 〉 = −〈R(X, Y )T, Z〉;

4. 〈R(X, Y )Z, T 〉 = 〈R(Z, T )X, Y 〉.

Closely related to the curvature operator is the sectional curvature. LetM be an n-dimensional Riemannian manifold.

Definition 1.21. Given a point p ∈ M and a two-dimensional subspaceπ ⊂ TpM , the real number

K(X, Y ) =〈R(X, Y )X, Y 〉|X ∧ Y |

(1.9)

does not depend on the choice of a basis X, Y of π and is called the sectionalcurvature of π at p.

To see that K(X, Y ) is invariant from the choice of the basis, observethat any other basis can be obtained by X, Y by iterating the followinglinear transformations X, Y → Y,X, X, Y → λX, Y and X, Y →X + λY, Y , which do not modify the sectional curvature of π.

Definition 1.22. Given a point p ∈ M and a vector X = Xn in TpM , takean orthonormal basis X1, . . . , Xn−1 of the hyperplane orthogonal to X. Then

Ricp(X) =n−1∑i=1

〈R(X,Xi)X,Xi)〉 =n−1∑i=1

K(X,Xi) (1.10)

and

K(p) =n∑j=1

Ricp(Xj) (1.11)

are independent of the choice of the corresponding orthonormal basis andare called respectively the Ricci curvature in the direction X and the scalarcurvature at p.


1.2.4 Second fundamental form and shape operator

We introduce now the second fundamental form and the mean curvature ofa surface. They are very important notions, which describe the geometryof the manifold. For sake of simplicity and since it will be the only case inwhich we consider them, we restrict our study to the surfaces embedded inR3.

LetM be a surface embedded in R3, that isM is a manifold of dimension2 and there exists an embedding F : M → F (M) ⊂ R3. Let us denote by Malso the manifold F (M) in R3. Let p ∈M be a point and take a coordinateneighborhood (U,ϕ) of p, with W = ϕ(U) a connected open subset of R2.Let (u, v) be the coordinates of R2. The function X : W → U ⊂ M isa parametrization of U and induces the vector frame Xu = ∂X/∂u andXv = ∂X/∂v which is a basis for the tangent space to U . The matrix of themetric g induced on M by R3 with respect to the basis Xu, Xv is given by

G =

(〈Xu, Xu〉〈Xu, Xv〉〈Xv, Xu〉〈Xv, Xv〉

)where 〈·, ·〉 denotes the Euclidean scalar product in R3. The matrix G iscalled the first fundamental form on M .

Suppose M is orientable, then it is well defined a unit normal vector fieldN on M with respect to the Riemannian metric g on M . Let us choosethe vector field N in such a way so that Xu, Xv and N form a frame withthe same orientation as the standard orthonormal frame (∂/∂x1), (∂/∂x2),(∂/∂x3) of R3. We will study the structure of M by means of the derivativeof N with respect to the various vectors in TpM .

Let v be a tangent vector in TpM and choose a C∞ curve γ : (−ε, ε)→ Usuch that γ(0) = p and γ(0) = v. Restricting N to γ we may differentiateN(γ(t)) as a vector field along a curve in R3. Since N has constant unitarynorm, we have

0 =d

dt〈N(t), N(t)〉g

∣∣∣∣t=0

= 2

⟨dN(t)

dt

∣∣∣∣t=0

, Np

⟩g

,

where 〈·, ·〉g denotes the scalar product on M induced from the Riemannianmetric g. This proves that (dN/dt)t=0 is orthogonal to N and therefore lies inTpM . The following theorem shows some more properties of this derivative.

Theorem 1.23. The vector (dN/dt)t=0 depends only on the vector Xp andnot on the chosen curve γ. The map

S : TpM → TpM with S(vp) := − dN(γ)

dt

∣∣∣∣t=0

(1.12)


is well defined and is a linear map.

Proof. Let vp = aXu+bXv be expressed in coordinates with respect toXu, Xv

and letγ(t) = (X1(u(t), v(t)), X2(u(t), v(t)), X3(u(t), v(t)))

be a differentiable curve such that γ(0) = p and γ(0) = vp. Computing γ(0)we have

γ(0) = u(0)Xu + v(0)Xv.

By the uniqueness of the coordinates, we then have a = u(0) and b = v(0).Let us express the vector field N with respect to the standard coordinate

system of R3 as follows

N = n1(u, v)∂

∂x1+ n2(u, v)

∂

∂x2+ n3(u, v)

∂

∂x3.

Then its derivative along the curve is given by

dN(γ)

dt

∣∣∣∣t=0

=3∑i=1

(∂ni

∂u

∣∣∣∣ϕ(0)

u(0) +∂ni

∂v

∣∣∣∣ϕ(0)

v(0)

)∂

∂xi

= a3∑i=1

∂ni

∂u

∣∣∣∣ϕ(0)

∂

∂xi+ b

3∑i=1

∂ni

∂v

∣∣∣∣ϕ(0)

∂

∂xi

= −S(vp).

This shows that S(vp) depends only on the coordinates (u(0), v(0)) of p andon the components (u(0), v(0)) of vp, and not on the chosen curve. The latterformula shows also that S(vp) depends linearly on the coordinates of vp, i.e.S is a linear map on TpM .

Let D be the Levi-Civita connection on R3 induced from the metric andnote that

S(v) = −dN(γ(t))

dt= −DvN, (1.13)

where γ is a C∞ curve with γ(0) = p and γ = v. It can be seen, byobserving that dN/dt is symmetric and compatible with the metric and usingthe uniqueness of the Levi-Civita connection. Then the operator S is intrinsicfor M and does not depend on the particular coordinate neighborhood.

Definition 1.24. The map S defined in (1.12) is called the shape operatoror Weingarten operator of M .

Proposition 1.25. The shape operator S is self-adjoint, i.e.

〈S(v), w〉g = 〈v, S(w)〉g for each v, w ∈ TpM.


Proof. We use the coordinate neighborhood (U,ϕ) of p and the induced frameXu, Xv introduced before. Then

Xu =3∑i=1

X iu

∂

∂xi, Xv =

3∑i=1

X iv

∂

∂xi, and N =

3∑i=1

ni∂

∂xi.

Since N is normal to the surface we have 〈Xu, N〉g = 0 = 〈Xv, N〉g. Differ-entiating them we obtain

−⟨∂N

∂u,Xu

⟩g

= 〈N,Xuu〉g =3∑i=1

niX iuu

−⟨∂N

∂v,Xu

⟩g

= 〈N,Xuv〉g =3∑i=1

niX ivu = 〈N,Xvu〉g = −

⟨∂N

∂u,Xv

⟩g

−⟨∂N

∂v,Xv

⟩g

= 〈N,Xvv〉g =3∑i=1

niX ivv.

These equations show that the components of the shape operator are C∞and the second relation in particular proves that S is symetric.

Definition 1.26. The symmetric bilinear form on TpM defined by

II(v, w) := 〈S(v), w〉g (1.14)

for each v, w ∈ TpM is called the second fundamental form on M .

Let IIp denote the matrix of the second fundamental form in p withrespect to Xu and Xv and S the matrix of the shape operator with respectto the same basis. Then for all vectors v, w ∈ TpM we have

vtIIpw = II(v, w) = 〈S(v), w〉g = 〈v, S(w)〉g = 〈v,GS(w)〉 = vtGSw.

Hence we have found the relation:

IIp = GS. (1.15)

Making use of equation (1.13) we can write explicitly the matrix of thesecond fundamental form with respect to a basis v, w as follows:

II =

(〈DvN, v〉〈DvN,w〉〈DwN, v〉〈DwN,w〉

). (1.16)

The shape operator of a surfaceM describes the geometry of the manifold.A clarifying result is the following theorem.


Theorem 1.27. Let M be a surface embedded in R3. At each point p ∈ Mthe eigenvalues of the linear transformation S are real numbers k1 and k2,k1 ≥ k2. If k1 6= k2, then the eigenvectors belonging to them are orthogonal; ifk1 = k2 = k at p, them S(vp) = kvp for every vector Xp in TpM . The numbersk1 and k2 are the maximum and minimum values of II(vp, vp) = (S(vp), vp)over all unit vectors vp ∈ TpM .

Proof. The statements to prove are consequences of the theory of linear al-gebra for symmetric operators.

Fix p ∈ M and consider S defined on the tangent space TpM . Since Sis a symmetric operator, it is diagonalizable and there exist real eigenvaluesk1, k2.

If k1 = k2 = k the eigenvalues of k span the tangent space TpM . Chooseany base v1, v2. Each vector v ∈ TpM can be written as v = av1 + bv2 and

S(v) = aS(v1) + bS(v2) = k(av1 + bv2) = kv.

The proof is complete in this case.If k1 6= k2, let v1, v2 be respective eigenvectors with unt norm. Then

k1〈v1, v2〉g = 〈S(v1), v2〉g = 〈v1, S(v2)〉g = k2〈v1, v2〉g,

implies that v1 and v2 are orthogonal since k1 6= k2. Replacing v2 with −v2

if necessary, we may suppose that v1, v2 have the same orientation of TpM .Then any vector v ∈ TpM may be written as v = cos θv1 + sin θv2. Letk(θ) := (S(v), v) = II(v, v). Then

k(θ) = k1 cos2 θ + k2 sin2 θ,

because v1, v2 are orthonotmal. (This equation is also known as Euler’sformula). Differentiating it we have

dk

dθ= 2(k2 − k1) sin θ cos θ = (k2 − k1) sin(2θ).

Hence the extrema of k(θ) occur when θ = 0, π/2, π, 3π/2, that correspondto v = ±v1, v = ±v2 as desidered.

The trace of the symmetric operator S and its determinant are the coeffi-cients of the characteristic polynomial of S. They are invariants that describethe geometry of the surface and take special names.

Definition 1.28. The Gaussian curvature of a surface is the determinant ofany representative matrix of S and is equal to

K = k1k2

1.3 Recalls on geometric measure theory 25

where k1 and k2 are the eigenvalues of S.The mean curvatureof a surface is the trace of S and is equal to

H = k1 + k2.

1.3 Recalls on geometric measure theoryWe take now into account the analytic aspects that will be used in the survey.To this end we explicite some notions given in the previous paragraph in theparticular case of a Ck submanifold of Rn, with k <∞ in general. We thendefine the Hausdorff measure in a metric space, some fundamental theoremsof geometric measure theory and their application to the calculus of the firstvariation. Finally we define a doubling measure and the Hardy-Littlewoodmaximal operator.

1.3.1 The divergence theorem for manifolds embeddedin Rn

Let us consider an n-dimensional Ck submanifoldM of Rn+r, with k ≥ 1 andr ≥ 0. As said in the previous paragraph, this means in particular that M isa subset of Rn+r such that for each y ∈M there exist open sets U, V ⊂ Rn+r

with y ∈ U , 0 ∈ V and a Ck diffeomorphism ϕ : U → V such that ϕ(0) = yand ϕ(M ∩U) = W = V ∩Rn. By Rn ⊂ Rn+r we mean the subspace of Rn+r

consisting of all points x = (x1, . . . , xn+r) such that xn+1 = . . . = xn+r = 0.Let ψ := ϕ−1|W . Then ψ : W → Rn+r is a diffeomorphism on the image

ψ(W ) = M ∩ V and ψ(0) = y, that is ψ is a local representation for M at y,and the vectors

∂ψ

∂x1

(0), . . . ,∂ψ

∂xn(0)

are linearly independent in Rn+r. In particular, this is a basis for the tangentspace TyM to M at y.

We want to give the definition of divergence on M . To this end, let usrecall that a function f : M → Rm is C l on a manifold M ⊂ Rn+r, withl ≤ k, if there exist an open set U ⊂ Rn+r such that M ⊂ U , and a functionf : U → Rm, such that f

∣∣∣M

= f .In case f : M → R is a C1 real-valued function, that is m = 1, we define

the gradient ∇Mf of f at y ∈M by

∇Mf(y) :=n∑j=1

(Dvjf)vj (1.17)


where v1, . . . , vn is any orthonormal base for the tangent space to M at y.Here Dvf denotes the directional derivative of f with respect to the vectorv ∈ TyM . More explicitly, given a C1 curve γ : (−ε, ε) → M with γ(0) = yand γ(0) = v, the directional derivative Dvf ∈ Rn+r of f is

Dvf :=d

dt(f γ(t))|t=0

and depends only on the vector v and the function f and not on the particularcurve γ that we have choosen. (Here ε > 0 is choosen so that γ is well-definedon M .)

Let e1, . . . , en+r be the standard basis of Rn+r and

∇jMf := 〈∇Mf, ej〉Rn+r

be the component of ∇Mf with respect to ej, for each j = 1, . . . , n+ r. Here〈·, ·〉Rn+r denotes the standard scalar product in Rn+r. Then the expressionof ∇Mf with respect to the standard basis is

∇Mf(y) =n+r∑j=1

∇jMf(y)ej.

Note that if f is the restriction to M of a C1 function f , defined on anopen subset U ⊂ Rn+r containing M , then from elementary calculus followsthat

Dvif =⟨

gradf , vi

⟩,

where gradf is the usual gradient (D1f , . . . , Dn+rf) on U ⊂ Rn+r, so thatwe can write equation (1.17) in the form

∇Mf(y) =n∑i=1

⟨gradf(y), vi

⟩vi = πM(gradf(y))

where πM : Rn+r → TyM is the orthogonal projection of Rn+r onto TyM . Inother words, the gradient of a function f with respect to M is simply theorthogonal projection of the classical gradient on the tangent space to M .

Let X = (X1, . . . , Xn+r) : M → Rn+r be a vector function with Xj ∈C1(M), j = 1, . . . , n + r. We define the divergence of X with respect to Mas

divMX :=n+r∑j=1

∇jMXj.


Notice that X can be more general than a vector field, since we do not requireXy ∈ TyM . If we write more explicitely the expression of ∇j

MXj we have

divMX =n+r∑j=1

〈∇MXj, ej〉 =n+r∑j=1

⟨n∑i=1

(DviXj)vi, ej

⟩

=n∑i=1

n+r∑j=1

〈(DviXj)vi, ej〉 =n∑i=1

n+r∑j=1

〈vi, (DviXj)ej〉 (1.18)

=n∑i=1

〈DviX, vi〉 , (1.19)

for any orthonormal basis v1, . . . , vn of TxM . The last equality follows fromthe expression X =

∑n+rj=1 Xjej.

We are now able to state the divergence theorem for manifolds.

Theorem 1.29 (Divergence theorem). Let M be a n-dimensional smoothcompact manifold. Let us denote by M the closure ofM and by ∂M = M \Mits boundary. Let X = (X1, . . . , Xn+r) : M → Rn+r be a vector function suchthat Xy ∈ TyM for each y ∈M . Then

ˆM

divMXdHn = −ˆ∂M

〈X, ν〉 dHn−1,

where ν is the inward pointing unit normal of ∂M , that is |ν| = 1, ν isnormal to ∂M , tangent to M , and points into M at each point of ∂M .

From this theorem follows immediately thatˆM

divMXdHn = 0

whenever X has compact support in M .To conclude this section, we give a Leibniz formula for the divergence.

Let X = (X1, . . . , Xn+r) : M → Rn+r be a vector function with Xj ∈ C1(M),j = 1, . . . , n + r and f : M → R be a C1 function. From equation (1.18)applied to the product fX we obtain

divM(fX) =n∑i=1

〈Dvi(fX), vi〉 =n∑i=1

(〈Dvi(f)X, vi〉+ 〈fDviX, vi〉)

=n∑i=1

〈X,Dvi(f)vi〉+ fdivMX

= Xtang(f) + fdivMX, (1.20)


where in the last equality, Xtang is the C1 vector field on M defined as thetangential component of X with respect to the tangent space toM . Xtang(f)denotes the derivative of f with respect to this vector field.

1.3.2 Hausdorff measures

Definition 1.30. Let (X, d) be a metric space and A ⊂ X. For any α ≥ 0,let

ωα :=πα2

Γ(1 + α2),

where Γ(t) :=´∞oxt−1e−xdx denotes the Euler function. For δ > 0 define

Hαδ (A) :=

ωα2α

inf

∑i∈N

(diam(Ai))α s.t. diam(Ai) < δ,A ⊂

⋃i∈N

Ai

. (1.21)

The α-dimensional Hausdorff measure of A is defined as

Hα(A) := supδ>0Hαδ (A). (1.22)

Notice that the Hausdorff measure is a metric concept and depends onlyon the given distance.

Remark 1.31. If X = Rn endowed with the usual euclidean distance andα = k is a positive integer, then

ωk = Lk(B(0, 1)),

where B(0, 1) is the unit ball in Rn centered in 0 and Lk is the k-dimensionalLebesgue measure in Rn.

Remark 1.32. Observe that the quantity in (1.21) is non-decreasing asδ → 0. Then the supremum in (1.22) is equivalent to

Hα(A) = limδ>0Hαδ (A).

Definition 1.33. If in Definition 1.30 we require that the covering sets Aiare balls, we obtain the spherical Hausdorff measure denoted by Sα(A).

Notice that for each A ⊂ X,

2−αSα(A) ≤ Hα(A) ≤ Sα(A).

We recall the following basic properties of the Hausdorff measure.


Proposition 1.34. Let 0 ≤ α < α′ <∞ and A ⊂ X, then

1. Hα(A) <∞ implies Hα′(A) = 0;

2. Hα′(A) >∞ implies Hα(A) = +∞.

Proof. To prove 1, choose δ > 0 and let A ⊂⋃iAi, with diam(Ai) < δ and

(ωα/2α)∑

i(diam(Ai))α ≤ Hα

δ (A) + 1. Then

Hα′

δ (A) ≤ C∑i

(diam(Ai))α′ ≤ Cδα

′−α∑i

(diam(Ai))α ≤ Cδα

′−α(Hαδ (A)+1),

where C is a suitable constant. As δ goes to zero, this estimate proves 1.Statement 2 is just the contrapositive of 1.

These properties of the Hausdorff measure allow us to give the followingdefinition.

Definition 1.35. Let A ⊂ X, we define the Hausdorff dimension of A as

H− dim(A) := infα ≥ 0|Hα(A) = 0:= supα ≥ 0|Hα(A) = +∞.

The Hausdorff dimension of a metric space (X, d) is defined as the Hausdorffdimension of its unit ball centered in 0.

Proposition 1.34 implies that if the dimension of A is α then Hα′(A) =+∞ for each α′ < α and Hα′(A) = 0 for each α′ > α, but we cannot deduceany information about the value of Hα(A). However if for a set A ⊂ X thereexists α ≥ 0 such that 0 < Hα(A) < +∞ then α is the Hausdorff dimensionof A.

1.3.3 Rademacher’s Theorem and Area and Coarea for-mulas

Definition 1.36. Let A ⊂ Rn. A function f : A→ Rm is called Lipschitz ifthere exists a constant L ≥ 0 such that

|f(x)− f(y)| ≤ L|x− y| for all x, y ∈ Rn. (1.23)

The smallest constant L such that (1.23) holds is called the Lipschitz constantof f and is denoted by

Lip(f) := sup

|f(x)− f(y)||x− y|

: x, y ∈ A, x 6= y

.


We recall the well-known extension theorem for Lipschitz functions.

Theorem 1.37. Let A ⊂ Rn and f : A → Rm be Lipschitz. Then thereexists a Lipschitz function f : Rn → Rm such that f is an extension of f ,that is f = f on A, and Lip(f) ≤

√mLip(f).

We state without proof the following important theorem:

Theorem 1.38 (Rademacher’s Theorem). Let f : A ⊂ Rn → Rm be aLipschitz function. Then f is differentiable Ln-a.e. in A, that is, there existsa linear mapping T : A→ Rm such that

limy→x

|f(y)− f(x)− T (x− y)||x− y|

= 0 for Ln-a.e.x ∈ A.

To state the Area and Coarea Formulas we need the notion of Jacobianof a function. Let k ≤ n and v1, . . . , vk linearly independent vectors in Rn.A k-dimensional parallelepiped P in Rn is the set defined by

P =

k∑i=1

λivi : 0 ≤ λi ≤ 1, for i = 1, . . . , k

.

Proposition 1.39. Let v1, . . . , vk be linearly independent vectors in Rn withk ≤ n, then the parallelepiped determined by those vectors has k-dimensionalarea

√det(V tV ), where V is the n×k matrix having v1, . . . , vk as its columns.

Definition 1.40. Let A ⊂ Rm and f : A → Rn be differentiable at x. Letk ≤ m. We define the k-dimensional Jacobian of f at x, denoted by Jkf(x),by setting

Jkf(x) = sup

Hk(Df(x)(P ))

Hk(P )

.

where the supremum is taken over all k-dimensional parallelepipeds P con-tained in Rm. Here Df(x) denotes the differential of f in x.

The following lemma simplifies the computation of the k-dimensional Ja-cobian in particular cases.

Lemma 1.41. Let f : Rm → Rn be differentiable at x ∈ Rm:

1. if m = n, then Jmf(x) = Jnf(x) = | det(Df(x))|;

2. if m ≤ n, then Jmf(x) =√

det ((Df(x))t(Df(x)));

3. if m ≥ n, then Jnf(x) =√

det ((Df(x))(Df(x))t).


We can now state the following important theorems of geometric measuretheory.

Theorem 1.42 (Area Formula). If f : Rm → Rn is a Lipschitz function andm ≤ n, then

Â

Jmf(x)dLm(x) =

ˆRnH0(A ∩ f−1(y))dHm(y)

for each Lebesgue measurable subset A ⊂ Rm.

The following is a generalization of the area formula, which can be provedby approximations by simple functions.

Corollary 1.43. If f : Rm → Rn is a Lipschitz function and m ≤ n, thenÂ

g(x)Jmf(x)dLm(x) =

ˆRn

∑x∈A∩f−1(y)

g(x)dHm(y)

for each Lebesgue measurable subset A ⊂ Rm and each nonnegative Lm-measurable function g : A→ R.

Theorem 1.44 (Coarea Formula). If f : Rm → Rn is a Lipschitz functionand m ≥ n, then

Â

Jnf(x)dLm(x) =

ˆRnHm−n(A ∩ f−1(y))dHm(y)

for each Lebesgue measurable subset A ⊂ Rm.

In casem = n, the n-dimensional Jacobian agrees with the usual Jacobian| det(Df)|, and the area and coarea formulas coincide. In case m > n, andf : Rm = Rn×Rm−n → Rn is the orthogonal projection onto the first factor,then the coarea formula simplifies to Fubini’s theorem. Hence one can thinkof the coarea formula as a generalization of Fubini’s theorem to functionsmore complicated than orthogonal projection.

We give a generalization of the co-area formula similar to that in Corollary1.43.

Corollary 1.45. If f : Rm → Rn is a Lipschitz function and m ≥ n, thenÂ

g(x)Jnf(x)dLm(x) =

ˆRn

Â∩f−1(y)

g dHm−ndLn(y)

for each Lebesgue measurable subset A ⊂ Rm and each nonnegative Lm-measurable function g : A→ R.


1.3.4 First variation formula

We apply the just defined notions, to compute the first variation of a 1-parameter family of diffeomerphisms.

Let M be a n-dimensional C1 submanifold in Rn+r and V be an opensubset of Rn+r sucht that M ∩U 6= ∅ and Hn(K ∩M) <∞ for each compactK ⊂ V .

A 1-parameter family of diffeomorphisms of V is a family ϕtt∈R ofdiffeomorphisms ϕt : V → V , such that

ϕt+s(x) = ϕt(ϕs(x))

for each t, s ∈ R and x ∈ V .Let ϕt be such a family of diffeomorphisms on V (it is enough to require

t ∈ (−1, 1)) such that ϕ(t, x) := ϕt(x) is a C1 map from (−1, 1) × V to V ,ϕ0 is the identity map and ϕt is the identity map out of a compact K ⊂ Vfor each t ∈ (−1, 1).

Since ϕ(t, x) is a C1 map, for each x it determines a C1 curve in V withstarting point x = ϕ(0, x). Let us define the vector field U of initial velocities:

Ux =∂ϕ(t, x)

∂t

∣∣∣∣t=0

for each x ∈ V.

Then U has compact support contained in K and

ϕt(x) = x+ tUx + o(t2).

Let us denote by Mt the image of M ∩ K under ϕt. In particular M0 =M ∩K. The computation of the first variation of the measure Hn(Mt) maybe simplified by an application of the area formula. Indeed, let ψt = ϕt|M∩V .Then by the area formula 1.42 we have

d

dtHn(Mt)

∣∣∣∣t=0

=d

dt

ˆMt

dHn

∣∣∣∣t=0

=d

dt

ˆM∩K

JψtdHn

∣∣∣∣t=0

.

Here we have omitted the indication of the dimension n in the Jacobian, forthe sake of simplicity. Differentiating under the integral, we may reduce tothe calculation of ∂

∂t(Jψt).

Fix t ∈ (−1, 1). In order to obtain the Jacobian, we have find the differ-ential of ψt. For each v ∈ TxM we have

dψt|x(v) = Dvψt(x) = Dv(x+ tU(x) + o(t2)) = v + tDvU(x) + o(t2).


So the matrix of dψt relative to an orthonormal basis v1, . . . , vn of TxM andthe standard basis e1, . . . , en+r of Rn+r is given by

[Dψt(x)]ki = v(k)i + tDviUk(x) + o(t2)

for i = 1, . . . , n and k = 1 . . . , n + r. Here v(k)i is the k-th component of vi.

Then (Dψt(x))t · (Dψt(x)) is the matrix with entries

bij = δij + t(〈vi, DvjU(x)〉+ 〈vj, DviU(x)〉

)+ o(t2).

Using the expansion formula det(I + tA) = 1 + t tr(A) + o(t2), where A isany square matrix and I is the identity matrix, we have

(Jψt)2 = 1 + t tr

(〈vi, DvjU〉+ 〈vj, DviU〉

)ij

+ o(t2)

= 1 + 2t

(n∑i=1

〈vi, DviU〉

)+ o(t2)

= 1 + 2t divMU + o(t2)

where the last equality follows from the definition of the divergence (1.18).Finally by the Taylor expansion

√1 + x = 1 + 1

2x + o(x2) we have Jψt =

1 + tdivMU + o(t2). Then the derivative of the Jacobian in t = 0 is

d

dtJψt

∣∣∣∣t=0

= divMU. (1.24)

We will make use of this formula in the computation of the first variationof the perimeter in Section 5.3.2.

1.3.5 Doubling measure and the Hardy-Littlewoodmaximal operator

We introduce here the doubling property of a measure and the related Hardy-Littlewood maximal operator. Its weak-type estimate will be fundamentalto state a Sobolev-Gagliardo-Nierenberg inequality in the Heisenberg group(see Section 5.2).

In this section let (X, d) be a metric space and let B(X) denote the σ-algebra of its Borel sets.

Definition 1.46 (Doubling measures). A measure µ : B(X) → [0,+∞] issaid to be doubling if µ is finite on bounded sets and there exists a constantC such that

µ(B(x, 2r)) ≤ Cµ(B(x, r)) for each x ∈ X, for each r > 0. (1.25)


The best constant C in the above inequality is called the doubling constantof µ.

The following theorem provides a lower bound for a doubling measure onballs and gives a characterization of the doubling property.

Theorem 1.47. A measure µ : B(X) → [0,+∞] finite on bounded sets, isdoubling if and only if there exist constants s, C ′ > 0 such that

µ(B(x, r))

µ(B(y,R))≥ C ′

( rR

)s(1.26)

for all x, y ∈ X and all R ≥ r > 0 such that x ∈ B(y,R).

Proof. On one hand, if (1.26) holds, we have in particular with R = 2r andy = x that

µ(B(x, 2r)) ≤ 2s(C ′)−1µ(B(x, r))

which proves the doubling property (1.25) with C := 2s(C ′)−1.For the other implication, suppose that µ is doubling and define for i ∈ N

Ri := 2ir and j := mini : B(x,Ri) ⊇ B(y,R).

Iterating (1.25) j times yields

µ(B(y,R)) ≤ µ(B(x,Ri)) ≤ Cjµ(B(x, r))

which implies thatµ(B(x, r))

µ(B(y,R))≥ C−j.

On the other hand, B(y,R) is not contained in B(x,Rj−1): hence Rj−1 ≤ 2R(since x ∈ B(y,R), by assumption) and j ≤ log2(4R/r). Then the proof iscompleted by letting C ′ := C−2 and s := − logC/ log 2.

Definition 1.48 (Hardy-Littlewood maximal operator). Given two non-negative measures µ and ν on the metric spaceX, define the Hardy-Littlewoodmaximal operator of the measure ν (with respect to the measure µ) as

Mν(x) := supr>0

ν(B(x, r))

µ(B(x, r))for each x ∈ X. (1.27)

Let X = Rn endowed with the ususal Euclidean distance. A Borel func-tion f in L1(Rn) defines a non-negative finite Borel measure

νf (A) =

ˆA

|f(y)|dy


for each Borel set A ⊂ Rn.The Hardy-Littlewood maximal operator of νf with respect to the mea-

sure µ is then given by

Mf(x) = supr>0

1

µ(B(x, r))

ˆB(x,r)

|f(y)|dy. (1.28)

Theorem 1.49 (Weak-type (1, 1) estimate). Suppose that µ is doubling andthat the support of µ coincides with the whole of X. Then there exists C > 0(depending only on the doubling constant of µ) such that

µ(x ∈ X : Mν(x) > λ) ≤ C

λν(X) (1.29)

for all λ > 0.

Proof. If ν(X) = +∞, there is nothing to prove. Otherwise, choose λ > 0,R > 0, and define for all x ∈ X

Mν,R(x) := sup0<r<R

ν(B(x, r))

µ(B(x, r))

AR := x ∈ X : Mν,R(x) > λ.(1.30)

Let F be the family of all open ballsB(x, r) such that x ∈ AR, 0 < r < R, andν(B(x, r)) > λµ(B(x, r)). F is a covering of AR, because from the definitionsin (1.30), for all x ∈ AR there exists 0 < rx < R so that B(x, rx) ∈ F . Bya Vitali-type covering theorem, there exists an at most countable disjointsubfamily F ′ ⊆ F such that AR ⊆

⋃B∈F ′ B, where B is the ball with the

same center as B and a radius five times that of B. Therefore, since µ isdoubling and 5 < 23, we have

µ(AR) ≤∑B∈F ′

µ(B) ≤ C3∑B∈F ′

µ(B)

≤ C3

λ

∑B∈F ′

ν(B) ≤ C3

λν(X).

Finally, since R was arbitrary, we obtain

µ(x ∈ X : Mν(x) > λ) = limR→+∞

µ(AR) ≤ C3

λν(X).

Chapter 2

The Isoperimetric problem inEuclidean Space

In this chapter we describe the classical isoperimetric problem. Being veryancient, it was approached many times and there are several proofs for theplanar Euclidean case; we will analyse some of them to give an idea of thevarious technics that may be involved in this problem. The last section isdevoted to the problem in the n-dimensional Euclidean space. In particularwe will introduce the generalized notion of perimeter given by De Giorgi, todeal with the largest class of sets.

2.1 The isoperimetric inequality in the planeThe isoperimetric problem in the plane has substantially three equivalentformulations: consider all bounded isoperimetric domains in R2, i.e. allconnected open sets of R2 with fixed given perimeter, and find that onewhich contains the greatest area. The answer, as suggested by the intuition,will be the disk; equivalently, one may fix the area of all bounded domainsand require that the perimeter is minimized; finally, we may express theproblem as an analytic inequality: we ask which is the greatest constant Csuch that the inequality L2 ≥ CA holds for every domain, where A denotesthe area of the domain under consideration and L deontes the length of itsboundary. We would like to show that the best constant is C = 4π and thatthe equality is achieved exactly when the domain is a disk.

Here we will recall some proofs for relatively compact domains with C1

boundary consisting of a single component, namely we will prove the follow-ing theorem.

Theorem 2.1 (Isoperimetric inequality in R2). Let Ω be a relatively compact

37

38 The Isoperimetric problem in Euclidean Space

domain, with boundary ∂Ω ∈ C1 consisting of one component. Then

L2(∂Ω) ≥ 4πA(Ω). (2.1)

First proof of (2.1): using complex variables. We identify R2 with the com-plex plane and denote any element with z = x + iy. The area measure isthen given by

dA = dx ∧ dy =i2dz ∧ dz.

Then we obtain

4πA(Ω) =

ˆΩ

4πdA =

ˆΩ

2πidz ∧ dz

By the residue theorem 2πi =´∂Ω

1ω−zdω for each z ∈ Ω. Then from this

observation, together with Fubini’s theorem it follows that

4πA(Ω) =

ˆΩ

ˆ∂Ω

1

ω − zdωdz ∧ dz =

ˆ∂Ω

ˆΩ

1

ω − zdz ∧ dzdω. (2.2)

Notice that d( ω−zω−zdz) = (ω − z)−1dz ∧ dz, then applying Green’s theorem to

the inner integral of (2.2) we have

4πA(Ω) =

ˆ∂Ω

ˆ∂Ω

ω − zω − z

dzdω.

Since∣∣ ω−zω−z

∣∣ = 1 a.e. on ∂Ω, the latter integral is less than or equal to L(∂Ω)2

and we get the desired conclusion.In particular, if Ω = B(0, ρ) is a disk with center in 0 and radius ρ, take

the substitutions ω = ρeiϕ and z = ρeiθ in the last step, then we haveˆ∂Ω

ˆ∂Ω

ω − zω − z

dzdω =

ˆ 2π

0

ˆ 2π

0

ρ2dθdϕ = 4πρ2 = L(∂B(0, ρ))2.

This proves that 4π is indeed the smallest suitable constant in (2.1).

The second proof makes use of Fourier series. First of all we need thefollowing

Theorem 2.2 (Wirtinger’s Inequality). If f is a C1, L-periodic function onR such that

´ L0f(t)dt = 0, then

ˆ L

0

|f ′(t)|2dt ≥ 4π2

L2

ˆ L

0

|f(t)|2dt,

with equality if and only if there exist constants a−1 and a1 such that f(t) =a−1e

−2πit/L + a1e2πit/L.

2.1 The isoperimetric inequality in the plane 39

Proof. This result follows from the theory of Fourier series: a function g,that is L-periodic and in L2(0, L), admits a Fourier expansion

g(t) =∞∑

k=−∞

cke2πikt/L,

where

ck =1

L

ˆ L

0

g(t)e−2πikt/Ldt

are the Fourier coefficients of g.Since f(t) is a C1, L-periodic function, it has a Fourier expansion together

with its derivative. Let

f(t) =∞∑

k=−∞

ake2πikt/L with ak =

1

L

ˆ L

0

f(t)e−2πikt/Ldt,

f ′(t) =∞∑

k=−∞

bke2πikt/L with bk =

1

L

ˆ L

0

f ′(t)e−2πikt/Ldt.

By hypothesis a0 = 0. Moreover, if we integrate by parts the formula for b0,the continuity of f implies b0 = 0. For each k 6= 0, an analogous integrationby parts implies bk = 2πikak/L. This computation together with Parseval’sidentity yields

ˆ L

0

|f ′(t)|2dt = L∑k 6=0

|bk|2 ≥ L4π2

L2

∑k 6=0

|ak|2 =4π2

L2

ˆ L

0

|f(t)|2dt,

which implies the desired inequality. The equality holds if and only if ak = 0for all |k| 6= 1.

We use this last result to give another proof of (2.1) which characterizesthe sets that realize the equality in (2.1).

Second proof of (2.1): Method of Fourier series. Without loss of generalitylet us assume

´∂Ωxds = 0, where x = (x1, x2) ∈ R2 and ds is the length

measure on ∂Ω (if necessary, translate Ω). Let us consider the vector field~x = x1e1 + x2e2. Note that div~x = 2, then by the divergence theorem wehave

2A(Ω) =

ˆΩ

div~xdA = −ˆ∂Ω

〈~x, ~n〉 ≤ˆ∂Ω

|~x|ds, (2.3)


where ~n is the inward pointing unit normal to ∂Ω. By Hölder’s inequality,we can majorize this last quantity with

2A(Ω) ≤(ˆ

∂Ω

|~x|2ds) 1

2(ˆ

∂Ω

12ds

) 12

= L(∂Ω)12

(ˆ∂Ω

|x1|2 + |x2|2ds) 1

2

.

Let us parametrize ∂Ω by arc length. By the hypothesis´∂Ωxds = 0, we can

apply Wirtinger’s inequality to each coordinate function x1 and x2, hence

2A(Ω) ≤ L(∂Ω)12

(L(∂Ω)2

4π

ˆ∂Ω

|x′|2ds) 1

2

=L(∂Ω)2

2π

as desired.Let us characterize the sets which achieve the equality in (2.1). To have

an equality in (2.3), we must require ~x in the same direction of ~n a.e. Sincethe boundary of Ω is C1, this holds for every ~x, hence ~n is radial. This impliesthat ∂Ω is a circumference and we have equality everywhere.

The last proof that we will present uses geometric arguments. It is basedon symmetry and convexity methods. However, it will give only a weak result,in the sense that it will find the nature of the minimizer, provided we assumeits existence. On the other hand, it is interesting because it generalizes toonly piecewise C1 boundaries.

Third proof of (2.1): Geometric methods. Let Ω be a minimizer such thatits boundary is piecewise C1. First notice that each line π which divides Ωin two pieces of equal area must separate also the boundary in two pieces ofequal length. In fact, let Ω1 and Ω2, be the two parts in which Ω is divided byπ, with A(Ω1) = A(Ω2). If, for example, L(∂Ω1) < L(∂Ω2), we could obtaina set with less perimeter than Ω by taking the union of Ω1 and its reflectionover the line π. This contradicts the hypothesis that Ω is an isoperimetricminimizer.

By carrying out this argument succesively, we may assume that thereexists a coordinate system in R2 such that Ω is symmetric with respect tothe two coordinate axes. Then every line through the origin cuts Ω into twoopen sets of equal area and hence equal bounding length.

Notice also that we may assume the closure of Ω, Ω, to be convex. Indeed,if it weren’t, we may replace it by its convex hull, which increases the areaand decreases the bounding length.

Since the origin is in Ω, for each y ∈ ∂Ω such that there exists the tangentline to Ω at y, consider the line l that passes through the origin and y. Weclaim that l is perpendicular to the tangent line at y. If not, by reflecting

2.2 The classical isoperimetric problem 41

over l one of the two parts in which Ω is divided by l, we obtain a non-convexdomain, with the same area and the same perimeter of Ω. Its convex hull hashowever greater area and smaller perimeter, which contradicts the hypothesison ∂Ω to be minimal.

Therefore, Ω is convex with every point of differentiability of ∂Ω havingtangent line orthogonal to the position vector ~x. Consider the unit circle S,denote by ~n its unitary position vector and by θ the local coordinate on S.For what we have just said we have ~x = τ~n, at all points of differentiabilityof ∂Ω, so that

∂~x

∂θ=∂τ

∂θ~n+ τ

∂~n

∂θ.

Since ~n is perpendicular to the tangent space to ∂Ω a.e., we have

0 = 〈∂~x∂θ, ~n〉 = 〈∂τ

∂θ~n, ~n〉+ 〈τ ∂~n

∂θ, ~n〉 =

∂τ

∂θ

and we conclude that τ is differentiable with ∂τ/∂θ = 0 at all points ofdifferentiability of ∂Ω; then τ is locally constant on the C1 arcs of ∂Ω. Hence∂Ω consists of a finite number of circular arcs connected by segments of radiallines. The convexity of Ω implies ∂Ω is a circle.

Remark 2.3. One could be tempted to carry out this argument also inhigher dimensions, but we have to be careful about the convexity. For ex-ample, passing to the convex hull of a domain does not in general decreaseits isoperimetric quotient. In fact, if we consider a standard imbedded solidtorus in R3, where the circle in the xy-plane has radius 1, and the rotatedcircle perpendicular to the xy − plane has radius ε << 1. Then its isoperi-metric quotient is asymptotic to a constant multiple of ε−1/3 as ε → 0. Onthe other hand, the convex hull has isopermetric quotient asymptotic to aconstant multiple of ε−2/3 as ε→ 0.

2.2 The classical isoperimetric problemThe classical formulation of the isoperimetric problem in the n-dimensionalEuclidean space, with n > 2, reads as follows:

Theorem 2.4 (Isoperimetric inequality). Let E ⊂ Rn be a Borel set withfinite perimeter P (E) and area denoted by |E|. Then

min|E|(n−1)/n, |Rn \ E|(n−1)/n ≤ Ciso(Rn)P (E), (2.4)

whereCiso(Rn) = (n1−1/nω

1/nn−1)−1. (2.5)


Moreover the equality in (2.4) holds if and only if E = B(x,R) for somex ∈ Rn and R > 0.

The proof was given by De Giorgi in 1958 and can be found in [9] pagg.185-197. There are also several works that prove the theorem in some restrictedclasses of sets. Some solutions of different kinds can be found for istance in[1], [7] or [14]. In this survey we will follow that one of [11].

2.2.1 BV functions and perimeter

Here we analyze the generalized concept of perimeter involved in this settingand study some properties that will be usefull also for the sub-Riemanniancase.

Let E be any measurable subset of Rn. If E were sufficiently smooth,that is if the boundary ∂E were C1, then one has the surface measure dσ of∂E, and

P (E) =

ˆ∂E

dσ.

For rougher domains we have to define the notion of perimeter introducedby De Giorgi.

Definition 2.5. Let Ω ⊂ Rn be an open set. A function u ∈ L1(Ω) hasbounded variation in Ω if

Var(u; Ω) := sup

ˆΩ

u divg dx| g ∈ C1c (Ω,Rn), |g| ≤ 1

< +∞. (2.6)

We denote the vector space of functions of bounded variation in Ω withBV (Ω).

A function u ∈ L1loc(U) is said to have locally bounded variation in Ω if

for each open set V , with compact closure in U ,

sup

ˆV

u divg dx| g ∈ C1c (V,Rn), |g| ≤ 1

< +∞.

The space of such functions will be denoted by BVloc(Ω).

Definition 2.6. Let E ⊂ Rn be a measurable set and Ω ⊂ Rn open. We saythat E has finite perimeter (or is a Caccipoli set) in Ω if

P (E,Ω) := Var(χE; Ω) < +∞, (2.7)

where χE is the characteristic function of E.E has locally finite perimeter in Ω if χE ∈ BVloc(Ω).


Remark 2.7. Note that if E is a bounded set with C1 boundary in Ω, wecan apply the divergence theorem to each g ∈ C1

c (Ω,Rn). So if ν is the outernormal to ∂E and σn−1 the surface measure, then

ˆE

divg dx =

ˆ∂E

〈g, ν〉dσn−1.

Taking the supremum we obtain the usual formula P (E,Ω) = σn−1(Ω∩∂E).

Theorem 2.8 (Lower semicontinuity of the variation functional). The vari-ation functional is lower semicontinuous with respect to the L1

loc(Ω) conver-gence.

Proof. Fix g ∈ C1c (Ω;Rn), let V ⊂ Ω open with compact closure in Ω and

let ukk be a sequence of BV (Ω) functions which converge to u ∈ BV (Ω)in the L1

loc(Ω) convergence, then∣∣∣∣ˆV

(u− uk)divg dx

∣∣∣∣ ≤ ||divg||∞ˆV

|u− uk|dx→ 0

as k tends to ∞. (Notice that ||divg||∞ is bounded because the derivative ofg is continuous and with compact support). This means that the functionalu 7→

´Ωudivg dx is continuous for each g. Now the variation functional is

lower semicontinuous because it is the supremum of these continuous func-tionals.

Theorem 2.9. Let u ∈ L1(Ω). Then u is in BV (Ω) if and only if thereexists a Radon measure Du = (D1u, . . . , Dnu) in Ω such that Var(u; Ω) =|Du|(Ω) ≤ M < +∞ and which represents the derivative of u in the senseof the distributions, i.e.

ˆΩ

u∂ϕ

∂xidx = −

ˆΩ

ϕdDiu for each ϕ ∈ C1c (Ω).

Here |µ|(B) denotes the variation of a scalar or vector measure µ on a Borelset B and is equal to

|µ|(B) := sup

∞∑i=0

|µ(Bi)| : B =∞⋃i=0

Bi

where the supremum is taken over all partitions Bii of B in Borel sets.


Proof. Let u ∈ L1(Ω). If the measure Du exists, then for each g ∈ C1c (Ω,Rn)

with |g| ≤ 1 we haveˆ

Ω

u divg dx = −ˆ

Ω

n∑i=1

gidDiu

then Var(u; Ω) = |Du|(Ω) ≤M .For the other implication, let u ∈ BV (Ω). From equation (2.6) we obtain∣∣∣∣ˆ

Ω

u divg dx

∣∣∣∣ ≤ Var(u; Ω)||g||∞ for each g ∈ C1c (Ω,Rn).

Let us extend by density the functional g 7→(´

Ωu divg dx

)to the whole

Cc(Ω,Rn), so that we obtain a linear bounded functional L(g) with normless or equal to Var(u; Ω). Then by Riesz theorem there exists a vectormeasure µ = (µ1, . . . , µn) such that

L(g) = −n∑i=1

ˆΩ

gi(x) dµi(x) for each g ∈ Cc(Ω,Rn)

with |µ|(Ω) ≤ Var(u; Ω). Hence the measure µ represents the derivative of uin the sense of the distributions.

We give for later purpose an useful approximation result.

Theorem 2.10 (Local approximation by smooth functions). Let Ω ⊂ Rn bean open set and u ∈ BV (Ω). There exist functions uk ∈ BV (Ω) ∩ C∞(Ω),k ∈ N, such that uk → u in L1(Ω) as k →∞ and

limk→∞

Var(uk,Ω) = Var(u,Ω).

Theorem 2.11 (Product with Lipschitz functions). Let Ω ⊂ Rn be an openset, u ∈ BV (Ω) and ψ a Lipschitz real function on Ω which gradient hascompact support where it exists. Then the product uψ has bounded variationon Ω and the associated measure is given by

D(uψ) = ψDu+ u∇ψLn. (2.8)

Proof. By Rademacher’s theorem ψ is differentiable a.e., therefore for everyg ∈ C1

c (Ω,Rn) the product law gives ψ∂gi/∂xi = ∂ψgi/∂xi− gi∂ψ/∂xi whereψ is differentiable. Then since u ∈ BV (Ω), there exists a scalar measureDu = (D1u, . . . , Dnu) such that

ˆΩ

uψ∂gi∂xi

dx = −ˆ

Ω

giψd(Diu)−ˆ

Ω

giu∂ψ

∂xidx. (2.9)


Hence the measure Di(uψ) := ψDiu+ u∂ψ/∂xiLn represent the i-derivativeof uψ in the sense of the distributions. Since the gradient of ψ has compactsupport, the measure Di(uψ) has bounded variation. So uψ ∈ BV (Ω) and

D(uψ) = ψDu+ u∇ψLn.

For later purpose we derive here a theorem of localization for the perime-ter measure. We will apply it in Section 6.3 in the proof of the existence ofminimizers for the Heisenberg isoperimeter problem.

Theorem 2.12. Let E be a set of finite perimeter in Ω, x0 ∈ Ω and δ =dist(x0, ∂Ω). Then

P (E ∩Bρ(x0),Rn) ≤ P (E,Bρ(x0)) +m′+(ρ)

holds for each ρ ∈ (0, δ), where m(ρ) = |E ∩ Bρ(x0)| and m′+ is the lowerright derivative of m.

Proof. By translation, it is not restrictive to assume x0 = 0. Let u(x) =χE(x). We will prove the following more general inequality:

Var(uρ,Rn) ≤ Var(u,Bρ(0)) +

(ˆBρ(0)

|u(x)|dx

)′+

(2.10)

for each ρ ∈ (0, δ) and for any u ∈ BV (Ω), where uρ = uχBρ(0). Given anyσ ∈ (0, δ − ρ) we construct the family of functions uσ ∈ BV (Rn) supportedin Bρ+σ(0) given by uσ(x) = u(x)γσ(|x|), where

γσ(t) :=

1 if t ≤ ρ1 + ρ−t

σif ρ ≤ t ≤ ρ+ σ

0 if t ≥ ρ+ σ.

Since the functions γσ are Lipschitz functions, by theorem 2.11 the uσ arestill of bounded variation in Ω and the respective variation measure is

Duσ = γσ(|x|)Du+ u(x)γ′σ(|x|) x|x|Ln.

Since γ′σ(t) = −σ−1 χBρ+σ(0)\Bρ(0) we have the estimate

|Duσ|(Rn) ≤ |Du|(Bρ+σ) + σ−1

ˆBρ+σ(0)\Bρ(0)

|u(x)|dx.

Finally, since uσ convergies to uρ in L1(Rn) as σ goes to 0, equation (2.10)follows from the semicontinuity of the variation.


2.2.2 Sobolev’s and Poincaré’s Inequalities for C1c func-

tions

The proof of the isoperimetric inequality (2.4) is a consequence of Sobolev’sand Poincaré’s inequalities for BV functions. In this section we will thereforeprove such inequalities and at the end we will show how the isoperimetricinequality follows from them.

Let us consider first a function f ∈ C1c (Rn). For each 1 ≤ p < n we define

the Sobolev conjugate of p as

p∗ :=np

n− p.

Notice that the Sobolev conjugate of p is such that 1/p− 1/p∗ = 1/n.

Theorem 2.13 (Gagliardo-Nirenberg-Sobolev Inequality). Let 1 ≤ p < n.There exists a constant C1, depending only on p and n, such that(ˆ

Rn|f |p∗dx

)1/p∗

≤ C1

(ˆRn|Df |pdx

)1/p

(2.11)

for all f ∈ C1c (Rn).

To prove the Gagliardo-Nirenberg-Sobolev inequality, we need the follow-ing theorem.

Theorem 2.14 (General Hölder inequality). Let U ⊂ Rn be an open setand let 1 ≤ p1 ≤ . . . ≤ pn ≤ ∞ be such that 1/p1 + . . . + 1/pn = 1. Givenfunctions u1, . . . , un such that uk ∈ Lpk(U) for each k = 1, . . . , n the followinginequality holds ˆ

U

|u1 · · ·un|dx ≤n∏i=1

||uk||Lpk (U). (2.12)

Proof. The proof is an application of the mathematical induction and theclassical Hölder inequality.

For n = 1, there’s nothing to prove.Let n > 1. If pn =∞, then 1/p1 + . . .+ 1/pn−1 = 1 and

ˆU

|u1 · · ·un|dx ≤ ||un||L∞(U)

ˆU

|u1 · · ·un−1|dx ≤n∏k=1

||uk||Lpk (U)

where the latter inequality follows from the inductive hypothesis.


In pn 6=∞, let q = pn and

1

p:= 1− 1

pn=

1

p1

+ . . .+1

pn−1

.

Then 1/p+ 1/q = 1 and we may apply the classical Hölder inequality to thefunctions u1 · · ·un−1 and un:

Û

|u1 · · ·un|dx ≤(ˆ

U

|u1 · · ·un−1|pdx)1/p(ˆ

U

|un|qdx)1/q

(2.13)

=

(Û

||u1|p · · · |un−1|p| dx)1/p

||un||Lpn (U). (2.14)

Let pk = pk/p for k = 1, . . . , n−1. Then 1/p1 + . . .+1/pn−1 = p(1/p1 + . . .+1/pn−1) = 1. Moreover |uk|p ∈ Lpk(U), becauseˆ

U

|uk|ppkdx =

Û

|uk|pkdx <∞.

Then we can apply the inductive hypothesis to the functions |u1|p, . . . , |un−1|pwith the new coefficients p1, . . . , pn−1 and we obtain(ˆ

U

||u1|p · · · |un−1|p| dx) 1

p

≤

(n−1∏k=1

(Û

|uk|ppkdx) 1

pk

) 1p

=

(Û

|u1|p1dx) 1

p1

. . .

(Û

|un−1|pn−1dx

) 1pn−1

.

This inequality, together with (2.14) proves inequality (2.12).

Proof of Theorem 2.13. For each 1 ≤ i ≤ n, we have

f(x1, . . . , xi, . . . , xn) =

ˆ xi

−∞

∂f

∂xi(x1, . . . , ti, . . . , xn)dti,

hence

|f(x1, . . . , xi, . . . , xn)| ≤ˆ +∞

−∞

∣∣∣∣ ∂f∂xi (x1, . . . , ti, . . . , xn)

∣∣∣∣ dti≤

ˆ +∞

−∞|Df(x1, . . . , ti, . . . , xn)| dti.

Multiplying for i = 1, . . . , n we obtain

|f(x)|nn−1 ≤

n∏i=1

(ˆ +∞

−∞|Df(x1, . . . , ti, . . . , xn)| dti

) 1n−1

. (2.15)


Notice that nn−1

= 1∗. Let us integrate (2.15) with respect to x1:

ˆ +∞

−∞|f |1∗dx1 ≤

ˆ +∞

−∞

n∏i=1

(ˆ +∞

−∞|Df(x1, . . . , ti, . . . , xn)| dti

) 1n−1

dx1

=

(ˆ +∞

−∞|Df | dt1

) 1n−1

ˆ +∞

−∞

n∏i=2

(ˆ +∞

−∞|Df | dti

) 1n−1

dx1

≤(ˆ +∞

−∞|Df | dt1

) 1n−1

n∏i=2

(ˆ +∞

−∞

ˆ +∞

−∞|Df | dx1dti

) 1n−1

where the latter inequality follows from (2.12) with

uk(x1) =

(ˆ +∞

−∞|Df(x1, . . . , tk, . . . , xn)| dtk

) 1n−1

and pk = n− 1 for k = 1, . . . , n− 1.Next integrate with respect to x2, so that it resultsˆ +∞

−∞

ˆ +∞

−∞|f |1∗dx1dx2

≤(ˆ +∞

−∞

ˆ +∞

−∞|Df | dx1dt2

) 1n−1

ˆ +∞

−∞

(ˆ +∞

−∞|Df | dt1

) 1n−1

dx2

·ˆ +∞

−∞

n∏i=3

(ˆ +∞

−∞

ˆ +∞

−∞|Df | dx1dti

) 1n−1

dx2

≤(ˆ +∞

−∞

ˆ +∞

−∞|Df | dx1dt2

) 1n−1(ˆ +∞

−∞

ˆ +∞

−∞|Df | dt1dx2

) 1n−1

·n∏i=3

(ˆ +∞

−∞

ˆ +∞

−∞

ˆ +∞

−∞|Df | dx1dx2dti

) 1n−1

Continue till xn and we obtainˆRn|f |1∗dx1 ≤

n∏i=1

(ˆ +∞

−∞· · ·

ˆ +∞

−∞|Df | dx1 . . . dti . . . dxn

) 1n−1

=

(ˆRn|Df |dx

) nn−1

.

So we have proven the inequality for p = 1:(ˆRn|f |1∗dx

)1/1∗

≤ˆRn|Df |dx. (2.16)


If 1 < p < n, let g = |f |γ with γ > 0 to be chosen later. Applying (2.16) tog, we get(ˆ

Rn|f |

γnn−1dx

)n−1n

≤ γ

ˆRn|f |γ−1|Df |dx

≤ γ

(ˆRn|f |(γ−1) p

p−1dx

) p−1p(ˆ

Rn|Df |pdx

) 1p

(2.17)

where the latter inequality follows from the classical Hölder inequality. Nowwe choose γ so that γn

n−1= (γ − 1) p

p−1, i.e.

γ =p(n− 1)

n− p≥ 1.

In this way we have γnn−1

= (γ − 1) pp−1

= npn−p = p∗. Then (2.17) becomes

(ˆRn|f |p∗dx

)n−1n

≤ γ

(ˆRn|f |p∗dx

) p−1p(ˆ

Rn|Df |pdx

) 1p

,

that is (ˆRn|f |p∗dx

)n−1n− p−1

p

≤ γ

(ˆRn|Df |pdx

) 1p

,

which is equivalent to (2.11).

We state now the Poincaré’s inequality for C1c functions on the balls. To

this end, we fix here the notation. We will denote by B(x, r) the open ballin Rn with center in x and radius r > 0. For an integrable function f on Rn

the average of f on a ball B(x, r) is given by

(f)x,r =

B(x,r)

f(y)dy =1

|B(x, r)|

ˆB(x,r)

f(y)dy.

Theorem 2.15 (Poincaré’s inequality). For each 1 ≤ p < n there exists aconstant C2, depending only on p and n, such that(

B(x,r)

|f − (f)x,r|p∗dx)1/p∗

≤ C2r

( B(x,r)

|Df |pdx)1/p

for all B(x, r) ⊂ Rn and f ∈ C1c (Rn).

The proof of this theorem is a consequence of the next lemma.


Lemma 2.16. For each 1 ≤ p < ∞ there exists a constant C, dependingonly on n and p, such that

ˆB(x,r)

|f(y)− f(z)|pdy ≤ Crn+p−1

ˆB(x,r)

|Df(y)|p|y − z|1−ndy

for all B(x, r) ⊂ Rn, f ∈ C1(B(x, r)), and z ∈ B(x, r).

Proof. If y, z ∈ B(x, r), then

f(y)− f(z) =

ˆ 1

0

d

dtf(z + t(y − z))dt =

ˆ 1

0

Df(z + t(y − z)) · (y − z)dt,

and so

|f(y)− f(z)|p ≤ |y − z|pˆ 1

0

|Df(z + t(y − z))|pdt.

Thus, for s > 0,ˆB(x,r)∩∂B(z,s)

|f(y)− f(z)|pdHn−1(y)

≤ spˆ 1

0

ˆB(x,r)∩∂B(z,s)

|Df(z + t(y − z))|pdHn−1(y)dt

≤ spˆ 1

0

1

tn−1

ˆB(x,r)∩∂B(z,ts)

|Df(ω)|pdHn−1(ω)dt

where we have used the change of variable ω = z+t(y−z) ∈ ∂B(z, ts). Since|ω − z| = st we further get

≤ sn+p−1

ˆ 1

0

ˆB(x,r)∩∂B(z,ts)

|Df(ω)|p|ω − z|1−ndHn−1(ω)dt

= sn+p−2

ˆB(x,r)∩B(z,s)

|Df(ω)|p|ω − z|1−ndω.

Hence the co-area formula impliesˆB(x,r)

|f(y)− f(z)|pdy ≤ Crn+p−1

ˆB(x,r)

|Df(ω)|p|ω − z|1−ndω.


Proof of Theorem 2.15. Using the previous lemma, we may compute

B(x,r)

|f − (f)x,r|pdy =

B(x,r)

∣∣∣∣ B(x,r)

f(y)− f(z)dz

∣∣∣∣p dy≤ B(x,r)

B(x,r)

|f(y)− f(z)|pdzdy

≤ C

B(x,r)

rp−1

ˆB(x,r)

|Df(z)|p|y − z|1−ndzdy

≤ Crp B(x,r)

|Df |pdz.

(2.18)

Moreover it holds a general inequality: there exists a constant C = C(n, p)such that for each g ∈ C1

c (B(x, r))

( B(x,r)

|g|p∗dy) 1

p∗

≤ C

(rp B(x,r)

|Dg|pdy +

B(x,r)

|g|pdy) 1

p

. (2.19)

Indeed, we may suppose x = 0. Let R such that spt(g) ⊂ B(0, R), then,by replacing g(y) by (R/r)g((r/R)y) if necessary, we may assume g = 0out of B(0, 1) and r = 1. Then the Gagliardo-Nirenberg-Sobolev inequality(2.11) implies

(ˆB(0,1)

|g|p∗dy) 1

p∗

≤(ˆ

Rn|g|p∗dy

) 1p∗

≤ C1

(ˆRn|Dg|pdy

) 1p

≤ C

(ˆRn|Dg|p + |g|pdy

) 1p

.

Now we may conclude, indeed (2.18) and (2.19) with g = f − (f)x,r give thedesired inequality.

2.2.3 Isoperimetric Inequalities

The relations proved in the latter section have done all the heavy job towardsthe isoperimetric inequality. From Theorem 2.13 and Theorem 2.15 followsthe next theorem, which proves an equivalent of the latter relations for BVfunctions.


Theorem 2.17 (Gagliardo-Nirenberg-Sobolev inequality for BV functions).There exists a constant C1 such that

||u||L

nn−1 (Rn)

≤ C1Var(u,Rn) (2.20)

for all u ∈ BV (Rn).

Proof. By the approximation theorem for BV functions, we may choose uk ∈C∞c (Rn), with k = 1, 2, . . ., such that uk converges to u in L1(Rn) and

limk→∞

Var(uk,Rn) = Var(u,Rn).

Passing to a subsequence we may further assume that uk converges to u a.e.Observe that the functions uk are also BV functions and their variation

Var(u,Rn) = |Du|(Rn) is equal to

Var(u;Rn) = sup

ˆRnu divg dx| g ∈ C1

c (Rn,Rn), |g| ≤ 1

= sup

−ˆRnDu · g dx| g ∈ C1

c (Rn,Rn), |g| ≤ 1

= ||Du||L1(Rn).

Then, keeping in mind this observation, we may apply Fatou’s lemmaand Theorem 2.13 to obtain

||u||Ln/n−1(Rn) ≤ lim infk→∞

||uk||Ln/n−1(Rn)

≤ limk→∞

C1Var(uk,Rn)

= C1Var(u,Rn).

Theorem 2.18 (Poincaré inequality for BV functions). There exists a con-stant C2 such that

||u− (u)x,r||L nn−1 (B(x,r))

≤ C2Var(u,B(x, r)) (2.21)

for all B(x, r) ⊂ Rn, u ∈ BVloc(Rn).

Proof. The proof follows similarly to the proof of Theorem 2.17, using thePoincaré inequality 2.15. Indeed, take the sequnce of functions uk ∈ C∞c (Rn)


as in the proof of Theorem 2.17 and apply Fatou’s lemma and Theorem 2.15to the functions uk − (uk)x,r to obtain

||u− (u)x,r||Ln/n−1(B(x,r)) ≤ lim infk→∞

||uk − (uk)x,r||Ln/n−1(B(x,r))

= lim infk→∞

(ˆB(x,r)

|uk − (uk)x,r|n/n−1dy

)n−1n

= lim infk→∞

(rn B(x,r)

|uk − (uk)x,r|n/n−1dy

)n−1n

≤ limk→∞

C2rn

B(x,r)

|Duk|dy

= limk→∞

C2Var(uk, B(x, r))

= C2Var(u,B(x, r)).

Now we are able to give a proof of the first part of Theorem 2.4 and wecan also prove something more, namely:

Theorem 2.19. Let E ⊂ Rn be a bounded Borel set of finite perimeter P (E)and area |E|. Then

|E|n−1n ≤ C1P (E). (2.22)

This relation is called isoperimetric inequality.Moreover, for each ball B(x, r) ⊂ Rn

min|B(x, r) ∩ E|, |B(x, r) \ E|n−1n ≤ 2C2P (E,B(x, r)) (2.23)

is called the relative isoperimetric inequality.

Proof. Equation (2.22) is the statement of Theorem 2.17 with u = χE.For equation (2.23), apply Theorem 2.18 with u = χB(x,r)∩E. So (u)x,r =

|B(x, r) ∩ E|/|B(x, r)| and the integrand |u − (u)x,r| assumes two differentvalues, depending on whether we integrate on B(x, r)∩E or on B(x, r) \E.Thus

ˆB(x,r)

|u− (u)x,r|nn−1dy =

(|B(x, r) \ E||B(x, r)|

) nn−1

|B(x, r) ∩ E|

+

(|B(x, r) ∩ E||B(x, r)|

) nn−1

|B(x, r) \ E|.


Now, if for example, |B(x, r) ∩ E| ≥ |B(x, r) \ E|, then |B(x,r)\E||B(x,r)| ≥

12and

applying Theorem 2.18 we have

C2P (E,B(x, r)) ≥(ˆ

B(x,r)

|u− (u)x,r|nn−1dy

)n−1n

≥(|B(x, r) \ E||B(x, r)|

)|B(x, r) ∩ E|

n−1n

≥ 1

2min|B(x, r) ∩ E|, |B(x, r) \ E|

n−1n .

The other case is similar.

Untill now we have analysed the isoperimetric problem in the Euclideanspace. Since the next chapter we will start the study of its extension to theHeisenberg group.

Chapter 3

The Heisenberg Group andSub-Riemannian Geometry

In this chapter we introduce the first Heisenberg group, the simplest exam-ple of space with a sub-Riemannian metric structure. We will start witha detailed description of its algebraic structure and differentiable structure,defining the horizontal subbundle. Next we present the Carnot-Carathéodorymetric as the least time required to travel between two given points at unitspeed along horizontal paths. Subsequently we introduce the notion of sub-Riemannian metric and show how it may be approximated by Riemannianmetrics. Finally we compute in these Riemannian approximants some of thestandard differential geometric tools that will be useful in later chapters.

3.1 The first Heisenberg group HThe first Heisenberg group admits several representations. The first that weconsider is given by a matrix model and the group with such a representationis called the polarized Heisenberg group: it is the following subgroup of thegroup of three by three upper triangular matrices equipped with the usualmatrix product:

H =

1 x1 x3

0 1 x2

0 0 1

∈ GL(3,R) : x1, x2, x3 ∈ R

. (3.1)

This is a three dimensional subgroup of the general linear group GL(3R)and it receives from it the structure of Lie group. Its Lie algebra has dimen-sion 3 and h can be considered as the tangent space at the identity, TIH, or

55

56 The Heisenberg Group and Sub-Riemannian Geometry

equivalently as the set of all left invariant tangent vectors, as seen in Section1.1.2.

Let us give a base for the tangent space of H in the point 1 u1 u3

0 1 u2

0 0 1

.

We will indicate this point with (u1, u2, u3). To this end we translate it bythree independent elements, to obtain curves in H. For example, we canchoose the curves γ1, γ2, γ3 : [−1, 1]→ H defined by

γ1(t) =

1 u1 u3

0 1 u2

0 0 1

1 t 00 1 00 0 1

=

1 u1 + t u3

0 1 u2

0 0 1

,

γ2(t) =

1 u1 u3

0 1 u2

0 0 1

1 0 00 1 t0 0 1

=

1 u1 tu1 + u3

0 1 t+ u2

0 0 1

,

γ3(t) =

1 u1 u3

0 1 u2

0 0 1

1 0 t0 1 00 0 1

=

1 u1 t+ u3

0 1 u2

0 0 1

.

If we derive these curves we obtain the tangent vectors

U1 =dγ1(t)

dt

∣∣∣∣t=0

=

0 1 00 0 00 0 0

,

U2 =dγ2(t)

dt

∣∣∣∣t=0

=

0 0 u1

0 0 10 0 0

,

U3 =dγ3(t)

dt

∣∣∣∣t=0

=

0 0 10 0 00 0 0

.

By construction, U1, U2 e U3 are left invariant vector fields and they spanthe tangent space to H at (u1, u2, u3). Computing the Lie brackets of thesevector fields, we observe that

[U1, U2] = U1U2 − U2U1 = U3

while all other brackets are zero.Using the basis U1, U2, U3 we may introduce a system of coordinates generally

3.1 The first Heisenberg group H 57

known as polarized coordinates or canonical coordinates of the second kind.We identify the point (u1, u2, u3) with the matrix product

exp(u3U3|I) exp(u2U2|I) exp(u1U1|I),

where we have denoted by exp(U) = I+U + 12U2 + . . . and by U |I the vector

field U evaluated at the identity. The group law in polarized coordinateswrites explicitly

(u1, u2, u3)(v1, v2, v3) = (u1 + v1, u2 + v2, u3 + v3 + u1v2).

The Heisenberg group is a nilpotent step two Lie group that means thatthe central series

e = G0 ≤ G1 ≤ G2 ≤ . . . ≤ Gn = G

is finite and stops for n = 2. Here Gi are such that [G,Gi+1] ≤ Gi and [G,H]represents the commutative subgroup of G generated by g−1h−1gh for g ∈ Gand h ∈ H. In H we can compute explicitly the series: G0 = (0, 0, 0);

G1 = c = (c1, c2, c3) ∈ H s.t. x−1c−1xc = (0, 0, 0) ∀x ∈ H= (c1, c2, c3) ∈ H s.t. (u1 + c1, u2 + c2, u3 + c3 + u1c2) =

= (c1 + u1, c2 + u2, c3 + u3 + c1u2) ∀(u1, u2, u3) ∈ H= (c1, c2, c3) ∈ H s.t. u1c2 = c1u2 ∀u1, u2 ∈ R= (0, 0, c) ∈ H : c ∈ R;

G2 = d = (d1, d2, d3) ∈ H s.t. x−1d−1xd = (0, 0, c) ∀x ∈ H for a c ∈ R= (d1, d2, d3) ∈ H s.t. (u1 + d1, u2 + d2, u3 + d3 + u1d2) =

= (d1 + u1, d2 + u2, d3 + u3 + d1u2 + c) ∀ x ∈ H for a c ∈ R= H.

In conclusion, the Heisenberg group H is the unique analytic nilpotentstep two Lie group whose background manifold is R3 and whose Lie algebrah has the following properties:

• h = V1 ⊕ V2, where V1 has dimension 2 and V2 has dimension 1

• [V1, V1] = V2, [V1, V2] = 0 and [V2, V2] = 0.

We are now ready to introduce a more intrinsic representation of H, usingthe structure of its Lie algebra, h. The key point of this presentation is that,via the exponential map and the Baker-Campell-Hausdorff formula, we canderive the Lie group from its Lie algebra.


First, note that since h is nilpotent the exponential map exp : h→ H is adiffeomorphism. Fix an arbitrary basis X1, X2 of V1 and let X3 = [X1, X2] ∈V2. The Baker-Campell-Hausdorff formula states that

exp−1(exp(X) exp(Y )) =

=∑n>0

(−1)n−1

n

∑ri+si>01≤i≤n

(∑n

i=1 ri + si)−1

r1!s1! · · · rn!sn![Xr1Y s1Xr2Y s2 · · ·XrnY sn ]

= X + Y +1

2[X, Y ] +

1

12[X, [X, Y ]]− 1

12[Y, [X, Y ]] + . . . (3.2)

where

[Xr1Y s1Xr2Y s2 · · ·XrnY sn ] =

= [X, [X, . . . [X,︸︷︷︸r1

[Y, [Y, . . . [Y,︸︷︷︸s1

. . . [X, . . . [X,︸︷︷︸rn

[Y, [Y, . . . , Y ]]︸︷︷︸sn

. . . ] .

It is a powerful formula that allows us to obtain all the information aboutthe group product by knowing the Lie algebra.

In this case, since the Lie algebra h is step two, the higher terms in (3.2)are zero, so equation (3.2) takes the simpler form:

exp−1(exp(x) exp(y)) = x+ y +1

2[x, y]. (3.3)

where we have denoted by x = x1X1 + x2X2 + x3X3 = (x1, x2, x3) a genericvector in h. Using the commutation relation X3 = [X1, X2] and [Xi, X3] = 0we obtain

x+ y +1

2[x, y] = (x1 + y1)X1 + (x2 + y2)X2 + (x3 + y3)X3

+1

2x1y1 [X1, X1] + (x1y2 − x2y1) [X1, X2] + x2y2 [X2, X2]

=

(x1 + y1, x2 + y2, x3 + y3 +

1

2(x1y2 − x2y1)

).

Then formula (3.3) reads more explicitly as

exp−1(exp(x) exp(y))=

(x1 + y1, x2 + y2, x3 + y3 +

1

2(x1y2 − x2y1)

). (3.4)

We will identify H with C×R by associating exp(x1X1 + x2X2 + x3X3) ∈ Hwith the point (z, x3) ∈ C × R, where z = x1 + ix2 ∈ C and x3 ∈ R. The


coordinates x = (x1, x2, x3) = (z, x3) ∈ H are called canonical coordinates ofthe first kind or simply exponential coordinates. Using these coordinates, by(3.4), the group law reads

(z, x3)(w, y3) = (z + w, x3 + y3 −1

2Im(zw)). (3.5)

From this equation we get immediately that the group identity is 0 = (0, 0, 0),while x−1 = (−x1,−x2,−x3).

An isomorphism between this latter model of H and the polarized Heisen-berg group defined in (3.1) is obtained by mapping the element exp(x1X1 +x2X2 + x3X3) to the matrix 1 x1 x3 + 1

2x1x2

0 1 x2

0 0 1

.

For use in next paragraphs, it is useful to write explicitly the vector fieldsX1, X2, X3 with respect to the standard derivations ∂x1 , ∂x2 , ∂x3 .

To this end let us choose for the tangent space at the identity the canonicalbase X1(0) = ∂x1 , X2(0) = ∂x2 and X3(0) = ∂x3 . The left invariant vectorfields X1, X2, X3 generated by this base are

Xi(x) = dLx(0)(Xi(0)) = dLx(0)(∂xi).

Now, since the operation of left translation, Lx(y) = xy, has differential withmatrix

dLx =

1 0 00 1 0−1

2x2

12x1 1

, (3.6)

the explicit form of the vector fields X1, X2, X3 is

X1 = ∂x1 −1

2x2∂x3 , X2 = ∂x2 +

1

2x1∂x3 e X3 = ∂x3 . (3.7)

Notice that the desired commutation relations hold.We want to give to H a structure of homogeneous group. Let us consider

the following family of non-isotropic dilations : namely, for each s ∈ R+ letus define

δs(x) = (sx1, sx2, s2x3).

These maps are bijections on H that preserve the group structure, indeed itis easy to see that

δs(xy) = δs(x)δs(y).


Moreoverδs δs′ = δss′ ∀s, s′ ∈ R+

and δs is the identity transformation exactly when s = 1.Let us observe that ordinary Euclidean volumes are preserved by Lx, that

means that the volume of a measurable set E is the same as the volume ofits image Lx(E) under Lx. This statement follows because the Jacobian ofthe mapping Lx, which is the determinant of the differential of Lx, is equalto 1 everywhere on H, as we can see from (3.6). In short, ordinary volumemeasure is invariant under left translations in the Heisenberg group, and inthe same way it can be seen that it is invariant also under right translation.In conclusion, the Lebesgue measure on R3 is the Haar measure of H.

If we consider the dilations, we observe that the volume of δs(E) is equalto s4 times the volume of E for all s > 0 and all measurable sets E ⊂ H.The number 4 = dimV1 + 2 · dimV2 is called the homogeneous dimension ofH and is one dimension greater than the topological dimension.

The homogeneous dimension takes an important role in the isoperimetricinequality. The reason why this dimension has not been introduced in theclassical isoperimetric problem is that in the Euclidean case, R3, there is nodifference between the topological and the homogeneous dimensions, sincethey are both equal to 3. In the Euclidean case, it is therefore insignifi-cant which dimension we take. We will see that this is not the case for theHeisenberg group.

3.1.1 The horizontal distribution and the horizontal lin-ear maps on H

The horizontal fibration H(x) of H is a distribution of planes on H. It isdefined as the subbundle of the tangent bundle spanned at every point bythe left invariant frame X1, X2. With an easy computation we can see thatH(x) = Ker[dx3 − 1

2(x1dx2 − x2dx1)], indeed if we define

ω = dx3 −1

2(x1dx2 − x2dx1) (3.8)

then ω(aX1 + bX2 + cX3) = c. Note that ω is a contact form in R3, i.e.,ω ∧ dω = −dx1 ∧ dx2 ∧ dx3 6= 0.

We can define on the horizontal fibration a new gradient, which considersonly the derivations with respect to the vector fields X1 and X2 that lie onthe horizontal fibration. Namely, for any C1 function φ defined in an openset of H, its horizontal gradient is

∇0φ = X1φX1 +X2φX2.


To conclude this section, we define the notion of a linear map on H.

Definition 3.1. Consider the Heisenberg group bbH with dilation δs. A mapL : H → H is a horizontal linear map if L is a group homomorphism whichrespects the dilations, i.e.

L(δsx) = δsL(x).

More explicitlyL(x · y) = L(x) · L(y) ∀x, y ∈ H (3.9)

and

Li(sx1, sx2, s2x3) = sLi(x1, x2, x3) for i = 1, 2

L3(sx1, sx2, s2x3) = s2L3(x1, x2, x3) (3.10)

where L = (L1, L2, L3).

The next Proposition gives a characterization of the linear maps.

Proposition 3.2. Each horizontal linear map L : H → H takes the formL(x) = Ax, where the matrix A is equal to a11 a12 0

a21 a22 00 0 a11a22 − a12a21

for some a11, a12, a21, a22 ∈ R.

Proof. The proof follows by some remarks about the definition and by someexplicite computations. Equations (3.9) and (3.10) are equivalent to require

Li(x · y) = Li(x) + Li(y) and Li(δs(x)) = sLi(x) (3.11)

for i = 1, 2, and

L3(x · y) = L3(x) + L3(y) + 12(L1(x)L2(y)− L2(x)L1(y))

L3(δs(x)) = s2L3(x) (3.12)

From (3.11) follows that Li is linear for i = 1, 2, hence we can writeLi(x1, x2, x3) = ai1x1 + ai2x2 + ai3x3. Rewrite equations (3.11) with x = y =(0, 0, c), so that

2Li(0, 0, c) = Li((0, 0, c) · (0, 0, c)) = Li(δ√2(0, 0, c) =√

2Li(0, 0, c).


Then we have found the desired conclusion about the first two componentsof L, that is:

Li(0, 0, c) = 0 for i = 1, 2 and a13 = a23 = 0. (3.13)

To reach the conclusion also for the third component, let x = y =(x1, x2, 0) in (3.12), then

2L3(x1, x2, 0) = L3((x1, x2, 0) · (x1, x2, 0)) = L3(δ2(x1, x2, 0)) = 4L3(x1, x2, 0).

ThenL3(x1, x2, 0) = 0 ∀x1, x2 ∈ R. (3.14)

Observe that for the dilation properties

L3(0, 0, c) =

cL3(0, 0, 1) if c ≥ 0−cL3(0, 0,−1) if c < 0,

and since L(0, 0, 0) = (0, 0, 0), it holds L3(0, 0, 1) = −L3(0, 0,−1) := a33.Then for every c ∈ R

L3(0, 0, c) = cL3(0, 0, 1) = ca33. (3.15)

Now, let x = (x1, x2, x3) = (x1, x2, 0) · (0, 0, x3), then the first equation of(3.12), together with (3.13), (3.14) and (3.15), yields

L3(x) = a33x3 and a31 = a32 = 0. (3.16)

Summing up equations (3.13) and (3.16), it follows that

L(x1, x2, x3) =

a11 a12 0a21 a22 00 0 a33

x1

x2

x3

.

The last step is to prove that a33 = a11a22 − a12a21, but this follows easilyusing the latter result to calculate properties (3.12) with x = (x1, x2, x3) andy = (y1, y2, y3).

3.2 Carnot-Carathéodory distance

3.2.1 Horizontal paths

We want to endow H with a notion of distance. We are particularly interestedin two metrics, the Carnot-Carathéodory distance and the Korányi metric.

3.2 Carnot-Carathéodory distance 63

The Carnot-Carathéodory distance between two points is defined as theshortest time required to go from one point to the other, travelling at unitspeed along an admissible path, an horizontal path.

Let x and y be points in H. For δ > 0 we define the class C(δ) ofabsolutely continuous paths γ : [0, 1] → R3 with endpoints γ(0) = x andγ(1) = y, so that

γ′(t) = a(t)X1|γ(t) + b(t)X2|γ(t) (3.17)

anda(t)2 + b(t)2 ≤ δ2 (3.18)

for a.e. t ∈ [0, 1]. Paths satisfying (3.17) are called horizontal or Legendrianpaths. In other words, Legendrian paths are paths for which the compo-nent of the speed vector γ′ in direction X3 vanishes. More explicitly fromexpression (3.7) we obtain

∂x1 = X1 +1

2x2X3 ∂x2 = X2 −

1

2x1X3 ∂x3 = X3,

then, given a curve γ = (γ1, γ2, γ3),

γ′(t) = γ′1(t)∂x1 + γ′2(t)∂x2 + γ′3(t)∂x3

= γ′1X1 + γ′2X2 + (γ′3 −1

2(γ1γ

′2 − γ2γ

′1))X3. (3.19)

So γ is a Legendrian path if and only if

ω(γ′) = γ′3 −1

2(γ1γ

′2 − γ2γ

′1) = 0 (3.20)

a.e., where ω is the contact form on R3 given in (3.8).Not all the paths are Legendrian, but, as we will see in a moment, it is

possible to join every pair of points in H with such paths.Let π : H → C denote the projection π(x) = x1 + ix2. Given any

absolutely continuous planar curve α : [0, 1]→ C and a point x = (α(0), h) ∈H it is possible to lift α to a Legendrian path γ : [0, 1]→ H starting at x andsatisfying π(γ) = α. To accomplish this we let γ1(t) = α1(t), γ2(t) = α2(t)and

γ3(t) = h+1

2

ˆ t

0

(γ1γ′2 − γ2γ

′1)(s)ds.

Given x = (x1, x2, x3) and y = (y1, y2, y3), let us lift the curves α, β :[0, 1] → C that join respectively x1 + ix2 and y1 + iy2 to the origin. Theproblem of joining x and y with a Legendrian curve is then equivalent toshow how to connect (0, 0, h) to (0, 0, k) with a horizontal path, for everychoice of h, k ∈ R, or also to connect (0, 0, 0) with (0, 0, c) for each c ∈ R.


Let us show a particular case, that is to go from the origin to the point(0, 0, 1). First, we travel in the X1 direction; as we begin at the origin, thisis simply travel along the x1 axis. From the point (1, 0, 0), we travel in theX2 direction to the point (1, 1, 1

2). We then travel from this point in the

−X1 direction to the point (0, 1, 1). Finally, we travel in the −X2 direction,arriving at the terminus (0, 0, 1).

To travel from the origin to a generic point (0, 0, c), we just have to takethe latter path multiplied by

√c if c > 0, by −

√−c if c < 0.

From this procedure it is clear that for any choice of x and y, the setC(δ) is nonempty for sufficiently large δ. So we can define the Carnot-Carathéodory (CC) metric

d(x, y) = infδ such that C(δ) 6= ∅.

A dual formulation is

d(x, y) = inf

T :

∃γ : [0, T ]→ R3, γ(0) = x, γ(T ) = y,and γ′ = a X1|γ + b X2|γ with a2 + b2 ≤ 1 a.e.

,

in other words, d(x, y) is the shortest time required to travel from x to y,travelling at unit speed along horizontal paths.

Since the vector fields X1 and X2 are left invariant, left translates ofhorizontal curves are still horizontal. In particular, given a horizontal curveγ,

(Lx(γ))′ (t) =2∑i=1

γ′i(t) Xi|Lxγ(t) ,

therefore also the distance is preserved by left translation and d(x, y) =d(y−1x, 0).

Note that if γ is a horizontal curve, then so is its dilation δsγ. In fact,the expression of (δsγ)′ with respect to the basis X1, X2, X3 is

(δsγ)′ =

(sγ′1, sγ

′2,

1

2s2(γ1γ

′2 − γ2γ

′1)

)=

2∑i=1

sγ′i Xi|δsγ .

Moreover, if γ ∈ C(δ) then δsγ ∈ C(sδ) (observe that the endpoints must bedilated as well). Consequently

d(δs(x), δs(y)) = sd(x, y).

The map x 7→ d(x, 0) is continuous, in fact from the definition of theCC-distance and the composition of paths, it follows that

d(x, 0) ≤ d(x, y) + d(y, 0) and d(y, 0) ≤ d(x, y) + d(x, 0),


then |d(x, 0) − d(y, 0)| ≤ d(x, y), that implies the continuity of the CC-distance.

The properties of the CC metric transfer directly to the Hausdorff measureHα of (H, d), with α > 0. Recall the definition of Hausdorff measure given inDefinition 1.30. The left invariance and scaling properties of the CC metricimply that

Hα(LyE) = Hα(E) and Hα(δsE) = sαHα(E),

for all s, α > 0, y ∈ H, and E ⊂ H. In particular, for each α there existsc(α) = Hα(B(x, 1)) ∈ [0,∞] so that

Hα(B(x, r)) = Hα(δrB(0, 1)) = c(α)rα (3.21)

for all x ∈ H and r > 0, where B(x, r) denotes the metric ball with center xand radius r in (H, d).

On the other hand, the Hausdorff measure of a ball is given by

Hα(B(x, r)) =

ˆB(x,r)

dHα = r4

ˆB(0,1)

dHα = c(α)r4, (3.22)

where r4 is the Jacobian of the dilations.By comparing (3.21) and (3.22), we must have c(α) = −∞ when 0 < α <

4, while for α > 4 we have c(α) = 0. In case α = 4,

c(4) = H4(B(0, 1)) ∈ (0,+∞).

Thus the Hausdorff dimension of (H, d) is 4, equal to the homogeneous di-mension.

3.2.2 The Korányi gauge and metric

An equivalent distance on H is defined by the so-called Korányi metric: foreach x, y ∈ H define

dH(x, y) =∥∥y−1x

∥∥H ,

where‖x‖4

H = (x21 + x2

2)2 + 16x23 = |x1 + ix2|4 + 16x2

3

is the Korányi gauge. To verify that dH is a metric, one needs to prove thetriangle inequality:

dH(x, y) ≤ dH(x, z) + dH(z, y),


or equivalently ∥∥y−1x∥∥H ≤

∥∥z−1x∥∥H +

∥∥y−1z∥∥H .

This can be done by direct computation as we now recall. By replacing z−1xwith y and y−1z with x, it sufficies to prove that

‖xy‖H ≤ ‖x‖H + ‖y‖H .

Writing x = (z, x3) and y = (w, y3) and using the group law (3.5), we find

‖xy‖4H = |z + w|4 + 16(x3 + y3 −

1

2Im(zw))2

=

∣∣∣∣|z + w|2 + 4i(x3 + y3 −1

2Im(zw))

∣∣∣∣2=

∣∣|z|2 + 4ix3 + 2zw + |w|2 + 4iy3

∣∣2≤

(∣∣|z|2 + 4ix3

∣∣+ 2|z||w|+∣∣|w|2 + 4iy3

∣∣)2

=(‖x‖2

H + 2|z||w|+ ‖y‖2H)2

≤ (‖x‖H + ‖y‖H)4 .

From the definition of dH follows easily that it is homogeneous of order 1with respect to the dilations (δs), i.e.

‖δsx‖H = s ‖x‖H .

Notice that both distances dH and the CC-metric d are anisotropic. Inparticular, both behave like the Euclidean distance in horizontal directions(X1 and X2), and behave like the square root of the Euclidean distance inthe missing direction (X3).

Indeed the two described distances are equivalent, i.e., there exist con-stants C1, C2 > 0 so that

C1 ‖x‖H ≤ d(x, 0) ≤ C2 ‖x‖H (3.23)

for any x ∈ H. To show this statement, recall that the Korányi unit spherex ∈ H : ‖x‖H = 1 is compact for the Riesz theorem. From the continuityof the function

(H, d) → Rx 7→ d(x, 0)

there existC1 := min

x∈SHd(x, 0) and C2 := max

x∈SHd(x, 0).


So (3.23) holds for each x ∈ SH and obviously for x = 0. Let y ∈ H \ 0,there exist x ∈ SH and s ∈ R+ such that y = δs(x), then

C1 ‖y‖H = sC1 ‖x‖H ≤ sd(x, 0) = d(y, 0) ≤ sC2 ‖x‖H ≤ C2 ‖y‖H ,

as desired.

3.2.3 Sub-Riemannian structure

Our next purpose is to endow H with a sub-Riemannian structure. This canbe done by choosing an inner product on the horizontal subbundle of theLie algebra h and extending it to the whole h. Using the sub-Riemannianstructure one may define the length of horizontal curves with respect to thesub-Riemannian product and equip H with the structure of a metric lengthspace. A metric space (X, d) is called length if the distance between anytwo points x and y is realized by the infimum of the lengths of rectificablepaths joining x to y. We will see that this metric turns out to agree with theCarnot-Carathéodory metric.

Let us give now a sub-Riemannian metric on H: since we have alreadymade an arbitrary choice of coordinates to present H, we may define the innerproduct at each point x so that X1 and X2 form an orthonormal basis of thehorizontal space H(x). We extend this inner product to an inner productdefined on the full tangent space, i.e., a Riemannian metric, by requiring thatthe two layers in the stratification of the Lie algebra are orthogonal and thatX1, X2 and X3 form an orthonormal system. We denote this extended innerproduct by g1 or 〈·, ·〉1 as dictation by specific situations.

Accordingly, we define the horizontal length of a curve γ to be

LengthH,CC(γ) =

ˆ 1

0

√⟨γ′(t), X1|γ(t)

⟩2

1+⟨γ′(t), X2|γ(t)

⟩2

1dt, (3.24)

and we claim that for the CC distance holds

d(x, y) = infγ

LengthH,CC(γ), (3.25)

where the infimum is taken over all horizontal curves joining x and y. Byproving (3.25), we will show that H with the CC distance has the structureof a metric length space.

Note that if π : H→ C denotes the projection π(x) = x1+ix2, dπ : h→ Cdenotes its differential, and LengthC,Eucl(·) denotes Euclidean length in theplane, then

LengthH,CC(γ) = LengthC,Eucl(π(γ)), (3.26)


indeed since γ′ = γ′1X1 + γ′2X2 + (γ′3 − 12(γ1γ

′2 − γ2γ

′1)X3, both the lengths

that we are considering are equal toˆ 1

0

√|γ′1(t)|2 + |γ′2(t)|2dt.

To prove (3.25) we fix x, y ∈ H and let d = infγ LengthH,CC(γ). For anyδ > d(x, y) we consider a curve γ ∈ C(δ) and note that

d ≤ LengthC,Eucl(π(γ)) ≤ˆ 1

0

√a2 + b2dt ≤ δ,

by (3.18). Thus d ≤ d(x, y).For the opposite inequality let ε > 0 and let γ : [0, 1] → H be a curve

connecting x and y so that LengthC,Eucl(π(γ)) = d + ε. If we reparametrizeγ to have constant velocity |dπ(γ′)| = d + ε, then γ ∈ C(d + ε). Henced+ ε ≥ d(x, y). Since ε > 0 was arbitrary, d = d(x, y).

The next lemma shows that the Korányi and CC metrics generate thesame infinitesimal structure. This fact follows from the definition of theKorányi metric. Indeed this behaves similarly to the CC metric in the X1

and X2 directions, while in the missing direction it behaves as the squareroot of the distance. An usefull exercise to understand the behaviour of thelength of a curve with respect to the square root of the norm is the following:

Example 1. Consider the real plane R2 with metric√|| · ||, i.e. the square

root of the Euclidean metric in R2, and let γ : [a, b] → R2 be a regular nonconstant curve. Then

Length√||·||(γ) =∞.

Proof. The length of a curve in the Euclidean plane is given by:

Length√|·||(γ) = sup

n∑i=1

√‖γ(xi)− γ(xi−1)‖ : a = x0 < . . . < xn = b

.

Let L := Length√|·||(γ) and suppose L < +∞. For each ε > 0 there exists apartition a = x0 < x1 < . . . < xn = b such that

n∑i=1

√‖γ(xi)− γ(xi−1)‖ > L− ε. (3.27)

For eachN ∈ N big enough, refining the partition if necessary, we can supposethat

1

N< ‖γ(xi)− γ(xi−1)‖ < 5

N(3.28)


for each i = 1, . . . , n. Note that by refining the partition, property (3.27)still holds. From (3.27) and (3.28) it follows that n > N

5(L− ε). Hence

Length√|·||(γ) ≥n∑i=1

√‖γ(xi)− γ(xi−1)‖ > n

1√N>N(L− ε)

5√N

.

If we send N to infinity (and consequently choose an appropriate n), thenLength√|·||(γ) tends to infinity and we reach a contraddiction. So L =∞.

Now we may state the Lemma about the Korányi and CC metric.

Lemma 3.3. If γ : [0, 1]→ R is a C1 curve and ti = in, for i = 1, . . . , n, is

a partition of [0, 1], then

lim supn→∞

n∑i=1

dH(γ(ti), γ(ti−1)) =

LengthH,CC(γ) if γ is horizontal∞ otherwise.

Proof. Let us denote γ(t) = (γ1(t), γ2(t), γ3(t)), γi = γ(ti) = (γ1i , γ

2i , γ

3i ), and

γji = (dγj

dt)(ti). Then

dH(γ(ti), γ(ti−1)) = ||γ−1i−1γi||H = |a2

in + 16b2i,n|

14

where ai,n = (γ1i − γ1

i−1)2 + (γ2i − γ2

i−1)2 and bi,n = γ3i − γ3

i−1 − 12[γ1i (γ

2i −

γ2i−1) − γ2

i (γ1i − γ1

i−1)]. Applying simple inequalities and then summing upfor i = 1, . . . , n we obtain

n∑i=1

|ai,n|12 ≤

n∑i=1

dH(γ(ti), γ(ti−1)) ≤n∑i=1

|ai,n|12 + 2

n∑i=1

|bi,n|12 (3.29)

n∑i=1

|bi,n|12 ≤

n∑i=1

dH(γ(ti), γ(ti−1)). (3.30)

Notice that limn→∞∑n

i=1 |ai,n|12 = LengthC,Eucl(π(γ)), as follows from the

classical properties of the Euclidean length. Hence it is enough to prove that

limn→∞

n∑i=1

|bi,n|12 =

0 if γ is horizontal,+∞ otherwise. (3.31)


To show (3.31) let us rewrite |bi,n| as

|bi,n| =

∣∣∣∣ˆ ti

ti−1

γ3(s)ds− 1

2

[γ1i

ˆ ti

ti−1

γ2(s)ds− γ2i

ˆ ti

ti−1

γ1(s)ds

]∣∣∣∣=

∣∣∣∣ˆ ti

ti−1

(γ3 − 1

2(γ1γ2 − γ2γ1))(s)ds

−1

2

[ˆ ti

ti−1

(γ1i − γ1)γ2ds− (γ2

i − γ2)γ1ds

]∣∣∣∣=

∣∣∣∣ˆ ti

ti−1

ω(γ)(s)ds

−1

2

ˆ ti

ti−1

ˆ ti

s

(γ1(σ)γ2(s)− γ2(σ)γ1(s)

)dσds

∣∣∣∣ . (3.32)

Let us consider the integral in (3.32). Since γ is continuous in [0, 1], it isalso bounded. Let M > 0 be such that |γj(s)| < M for each s ∈ [0, 1] andj = 1, 2. Since [0, 1] is compact, γ is uniformly continuous and for each ε > 0there exists δ > 0 such that |γj(t) − γj(s)| < ε for each |t − s| < δ in [0, 1]and for each j = 1, 2. Let us take n so that 1

n< δ. Hence the second integral

in (3.32) is controlled by

|ˆ ti

ti−1

ˆ ti

s

(γ1(σ)γ2(s)− γ2(σ)γ1(s)

)dσds|

≤ˆ ti

ti−1

ˆ ti

s

|(γ1(σ)− γ1(s))γ2(s)− (γ2(σ)− γ2(s))γ1(s)|dσds

≤ˆ ti

ti−1

ˆ ti

ti−1

2Mε dσds =2Mε

n2.

Now, if γ is horizontal, then ω(γ) = 0 and

limn→∞

n∑i=1

|bi,n|12 ≤ lim

n→∞

n∑i=1

√Mε

n2=√Mε.

Since ε was arbitrary, the claim is proved in case γ horizontal.If γ is not horizontal, we have to prove that limn→∞

∑ni=1 |bi,n|

12 = +∞.

Notice that

√|bi,n| =

√∣∣∣∣ˆ ti

ti−1

ω(γ)(s)ds− ci,n∣∣∣∣ ≥

√∣∣∣∣ˆ ti

ti−1

ω(γ)(s)ds

∣∣∣∣−√|ci,n|

3.3 Geodesic and bubble sets 71

where

ci,n =1

2

ˆ ti

ti−1

ˆ ti

s

(γ1(σ)γ2(s)− γ2(σ)γ1(s)

)dσds

is as in (3.32). Since we have just shown that limn→∞∑n

i=1

√|ci,n| = 0, let

us prove that

limn→∞

n∑i=1

√∣∣∣∣ˆ ti

ti−1

ω(γ)(s)ds

∣∣∣∣ = +∞.

Since γ is not horizontal, there exists t ∈ (0, 1) such that ω(γ)(t) 6= 0, forexample ω(γ)(t) > 0. Then for continuity, there esists a neighborhood of twhere ω(γ) is positive. Let us suppose for simplicity that ω(γ) > 0 on thewhole [0, 1]. Since ω(γ) is continuous, there exists c > 0 such that ω(γ) ≥ c.Then

limn→∞

n∑i=1

√∣∣∣∣ˆ ti

ti−1

ω(γ)ds

∣∣∣∣ ≥ limn→∞

n∑i=1

√c

n= +∞

and the proof is complete.

3.3 Geodesic and bubble sets

In this section we describe the length minimizing curves joining pairs of pointsin the Heisenberg group and define the so-called bubble sets which appear inPansu’s conjecture on the isoperimetric profile of H.

Without loss of generality, and thanks to left-invariance of the CC metric,we may assume that the two points are the origin 0 = (0, 0, 0) and x =(x1, x2, x3). Let γ : [0, 1] → H denote a Legendrian curve joining 0 and x.Let S be the region in the (x1, x2)-plane bounded by π(γ) and by the segmentjoining π(x) to the origin, and let γ be the closed curve obtained by closingπ(γ) with the segment. By Stoke’s theorem we have

x3 =

ˆ 1

0

γ′3(t)dt =1

2

ˆ 1

0

(γ1γ′2 − γ2γ

′1)(t)dt

=1

2

ˆγ

x1dx2 − x2dx1 =

ˆS

dx1 ∧ dx2 = Area(S), (3.33)

where the equality from the first to the second row holds because the integralon the segment joining 0 to π(x) is zero. In view of (3.33), since the horizon-tal length of a Legendrian curve γ depends only on the length of π(γ), wecan rephrase the problem of finding the Legendrian curve from 0 to x withminimal length with the following problem: Find the plane curve from the


origin to (x1, x2) with minimum length, subject to the constraint that theregion S delimited by the curve and the segment joining (0, 0) to (x1, x2) hasfixed area. This is one formulation of Dido’s problem, closely related to theisoperimetric problem, and is solved by choosing the plane curve to be anarc of a circle.

In conclusion, a length minimizing curve between 0 and x is the lift of acircular arc joining the origin in C with (x1, x2), whose convex hull has areax3. The family of such curves emanating from 0 is parametrized by eiφ ∈ S1

and c ∈ R and is given explicitly in the form

γc,φ(s) =

(eiφ

1− e−ics

c,cs− sin(cs)

2c2

); (3.34)

it is length minimizing over any interval of length 2π|c| . These curves are all

the Legendrian curves that have as projection, π(γ), the arc of a circle withcenter in eiφ

cand radius 1

c, which passes through the origin and is clockwise

parametrized if c > 0, anticlockwise otherwise. If c = 0, γc,φ is a straight linethrough 0 in the xy-plane. We call c the curvature of the geodesic arc γc,φ.

If x = (0, 0, x3) lies on the x3-axis, then the projection of the geodesicin H joining 0 to x is a circle of area x3 passing through the origin. Clearlythere are infinitely many such circles, so geodesics are not unique in thiscase. Choosing t = cs, R = 1/c and φ = 0 in (3.34) gives the followingrepresentation:

γ(t) = (R(1− e−it), 1

2R2(t− sin t)), (3.35)

with 0 ≤ t ≤ 2π. The request that the area is equal to x3, gives R2 = x3/π.Rotating such a geodesic around the x3-axis produces a surface of revo-

lution Σ whose profile curve is given parametrically as

t 7→(

2R sin(t

2),

1

2R2(t− sin t)

), (3.36)

that is determined by taking (||π(γφ(t))||, γ3,φ(t)).For each z ∈ C with |z| < 2R there are two distinct points (z, x3,1) and

(z, x3,2) on Σ, determined respectively by

t12

= arcsin

(|z|2R

)=π

2− arccos

(|z|2R

)and

t22

= π − arcsin

(|z|2R

)=π

2+ arccos

(|z|2R

).

3.3 Geodesic and bubble sets 73

So Σ may be seen as the boundary of the open set Ω consisting of all thepoints (z, x3) ∈ H such that

R2

(π

2− arccos

(|z|2R

))− |z|

2

√R2 − |z|

2

4< x3

< R2

(π

2+ arccos

(|z|2R

))+|z|2

√R2 − |z|

2

4. (3.37)

To simplify the notation we dilate the ball by a factor of 2 and translatevertically by −πR2/2. The resulting domains

B(0, R) := (z, x3) ∈ H : |x3| < fR(|z|), (3.38)

fR(r) =1

4

(R2 arccos(

r

R) + r

√R2 − r2

),

are the conjectures extremals for the sub-Riemannian isoperimetric problemin H and are often called bubble sets.

We conclude this section with the following result on the regularity ofthe boundary of the bubble sets. Unlike the classical case, the conjecturedsolution for the sub-Riemannian isoperimetric problem in H is not C∞.

Theorem 3.4. The boundary of the set B(0, R) is C2 but not C3. Moreprecisely, ∂B(0, R) is not C3 in (0, 0,±fR(0)) and is C∞ away from thesepoints.

Remark 3.5. Notice that (0, 0,±fR(0)) are the unique points on ∂B(0, R)such that the tangent space to the surface coincides with the horizontal plane.Points with this property take a special name. We will call them character-istic points (see Definition 4.2) and we will discuss them more deeply in thenext chapters.

Proof. By smooth dilating the bubble set, we can restrict our proof to theunitary bubble set B(0, 1) and the points (0, 0,±π

8).

Let us denote f1(r) by f(r) in what follows. The regularity of ∂B(0, 1)at (0, 0,±π

8) is equivalent to verifying up to what order of derivatives n we

have

limr→0+

f (n)(r) = limr→0−

f (n)(−r). (3.39)


Then let us compute the derivatives:

f ′(r) = − r2

2√

1− r2f ′(−r) = −f ′(r);

f ′′(r) =r3 − 2r

2√

1− r2(1− r2)f ′′(−r) = (f ′(−r))′ = −f ′′(r);

f ′′′(r) = − r2 + 2

2√

1− r2(1− r2)f ′′′(−r) = (f ′′(−r))′ = −f ′′′(r).

Hence, equation (3.39) holds for n = 0, 1, 2, whereas for n = 3 we havedifferent limits. This shows that the boundary is C2 but not C3 near (0, 0, π

8).

The boundary is C∞ away from these points because, after a dilation anda translation, it is obtained by revolution around the x3-axis of the C∞ curvegiven by (3.36), that is a C∞ surface away from (0, 0,±π

8).

3.4 Riemannian approximants to HIn this section we will study the connection between the sub-Riemanniangeometry of the Heisenberg group and its Riemannian approximants. Forthis purpose we will introduce the approximant Riemannian metrics and adefinition of convergence between metric spaces, the Gromov-Hausdorff con-vergence. The main result is that the Heisenberg group equipped with theCC metric may be realized as the Gromov-Hausdorff limit of a sequence ofRiemannian manifolds (R3, gL), as L → ∞. This Riemannian approximantscheme plays a central role in the development of sub-Riemannian subman-ifold geometry of Chapter 4.

3.4.1 The gL metrics

To describe the Riemannian approximants to H, let L > 0 and define a metricgL on R3 so that the left invariant basis X1, X2, X3/

√L of h is orthonormal.

This family of metrics is essentially obtained as an anisotropic blow-up ofthe Riemannian metric g1 defined in section 3.2.3.The left invariant propertyfor the scalar product, requires that

〈Xi(x), Xj(x)〉x = 〈(dLx)−1Xi(x), (dLx)−1Xj(x)〉e = 〈Xi,e, Xj,e〉e,

for each x ∈ R3, where Xi(x) and Xi,e, 〈·, ·〉x and 〈·, ·〉e denote the canonicalvector fields and the scalar products evaluated in x and the neutral element

3.4 Riemannian approximants to H 75

respectively. This leads to the conclusion that the matrix of the scalar prod-uct gL(x) must be equal to

gL(x) = CT ILC,

where

C = (dLx)−1 =

1 0 00 1 0

12x2 −1

2x1 1

is the inverse of the matrix defining left translation (see (3.6)) and

IL =

1 0 00 1 00 0 L

is the matrix of the gL metric in the neutral element. We may represent gLexplicitly in exponential coordinates x = (x1, x2, x3) via the positive definitematrix

gL(x) =

1 + 14x2

2L −14x1x2L

12x2L

−14x1x2L 1 + 1

4x2

1L −12x1L

12x2L −1

2x1L L

. (3.40)

Observe that the Riemannian volume element in (H, gL) is√det gLdx1 ∧ dx2 ∧ dx3 =

√Ldx1 ∧ dx2 ∧ dx3. (3.41)

When calculating with the Riemannian metric gL, we will sometimesuse 〈·, ·〉L to denote the inner product on vectors. In other words, for~a =

∑3i=1 aiXi and ~b =

∑3i=1 biXi in TH,

〈~a,~b〉L = a1b1 + a2b2 + La3b3. (3.42)

As usual, the length of a vector is given as

|~a|L = 〈~a,~a〉1/2L .

The metric 〈·, ·〉L restricted to the horizontal directions gives the sub-Riemannian inner product on H. Moreover, suppose γ : [0, 1] → H is a C1

curve and that γ′(t) = a1(t)X1 + a2(t)X2 + a3(t)X3. Then

LengthL(γ) =

ˆ 1

0

〈γ′(t), γ′(t)〉12Ldt (3.43)

=

ˆ 1

0

(a1(t)2 + a2(t)2 + La3(t)2)12dt, (3.44)

where we have written LengthL = LengthdL for simplicity. Note that γ hasfinite length in the limit as L→∞ if and only if a3 = 0, i.e., γ is a horizontalcurve. We define dL to be the standard path distance associated to gL.


3.4.2 Levi-Civita connection and curvature in theRiemannian approximants

In this section, we compute the sectional, Ricci and scalar curvatures of theHeisenberg group with respect to gL. This is a quite computational section,but we will see that such explicite results are necessary to carry out theproof of the isoperimetric problem. In particular we will use these formulaein the computation of the generalized first variation of the perimeter (seeProposition 5.24).

Let us denote by D the Levi-Civita connection on (H, gL). Given vectorfields U, V,W on a Riemannian manifold (M, g), the Kozul identity definedin (1.8) is

〈∇UV,W 〉L =1

2U〈V,W 〉L + V 〈W,U〉L −W 〈U, V 〉L−〈W, [V, U ]〉L − 〈[V,W ], U〉L − 〈V, [U,W ]〉L (3.45)

where 〈·, ·〉L is the inner product associated to gL. To simplify the notationlet us introduce the functions

αijk = 〈Xi, [Xj, Xk]〉L

where Xi = Xi for i = 1, 2 and X3 = L−1/2X3. Using the Kozul identity andconsidering that X1, X2, X3 is an orthonormal basis with respect to gL, wehave

〈DXiXj, Xk〉L = −1

2(αkji + αijk + αjik).

Since the only non-trivial Lie bracket is [X1, X2] = X3 =√LX3, we have

α312 =√L, α321 = −

√L, and αijk = 0 for all other triples (i, j, k). Then

DX1X2 = 1

2X3 DX2

X1 = −12X3

DX1X3 = DX3

X1 = −√L

2X2 DX2

X3 = DX3X2 =

√L

2X1

DXiXj = 0 for all other (i, j).

(3.46)

Using these results we are now able to compute the curvatures introducedin Subsection 1.2.3. The curvature tensor for a Riemannian manifold (M, g)and Riemannian connection D as defined in Definition 1.17 is

R(U, V )W = DUDVW −DVDUW −D[U,V ]W.

Then the sectional curvatures of the two-planes spanned by the basis vectors


Xi and Xj are (see equation (1.9)):

K12 = 〈R(X1, X2)X1, X2〉L = 〈DX1DX2X1 −DX2DX1X1 −DX3X1, X2〉L

=

⟨DX1

(−1

2X3

)−DX2(0) +

L

2X2, X2

⟩L

=L

4+L

2=

3

4L, (3.47)

K13 = 〈R(X1, X3)X1, X3〉L = 〈DX1DX3X1 −DX3

DX1X1, X3〉L

=

⟨DX1

(−√L

2X2

)−DX3

(0), X3

⟩L

=

⟨−√L

4X3, X3

⟩L

= −1

4L, (3.48)

K23 = 〈R(X2, X3)X2, X3〉L = 〈DX2DX3X2 −DX3

DX2X2, X3〉L

=

⟨DX2

√L

2X1 −DX3

(0), X3

⟩L

=

⟨−√L

4X3, X3

⟩L

= −1

4L. (3.49)

More generally, denote by Rijkl = 〈R(Xi, Xj)Xk, Xl〉 the full Riemanniancurvature tensor, then

Rijkl =

34L if (ijkl) = (1212) or (2121),−3

4L if (ijkl) = (1221) or (2112),

−14L if (ijkl) = (1313), (3131), (2323) or (3232),

14L if (ijkl) = (1331), (3113), (2332) or (3223),

0 otherwise.

Finally, the Ricci curvatures Rici = Ki1 +Ki2 +Ki3 are

Ric1 = Ric2 =1

2L and Ric3 = −1

2L,

and the scalar curvature σ = Ric1 + Ric2 + Ric3 is

σ =1

2L.


3.4.3 Gromov-Hausdorff convergence

A way to value the distance between metric spaces is the Gromov-Hausdorffdistance. In this section we will see its definition and the correspondingnotion of convergence.

First we give a preliminary definition.

Definition 3.6. Let (Z, d) be a metric space. The Hausdorff distance be-tween E1 and E2 is given by

HausZ(E1, E2) = infε > 0|E1 ⊂ (E2)ε, E2 ⊂ (E1)ε

where (Ei)ε = z ∈ Z : dist(z, Ei) < ε is the ε-neighborhood of Ei in (Z, d).

Definition 3.7. Let (X, dX) and (Y, dY ) be metric spaces. The Gromov-Hausdorff distance between X and Y is given by

dGH(X, Y ) = inff,g,Z

HausZ(f(X), g(Y )),

where the infimum is taken over all metric spaces Z and isometric embeddingsf and g of X and Y respectively into Z.

Using this metric, we have a corresponding notion of convergence:

Definition 3.8. A sequence of compact metric spaces (Xn) Gromov-Haus-dorff converges to a compact metric space X if dGH(Xn, X)→ 0 as n→∞.

We would like to apply this notion of convergence to the Riemannianapproximants (R3, gL) and the limit (H, d), but this definition is unnatu-ral for noncompact spaces and limit. Indeed, consider the dilated spheres(Sn, λdSn), where dSn is the geodesic distance on Sn. We would like to statethat they converge to Rn as λ → ∞. However, according to Definition 3.7,dGH((Sn, λdSn),Rn) = +∞ for all λ > 0.

To generalize this notion of convergence to non-compact metric spaces wehave to restrict ourselves to proper pointed metric length spaces. A metricspace (X, d) is called proper if all closed balls in X are compact, while apointed metric space (X, d, x) is a metric space (X, d) equipped with a fixedbasepoint x ∈ X. The generalized definition is the following:

Definition 3.9. A sequence of proper pointed length spaces (Xn, dn, xn)Gromov-Hausdorff converges to a proper pointed length space (X, d, x) ifthe sequence of closed balls BXn(xn, r) Gromov-Hausdorff converges (in thesense of definition 3.8) to BX(x, r), uniformly in r.


The following proposition gives a sufficient condition for the Gromov-Hausdorff convergence, which will guaranty that the Riemannian approxi-mants scheme Gromov-Hausdorff convergies to the sub-Riemannian Heisen-berg group.

Proposition 3.10. Let X be a set equipped with a family of metrics (dt)t≥0

generating a common topology. For K compact in X, let:

ωK(ε) := supx,y∈K,t≥0

dt(x, y)− dt+ε(x, y).

Assume that:

(i) For each t ≥ 0, (X, dt) is a proper length space.

(ii) For fixed x, y ∈ X, the function t 7→ dt(x, y) is non-increasing.

(iii) For each compact set K in X, ωK(ε)→ 0 as ε→ 0.

Then (X, dt) converges, in the sense of pointed Gromov-Hausdorff conver-gence, to (X, d0).

Proof. By hypothesis (ii) and the definition of ωK we easily verify the fol-lowing additional facts for each compact set K:

(iv) the map ε 7→ ωK(ε) is non-decreasing in ε: indeed, let ε1 < ε2. Byhypothesis (ii) dt+ε1(x, y) ≥ dt+ε2(x, y) for each t ≥ 0, x, y ∈ K, then

ωK(ε2) = supx,y∈K,t≥0

dt(x, y)− dt+ε2(x, y)

≥ supx,y∈K,t≥0

dt(x, y)− dt+ε1(x, y) = ωK(ε1);

(v) ωK is sublinear, i.e. ωK(a+ b) ≤ ωK(a) + ωK(b) for all a, b ≥ 0;

(vi) if we denote by Bt(x0, R) the closed metric ball with center x0 andradius R in the metric (X, dt), t ≥ 0, then

B0(x0, R) ⊂ Bt(x0, R) ⊂ B0(x0, R + ωK(t))

for any x0, R > 0 and t ≥ 0 so that Bt(x0, R) ⊂ K: the first inclusionholds because d0(x0, x) ≥ dt(x0, x) for each t ≥ 0, x ∈ K. The secondinclusion follows because d0(x0, x) = d0(x0, x)− dt(x0, x) + dt(x0, x) ≤dt(x0, x) + ωK(t), then dt(x0, x) ≥ d0(x0, x)− ωK(t).

From (vi) and (i) we further conclude


(vii) For any x0, R > 0 and t ≥ 0, to each y ∈ Bt(x0, R) there correspondsa point x ∈ B0(x0, R) with dt(x, y) ≤ ωK(t): in fact, if y ∈ B0(x0, R)it is trivial. Let y ∈ Bt(x0, R) \ B0(x0, R), then y ∈ B0(x0, R + ωK(t))and for each ε > 0, by (i), there exists γ : [0, 1] → (X, d0) such thatγ(0) = x0, γ(1) = y and Length0(γ) ≤ d0(x0, y) + ε ≤ R + ωK(t) + ε.Let t0 = maxt : γ(t) ∈ B0(x0, R) (it is indeed a maximum becauseB0(x0, R) is closed by (i)) and let x := γ(t0) ∈ B0(x0, R). We claimthat Length0(γ|[0,t0]) ≥ R: if it weren’t, there would exist t > t0 suchthat Length0(γ|[0,t]) ≤ R and γ(t) ∈ B0(x0, R), which contradicts themaximality of t0. Hence

R + ωK(t) + ε ≥ Length0(γ)

≥ Length0(γ|[0,t0]) + Length0(γ|[t0,1])

≥ R + d0(x, y).

Since ε was arbitrary dt(x, y) ≤ d0(x, y) ≤ ωK(t) and statement (vii)follows.

We now establish the desired conclusion. Fix a basepoint x0 ∈ X. Itsufficies to prove that the compact metric balls Bt(x0, R) converge (in thesense of Definition 3.8) to B0(x0, R), for each R > 0. We restrict our attentionto t ∈ [0, 1], let K = B1(x0, R), and equip the space

Z = K × [0, 1]

with the metric

dZ((x, t), (x′, t′)) = dmaxt,t′(x, x′) + ωK(|t− t′|) + |t− t′|.

We claim that dZ is a metric. If dZ((x, t), (x′, t′)) = 0 then t = t′ anddt(x, x

′) = 0, hence also x = x′ (since dt is a metric).Next we verify the triangle inequality. Let (x, t), (x′, t′), (x′′, t′′) ∈ Z. If

maxt, t′, t′′ 6= t′′, then

dZ((x, t), (x′, t′)) = dmaxt,t′,t′′(x, x′) + ωK(|t− t′|) + |t− t′|

≤ dmaxt,t′′(x, x′′) + dmaxt′′,t′(x

′′, x′) + ωK(|t− t′|)+ ωK(|t′′ − t′|) + |t− t′′|+ |t′′ − t′|

= dZ((x, t), (x′′, t′′)) + dZ((x′′, t′′), (x′, t′))

where we used (ii) and (v) in the middle step.If maxt, t′, t′′ = t′′, then

dZ((x, t), (x′, t′)) = dmaxt,t′(x, x′) + ωK(|t− t′|) + |t− t′|

≤ dt′′(x, x′) + ωK(t′′ −maxt, t′) + ωK(|t− t′|) + |t− t′|


by the definition of ωK , and by (iv) follows

≤ dt′′(x, x′′) + dt′′(x

′′, x′) + ωK(t′′ − t) + ωK(t′′ − t′) + |t− t′′|+ |t′′ − t′|= dZ((x, t), (x′′, t′′)) + dZ((x′′, t′′), (x′, t′)).

It is clear that the map x 7→ (x, t) is an isometric embedding of Bt(x0, R)into Z. To complete the proof, it sufficies to verify that the Hausdorff distance(in Z) between Bt(x0, R)× t and B0(x0, R)× 0 tends to zero as t→ 0.In fact, we claim that

HausZ(Bt(x0, R)× t, B0(x0, R)× 0) ≤ 2ωK(t) + t; (3.50)

the result then follows from assumption (iii).To see that (3.50) is true: if x ∈ B0(x0, R), then x ∈ Bt(x0, R) and

dZ((x, t), (x, 0)) = ωK(t) + t ≤ 2ωK(t) + t,

while if y ∈ Bt(x0, R) we choose x as in (vii) and conclude

dZ((y, t), (x, 0)) = dt(x, y) + ωK(t) + t ≤ 2ωK(t) + t.

We now state the main result of this section.

Corollary 3.11. The sequence of metric spaces (R3, dL) converges to (H, d)in the pointed Gromov-Hausdorff sense as L→∞.

The proof follows from Proposition 3.10 where we choose X = R3, d0

equal to the Carnot-Carathéodory metric d on X for the usual Heisenbergstructure, and dt the distance function associated to the Riemannian metricgL, where L = 1/t. The third hypothesis remains to be proved and it is thecontent of the next section.

3.4.4 Carnot-Carathéodory geodesics andGromov-Hausdorff convergence

The CC geodesics in the Heisenberg group can be obtained also throughthe approximation scheme, using the geodesics in the Riemannian manifolds(R3, gL).

Let γ : [0, 1]→ H be a Lipschitz curve and let ω = dx3− 12(x1dx2−x2dx1)

denote the contact form defined in (3.8). The gL-length of γ is given by

LengthL(γ) =

ˆ 1

0

√|γ′1(t)|2 + |γ′2(t)|2 + L

∣∣∣ω(γ′(t))|γ(t)

∣∣∣2dt. (3.51)


In our computation, however, we will consider the “penalized” energy of γ:

EL(γ) =

ˆ 1

0

|γ′1(t)|2 + |γ′2(t)|2 + L∣∣∣ω(γ′(t))|γ(t)

∣∣∣2 dt. (3.52)

Since minimizing curves of (3.52) are also minimizing curve of (3.51) andviceversa, we will find the solutions of the Euler-Lagrange equations of theenergy rather than of the length.

Let us substitute for γ a one-parameter family of compactly supportedperturbations γλ, where

γλi (s) = γi(s) + λfi(s),

with fi ∈ C∞0 ([0, 1]) for i = 1, 2, 3, and compute the derivative of EL(γλ)with respect to λ at λ = 0 to obtain the Euler-Lagrange equations for thecritical points of the penalized energy. The equations are given by

d

dt

[d

dγ′ig(t, γ, γ′)

]=

d

dγig(t, γ, γ′), (3.53)

with i = 1, 2, 3, where g(t, γ, γ′) = |γ′1(t)|2 + |γ′2(t)|2 +L∣∣∣ω(γ′(t))|γ(t)

∣∣∣2. Com-puting explicitely equations (3.53) we obtain the following three equationsfor the critical points of the penalized energy:

γ′′1 = −Lωγ′2, γ′′2 = Lωγ′1, (ω(γ′)|γ)′ = 0, (3.54)

where to obtain the first two equations, we have to use the third one. Theright-hand side of the first two equations contains L and hence could po-tentially blow up as L → ∞. However, note that ω(γ′)|γ equals a constantdepending only on the initial data γ(0) and γ′(0) by the third equation. Ifwe choose γ(0) = 0 and γ′(0) = (h1, h2, aL/L), with h1, h2, aL ∈ R, thenω(γ′(t))|γ(t) = aL/L for all t and the (3.54) yield

(γ1 + iγ2)′′ = iaL(γ1 + iγ2)′, (3.55)

with (γ1, γ2)(0) = (0, 0) and (γ′1, γ′2)(0) = (h1, h2). Comparing (3.34) with

(3.55) indicates that solutions to (3.55) corresponding to h21 + h2

2 6= 0 (whichare arcs of circles) are projections of length minimizing arcs emanating fromthe origin with initial velocity h1X1 + h2X2, parametrized by aL ∈ R. Inparticular, gL-geodesic arcs with horizontal initial velocity (aL = 0) are alsolength minimizing arcs in (H, d), the horizontal segments. In general, ifaL 6= 0, the solutions of (3.54) may be not horizontal. However, choosing


a sequence aL → a∞ ∈ R as L → ∞, one can easily prove, using energyestimates for the ODE (3.55), that the corresponding solutions γL of (3.54)converge uniformly to a length minimizing arc in (H, d) as described in (3.34).

As a result of the above observations, we can state the following corollary,which completes the proof of Corollary 3.11, showing that we can indeedapply Proposition 3.10.

Proposition 3.12. Given x ∈ H, any length minimizing horizontal curve γjoining x to the origin 0 ∈ H is the uniform limit as L→∞ of geodesic arcsfrom 0 to x in the Riemannian spaces (R3, gL). Moreover, the convergence isuniform both in x and in the parameter L, in the sense that

limε→0

sup0<L≤∞,x∈K

dL(0, x)− dL−ε(0, x) = 0

for any compact K ⊂ R3.

Chapter 4

Horizontal geometry ofhypersurfaces in H

We now focus our attention to the sub-Riemannian geometry of hypersurfacesin H, introducing the main geometric notions, that are necessary to study theisoperimetric problem. We introduce them as a limit of the correspondentRiemannnian objects.

4.1 The second fundamental form in (R3, gL)

Let us consider a C2 regular surface S in the Heisenberg group, given as thelocus of the zeros of a function u ∈ C2(R3) with nonvanishing gradient alongS:

S =x ∈ R3 : u(x) = 0

. (4.1)

Let us introduce some additional useful notation. Let X1, X2 be as in(3.7) and X3 = X3√

L. Set p = X1u, q = X2u and r = X3u. To simplify the

upcoming formulas, we set

l =√p2 + q2, lL =

√p2 + q2 + r2,

and p = p/l, q = q/l, pL = p/lL, qL = q/lL, rL = r/lL. Observe that theRiemannian normal to S is

νL = pLX1 + qLX2 + rLX3,

while the tangent space to S is spanned by the orthonormal basis

e1 = qX1 − pX2, and e2 = rLpX1 + rLqX2 −l

lLX3. (4.2)

85

86 Horizontal geometry of hypersurfaces in H

As seen in Section 1.2.4, in particular in equation (1.16), the secondfundamental form of S with respect to this orthonormal frame is

IIL =

(〈−De1νL, e1〉L 〈−De1νL, e2〉L〈−De2νL, e1〉L 〈−De2νL, e2〉L

),

where D denotes the Levi-Civita connection associated to gL. Since e1, e2form an orthonormal system, the matrix G of the metric with respect to thisbasis is the identity matrix. Then the relation between the matricies of theshape operator and of the second fundamental form (equation (1.15)) impliesthat the two matricies are equal and we can compute the Gaussian and meancrvature of S with respect to the second fundamental form.

We explicite in the next theorem the geometric quantities of S ⊂ (R3, gL),but, since the proof implies only long computations, we refer to [6] (Theorem4.3) for the whole calculus.

Theorem 4.1. Let S ⊂ (R3, gL) be a regular C2 surface defined as in (4.1).Relative to the orthonormal frame e1, e2 defined in (4.2), the second fun-damental form of S is

IIL =

(− llL

(X1p+X2q)lLl〈e1,∇0rL〉L +

√L

2lLl〈e1,∇0rL〉L +

√L

2l2

l2L〈e2,∇0( r

l)〉L − X3rL

). (4.3)

In particular, the main curvature HL = trIIL of S is

− l

lL(X1p+X2q) +

l2

l2L〈e2,∇0(

r

l)〉L − X3rL, (4.4)

while the Gauss curvature KL = det IIL is

l

lL(X1p+X2q)

(− l

2

l2L〈e2,∇0(

r

l)〉L + X3rL

)−

(lLl〈e1,∇0rL〉L +

√L

2

)2

.

(4.5)

4.2 Horizontal geometry of hypersurfacesWe want to study the behaviour of these classical differential geometric ob-jects as L→∞. We will see that the horizontal components have well-definedlimits, which are natural candidates for sub-Riemannian analogs. On theother hand, the vertical components are unbounded.

Before initiating this analysis we introduce a fundamental notion in thestudy of sub-Riemannian submanifold geometry.

4.2 Horizontal geometry of hypersurfaces 87

Definition 4.2. Let S ⊂ H be a C1 surface defined as in (4.1). The charac-teristic set of S is the closed set

Σ(S) = x ∈ S : ∇0u(x) = 0. (4.6)

In other words, Σ(S) is the set of points x ∈ S where the tangent spacecoincides with the horizontal plane H(x).

Note that r, rL and X3 all converge to zero as L→∞ at rates of the orderof L−1/2. On the other hand, qL → q, pL → p, lL → l, and e2 → 0. Hencethe Riemannian normal νL converges to the so-called horizontal normal

νH =X1uX1 +X2uX2

|∇0u|= pX1 + qX2 ∈ L∞(S \ Σ(S)) (4.7)

in the complement of the characteristic set. Note that νH is simply the pro-jection of νL onto the horizontal subbundle rescaled to have unitary norm.The vectors νH and e1 form an orthonormal frame of the horizontal subbun-dle.

A direct computation shows that the Gauss curvature KL in (4.5) divergesas L → ∞ (similarly to the behaviour of the sectional, Ricci and scalarcurvatures discussed in Section 3.4.2). Indeed

limL→∞

KLL

= −1

4.

This reflects the fact that νL → pX1 + qX2 = νH as L→∞, i.e. the tangentplane to S tends, as L → ∞, towards a plane, that is orthogonal to thehorizontal plane. As (3.48) and (3.49) show, the curvature of such planescomputed with respect to gL equals −L/4.

Surprisingly, while the Gauss curvature does not have a limit as L→∞,the mean curvature presents a rather different behaviour. The followinglemma is an immediate consequence of (4.4).

Lemma 4.3. Let S be a C2 regular surface defined as in (4.1). Then

limL→∞

trIIL = limL→∞

HL = −(X1p+X2q) (4.8)

at noncharacteristic points.

This behaviour of the mean curvature suggests us the definition of itssub-Riemannian analogous.

Definition 4.4. Let S ⊂ H be a C2 regular surface, given as a level set of afunction u. The horizontal mean curvature of S at a noncharacteristic pointis

H0 = −X1p−X2q.


We can rewrite the horizontal mean curvature H0 in several ways. First,its explicite expression is:

H0 = −2∑i=1

Xi

(Xiu

|∇0u|

). (4.9)

Next, a direct computation shows that

H0 =X1uX1(|∇0u|) +X2uX2(|∇0u|)

|∇0u|2− (X2

1u+X22u)|∇0u|

|∇0u|2

with Xi(|∇0u|) = (|∇0u|)−1(X1uXiX1u+X2uXiX2u). Then, if we set Lu =X2

1u+X22u for the Heisenberg Laplacian of u and

L∞u :=2∑

i,j=1

XiuXjuXiXju (4.10)

for the Heisenberg infinite Laplacian of u : H → R, the horizontal meancurvature can also be expressed via the identity

H0|∇0u| =L∞u|∇0u|2

− Lu. (4.11)

Finally, if u(x) = x3 − f(|z|) and we let r = |z|, then direct computationsshow:

H0 =14r2f ′′ + (f ′)3

r

((f ′)2 + 14r2)

32

. (4.12)

The horizontal mean curvature turn out to have a fundamental role inthe isoperimetric problem, indeed we will see in section 5.3 that the firstvariation of the perimeter functional among all horizontal perturbations isdetermined by the horizontal mean curvature. We emphasize this relationwith the following definition.

Definition 4.5. A C2 regular surface S ⊂ H is called a horizontal minimalsurface if it has vanishing horizontal mean curvature along its noncharacter-istic locus.

We conclude by giving a kind of extension of the horizontal mean curva-ture to the characteristic locus. Rearranging equations (4.4) and (4.9) usingthe relation between the coefficients rL, l and lL, we can express the meancurvature HL as

HL = −divgL(νL) = −3∑i=1

Xi

(Xiu

|∇Lu|

). (4.13)

4.2 Horizontal geometry of hypersurfaces 89

Direct computations as those done to obtain (4.11) show that

HL|∇Lu| = −2∑

i,j=1

(δij −

XiuXju

|∇Lu|2

)XiXju.

Since Xiu = 0 for i = 1, 2 at the characteristic locus, both sides of thisequation converge at every point, characteristic or not, as L→∞, giving

limL→∞

HL|∇Lu| =H0|∇0u| = L∞u

|∇0u|2 − Lu on S \ Σ(S),

−Lu on Σ(S),(4.14)

where Lu = X21u + X2

2u and L∞ as in (4.10) denote the sub-Laplacian andthe infinite Laplacian in H, respectively.

Let πH denote the orthogonal projection of Lie algebra vectors onto thehorizontal bundle. In view of equation (4.14), we can extend the functionH0|πH(ν1)| from S \ Σ(S) to all of S as a continuous function.

4.2.1 The Legendrian foliation

Let S = x ∈ H : u(x) = 0 be a C2 regular surface with characteristic setΣ(S).

For every point x ∈ S \ Σ(S) consider the intersection of the horizontalplane H(x) with the tangent space TxS. Since x is noncharacteristic, thisinteresection is a line HxS, that is horizontal and tangent; we denote by HSthe distribution of these lines and call it the horizontal line subbundle.

The unit vectors e1 defined in (4.2) form a tangent vector field inHS, thatat each point is orthonormal to νH with respect to the g1-metric. For eachx ∈ S \Σ(S) define the curve γx = (γ1

x, γ2x, γ

3x) determined by the intersection

S ∩H(x), with γx(0) = x and γ′x(0) = e1(x). We call the family of all suchcurves the Legendrian foliation of S.

As we are going to see, the Legendrian foliation describes well the hori-zontal mean curvature of the relative surface. Let us investigate more deeplyone of its curve.

The curve γx has tangent vector γx = (γ1x, γ

2x, γ

3x) which in general is not in

the horizontal distribution (notice indeed that γx is not in general horizontalaway from 0). The components of γx with respect to the horizontal baseX1, X2 are (γ1

x, γ2x) = π(γx), as computed in (3.19), where π denotes the

projection of a vector v = v1∂x1 + v2∂x2 + v3∂x3 on the first two components.Let us parametrize γx so that (γ1

x, γ2x) is a unitary vector. Then (γ1, γ2)

defines a planar curve on C parametrized by arc length.


The horizontal normal unitary vector to S at x was defined as

νH =X1uX1 +X2uX2

|∇0u|=X1u∂x1 +X2u∂x2 + 1

2(x1X2u− x2X1u)∂x3

|∇0u|,

then the projection ~n(t) := π(νH(γ(t))) ∈ C is a unitary vector normal tothe planar curve (γ1

x, γ2x) in 0.

The vector π(e1) = (γ1x, γ

2x)(0) is tangent to the curve (γ1

x, γ2x) in 0 and is

orthogonal to ~n. More precisely, with respect to the defining function u wehave

π(e1) = (q,−p) γ =(X2u,−X1u)

|∇0u| γ = −i~n.

Recall from equation (1.3) that the curvature vector of the planar curve(γ1, γ2) is defined as

~k = (γ′′1 , γ′′2 ) = ki(γ′1, γ

′2) = k~n.

We are now going to see that it is deep related with the horizontal meancurvature of the surface. Indeed, let us compute it with respect to u:

~k = (γ′′1 , γ′′2 ) =

((X2u

|∇0u|,−X1u

|∇0u|

) γ)′

=

((dγ1

dtX1 +

dγ2

dtX2

)(X2u

|∇0u|

),

(dγ1

dtX1 +

dγ2

dtX2

)(− X1u

|∇0u|

)) γ

This is equal to

1

|∇0u|

((X2uX1 −X1uX2)

(X2u

|∇0u|

), (X2uX1 −X1uX2)

(− X1u

|∇0u|

)) γ

and by a direct computation we obtain

−(X2u)2X1X1u+X1uX2u(X1X2u+X2X1u)− (X1u)2X2X2u

|∇0u|4(X1u,X2u)γ.

So

~k =

(L∞u|∇0u|2

− Lu)

(X1u,X2u)

|∇0u|2 γ

= H0π(νH) γ = H0~n.

Since ~k = k~n we have proved the following proposition.

Proposition 4.6. Let S be a C2 regular surface in H, and let γ = (γ1, γ2, γ3)be a curve in the Legendrian foliation of S \ Σ(S). Then the curvaturek of γ at (γ1(t), γ2(t)) equals the horizontal mean curvature H0 of S at(γ1(t), γ2(t), γ3(t)).

Chapter 5

Sobolev and BV Spaces and firstvariation of the perimeter in H

In this chapter we introduce the notions of Sobolev spaces, BV functions andperimeter of a set with respect to the sub-Riemannian structure of H. Theseare crucial notions in the sub-Riemannian geometric measure theory. At theend, we will conclude by giving an explicite formula for the first variation ofthe perimeter.

5.1 Sobolev spaces, perimeter measure and to-tal variation

We begin by introducing the sub-Riemannian analog of the classical first-order Sobolev spaces. Let Ω ⊂ H be an open set and p ≥ 1. We definethe Sobolev space S1,p(Ω) as the set of functions f ∈ Lp(Ω) such that ∇0fexists in the sense of distributions with |∇0f | ∈ Lp(Ω). We endow it withthe following norm

||f ||S1,p(Ω) = ||f ||Lp(Ω) + ||∇0f ||Lp(Ω),

where

||f ||Lp(Ω) =

(´Ω|f |pdx

) 1p <∞ if p <∞

infM : |f(x)| ≤M <∞ a.e. x ∈ Ω if p =∞

and if ∇0f = (y1, y2), ||∇0f ||Lp(Ω) = ||√y2

1 + y22||Lp(Ω. The underlying mea-

sure in use here to compute these norms is the Haar measure in H. As seenin Sections 3.1 and 3.2.1, it agrees with the exponential of the Lebesgue mea-sure on h or equivalently with the Hausdorff 4-measure associated with theCarnot-Carathéodory metric.

91

92 Sobolev and BV Spaces and first variation of the perimeter in H

Using group convolution

f ∗ g(x) =

ˆHf(xy−1)g(y)dy

and following the out-line of the Euclidean argument one can easily verifythat S1,p(H) is the closure of C∞0 (H) in the norm || · ||S1,p .

We may also define the local Sobolev space S1,ploc (Ω) by replacing Lp with

Lploc in the above definition.Let us now introduce the sub-Riemannian analogs of the classical notions

of variation of a function and perimeter of a set.Let Ω ⊂ H. We will denote by F(Ω) the class of R2-valued functions

φ = (φ1, φ2) : Ω→ R2 such that φ ∈ C10(Ω,R2) and |φ| ≤ 1.

Definition 5.1. Let Ω ⊂ H be open and f ∈ L1loc(Ω). We define the variation

of f in Ω as

VarH(f,Ω) = supφ∈F(Ω)

ˆΩ

f(x)(X1φ1 +X2φ2)(x)dx.

Notice that if f is C1, we can integrate by parts the above formula, sothat


ˆΩ

(X1f(x))φ1(x) + (X2f(x))φ2(x)dx =

ˆΩ

|∇0f |.

Remark 5.2. Let us denote by A the 2× 3 smooth matrix whose rows arethe coefficients of the vector fiels Xj =

∑3i=1 aji∂xi :

A =

(1 0 −1

2x2

0 1 12x1

). (5.1)

Then

ATφ =

φ1

φ212(x1φ2 − x2φ1)

,

so that div(ATφ) = ∂x1φ1 + ∂x2φ2 − 12x2∂x3φ1 + 1

2x1∂x3φ2 = X1φ1 +X2φ2 for

all φ ∈ F(Ω). Then the above definition is equivalent to


ˆΩ

f(x)(X1φ1 +X2φ2)(x)dx

= supφ∈F(Ω)

ˆΩ

f(x)div(ATφ)dx.

5.1 Sobolev spaces, perimeter measure and total variation 93

Definition 5.3. The space BV (Ω) of functions with bounded variation in Ωis the space of all functions f ∈ L1(Ω) such that

||f ||BV (Ω) := ||f ||L1(Ω) + VarH(f,Ω) <∞.

Clearly if f ∈ S1,1(Ω), then f ∈ L1(Ω) and there exists ∇0f in the senseof distributions, such that

ˆΩ

f(x)(X1φ1 +X2φ2)(x)dx =

ˆΩ

〈∇0f, φ〉 (x)dx,

which is bounded because |∇0f | ∈ L1(Ω). Therefore S1,1(Ω) ⊂ BV (Ω).In the next chapters we will need the following useful approximation

results.

Lemma 5.4. Let Ω ⊂ H be an open and bounded set and let p ≥ 1. For anyu ∈ S1,p(Ω) there exists a sequence ukk∈N ⊂ C∞(Ω) such that uk → u inL1(Ω) as k →∞ and

limk→∞

ˆΩ

|∇0uk −∇0u|p = 0.

Lemma 5.5. Let Ω ⊂ H be an open set. For any u ∈ BV (Ω) there exists asequence ukk∈N ⊂ C∞(Ω) such that uk → u in L1(Ω) as k →∞ and

limk→∞

VarH(uk,Ω) = VarH(u,Ω).

Definition 5.6. Let E ⊂ H be a measurable set and Ω ⊂ H be an open set.The horizontal perimeter of E in Ω is given by

PH(E,Ω) = VarH(χE,Ω),

where χE denotes the characteristic function of E. A set with finite perimeteris called Cacciopoli set. For Ω = H we let PH(E,Ω) = PH(E).

Next lemma is a consequence of the properties of the vector fields Xi.

Lemma 5.7. For any y ∈ H, s > 0, Ω ⊂ H open and E ⊂ H Cacciopoli,

PH(δsE, δsΩ) = s3PH(E,Ω) and PH(Ly(E), Ly(Ω)) = PH(E,Ω),

where Ly(x) = yx denotes the operation of left translation by y. In otherwords, the perimeter is homogeneous under dilations and invariant underleft translation.


Proof. To prove the homogeneity property, recall that δs(x) = δs(x1, x2, x3) =(sx1, sx2, s

2x3). Then dδsX1 = sX1, dδsX2 = sX2 and |J(dδs)| = s4. There-fore, with the change of variable y = δs(x), we obtain

PH(δsE, δsΩ) = supφ∈F(δsΩ)

ˆδs(E∩Ω)

(X1φ1 +X2φ2)(y)dy

= supφ∈F(δsΩ)

s3

Ê∩Ω

s(X1φ1 +X2φ2)(δsx)dx

= supφ∈F(Ω)

s3

Ê∩Ω

(X1φ1 +X2φ2)(x)dx

= s3PH(E,Ω).

For the other equation, since the vector fields Xi are left invariant, thendLyXi = Xi, hence, using the change of variable z = Lyx, we compute

PH(Ly(E), Ly(Ω)) = supφ∈F(Ly(Ω))

ˆLy(E∩Ω)

(X1φ1 +X2φ2)(z)dz

= supφ∈F(Ly(Ω))

Ê∩Ω

(dLyX1φ1 + dLyX2φ2)(x)dx

= supφ∈F(Ω)

Ê∩Ω

(X1φ1 +X2φ2)(x)dx = PH(E,Ω).

The perimeter of smooth sets has a very explicit integral representationin terms of the underlying Euclidean geometry.

Proposition 5.8. Let E be a C1 set, dσ the surface measure on ∂E, and ~nthe outer unit normal. Then

PH(E,Ω) =

ˆ∂E∩Ω

(〈X1, ~n〉2 + 〈X2, ~n〉2

)1/2dσ(x).

Proof. In view of Definition 5.6 and Remark 5.2 and by an application of thedivergence theorem 1.29, we have

PH(E,Ω) = supφ∈F(Ω)

ˆΩ∩E

div(ATφ)dx = supφ∈F(Ω)

ˆΩ∩∂E〈ATφ,~n〉dσ

= supφ∈F(Ω)

ˆΩ∩∂E〈φ,A~n〉dσ =

ˆΩ∩∂E

|A~n|dσ.

5.1 Sobolev spaces, perimeter measure and total variation 95

Since A~n = (〈X1, ~n〉, 〈X2, ~n〉)T we conclude that

PH(E,Ω) =

ˆΩ∩∂E

|A~n|dσ =

ˆΩ∩∂E

(2∑i=1

〈Xi, ~n〉2)1/2

dσ.

Corollary 5.9. If S = u = 0 is a C1 hypersurface in H which bounds anopen set E = u < 0, then

PH(E,Ω) =

ˆS∩Ω

dµ (5.2)

for every domain Ω ⊂ H, where

dµ =|∇0u||∇u|

dσ = |πH(ν1)|dσ = |A~n|dσ.

Proof. From the previous Lemma, dµ = |A~n|dσ. As computed in Section 4.2the Riemannian unitary normal to S is

~n =X1uX1 +X2uX2 +X3uX3

|X1uX1 +X2uX2 +X3uX3|= ν1,

thus |πH(ν1)| = |X1uX1 +X2uX2|/|X1uX1 +X2uX2 +X3uX3| = |A~n|, whichis equal to |πH(ν1)| = |∇0u|/|∇u|.

Note that, since the value of |∇0u|/|∇u| does not change if we choose adifferent parametrization of the surface, S = v = 0, then the perimeterwill not change under a reparametrization of S.

5.1.1 Riemannian perimeter approximation

We may also obtain the representation formula (5.2) via the Riemannianapproximation scheme introduced in Subsection 3.4.1, with the Heisenbergperimeter being the limit of the Riemannian perimeters, provided we rescalethem.

Let us consider a C1 parametrized surface S = f(D) in R3, where f =(f1, f2, f3) : D ⊂ R2 → R3. Recall from elementary calculus that if g : S → Ris a continuous function then

ˆS

gdσ =

ˆD

g f(u, v)| ~N(u, v)|dudv (5.3)


where ~N(u, v) = fu × fv is the Euclidean normal vector to S determined bythe parameterization. Consider the sequence of Riemannian approximatingspaces (R3, gL) introduced in Section 3.4.1. Recall that the vector fieldsX1, X2, and X3 form an orthonormal frame with respect to the metrics gL.Let us call this frame F = X1, X2, X3. In these coordinates the standardderivations in R3 are expressed by

∂x1 = X1 +

√L

2x2X3 ∂x2 = X2 −

√L

2x1X3 ∂x3 =

√LX3

so that the basis of the tangent bundle of S with respect to the frame F is

[∂uf ]F =

(∂uf1, ∂uf2,

√L

[∂uf3 −

(∂uf2x1 − ∂uf1x2

2

]),

[∂vf ]F =

(∂vf1, ∂vf2,

√L

[∂vf3 −

(∂vf2x1 − ∂vf1x2

2

]).

Let νL denote the Riemannian normal to S in (R3, gL), given by [νL]F =[∂uf ]F × [∂vf ]F , and let us call with CT

L the 3× 3 matrix whose rows are thecoefficients of the vector fields X1, X2, and X3:

CTL =

1 0 −12x2

0 1 12x1

0 0 L−1/2

.

Simple computations show that νL =√LCT

L~N . On the other hand, recall

that for any vector w1, w2 ∈ R3 hold

||w1 ∧ w2||2 = det (Wij) ,

where the entries of the matrix W are the scalar products Wij = wi · wj. Inparticular, we can express the norm of νL as

||νL||2L = detG,

where G = (gij) is the 2×2 matrix of the first fundamental form with entriesgij = 〈∂if, ∂jf〉L. Thus

√detG = ||νL||L =

√L|CT

L~N | and the change of

variable formula (5.3) on (R3, gL) yieldsˆS

dσL =

ˆD

||νL||Ldudv =

ˆD

√L|CT

L~N |dudv. (5.4)

Here we have denoted by dσL the Riemannian surface area element on Sinduced by the metric gL on H.

5.2 Embedding theorems for the Sobolev and BV spaces 97

Note that if we send L to∞, the matrix CTL becomes the matrix A defined

in (5.1) with a row of zeros added, thus

limL→∞

1√L

ˆS

dσL = limL→∞

ˆD

|CTL~N |dudv

=

ˆD

|A ~N |dudv =

ˆS

|A ~N || ~N |

dσ, (5.5)

where for the last equation we have used the change of variable formula (5.3).In particular, if S = u = 0 is given as in Corollary 5.9 as a level set,

we have |A ~N | = |∇0u| and | ~N | = |∇u|, thus

limL→∞

1√L

ˆS

dσL =

ˆS

dµ =

ˆS

|∇0u||∇u|

dσ. (5.6)

We conclude that the perimeter measure in H can be obtained as the limitas L→∞ of the Riemannian surface measures in S, computed with respectto gL and rescaled by the factor 1/

√L.

5.2 Embedding theorems for the Sobolev andBV spaces

As seen in Chapter 2, the Sobolev inequalities are strong related to theisoperimetric one. For this reason, we want to study here an analog relationfor H.

The Sobolev conjugate of a number p ≥ 1 is no more determined by thetopological dimension of the space, but it turns out that the right dimensionis the homogeneous one. Then the general Sobolev embedding theorem in Htakes the following form.

Theorem 5.10 (Sobolev embedding theorem in the Heisenberg group). Letp ≥ 1. Then

S1,p(H) → L4p4−p (H) for 1 ≤ p < 4

andS1,p(H) → C0,1− 4

p (H) for p > 4.

More precisely, there exist constants Cp(H) <∞ for each p 6= 4 so that

||f ||4p/(4−p) ≤ Cp(H)||∇0f ||p (5.7)


for all f ∈ S1,p(H) if 1 ≤ p < 4, while if p > 4 and f ∈ S1,p(H), then thereexists a representative f of f satisfying the Hölder condition

|f(x)− f(y)| ≤ Cp(H)d(x, y)1−4/p||∇0f ||p

for all x, y ∈ H.As seen in Chapter 2, the case p = 1 is the most interesting from the

point of view of the isoperimetric inequality. Consequently, we will restrictour attention to this case.

5.2.1 The geometric case (Sobolev-Gagliardo-Nirenberginequality)

We prove Theorem 5.10 in the geometric case p = 1. To obtain it we willfirst prove a weak-type estimate.

Let B(0, r) ⊂ H be a metric ball1, and let q > 1. The weak Lq spaceLq,∗(B(0, r)) consists of all measurable functions f : B(0, r) → R for whichthe quantity

||f ||qLq,∗(B(0,r)) = supλ>0

λq|x ∈ B(0, r)| |f(x)| > λ|

is finite. (We have denoted by |A| the Haar measure of A ⊂ H).Recall the Hardy-Littlewood maximal operator defined in Section 1.3.5

with respect to Carnot-Carathéodory metric balls is

Mf(x) = supr>0

1

|B(x, r)|

ˆB(x,r)

|f(y)|dy,

where f ∈ L1(H). By equation (3.21), |B(x, 2r)| = 24|B(x, r)|. This provesthat the Carnot-Carathéodory metric d is doubling and we can apply a weak-type (1, 1) estimate (Theorem 1.49): there exists a constant C > 0 such that

|x ∈ H : |Mf(x)| > λ| ≤ C

λ||f ||L1(H) (5.8)

for all f ∈ L1(H) and λ > 0.The weak-type Sobolev-Gagliardo-Nirenberg inequality reads as follows.

1By homogeneity we could simply assume r = 1 and then rescale the resulting estimates.Similarly, by translation invariance the same estimates hold for balls centered at any pointin H.


Proposition 5.11. There exists a constant C > 0 such that

|x ∈ B(0, r) : |f(x)| > λ| ≤ Cλ−43 ||∇0f ||4/3L1(B(0,r))

for all Lipschitz functions f compactly supported in B(0, r) and for all λ > 0.

Before giving the proof of this inequality, let us present a singular integralrepresentation formula for an f ∈ C∞0 (H). Such formula derives from thesub-Riemannian Green’s formula, that we are going to introduce.

Let D be a bounded C1 domain in R3 equipped with the approximantmetric gL. Let us recall from equation (3.41), that the respective Riemannianvolume element is

√Ldx1∧dx2∧dx3. Then Green’s formula in (R3, gL) takes

the formˆD

(f∆Lg − g∆Lf)√Ldx1 ∧ dx2 ∧ dx3 =

ˆ∂D

(f〈∇g, νL〉L − g〈∇f, νL〉L)dσL

for each f, g ∈ C2(D), where ∆L = X21 + X2

2 + X23 = X2

1 + X22 + L−1X2

3

denotes the Laplacian in (R3, gL) and νL is the outer unit normal to ∂D.Dividing both sides of the equation by

√L and using (5.6), we obtain the

sub-Riemannian Green’s formulaˆD

(f∆g − g∆f)dx =

ˆ∂D

(f〈∇0g, νH〉1 − g〈∇0f, νH〉1)dµ (5.9)

where ∆ = X21 +X2

2 is the Heisenberg laplacian introduced in Section 4.2.Let us consider the case when ∂D is the boundary Sε of the Heisenberg

ball Bε := x ∈ H : ||x||H = ε. We may parameterize Sε as Sε = f(D),where f = (f1 + if2, f3) is given by

(f1 + if2)(ϕ, θ) = ε√

cosϕ exp(iθ)

andf3(ϕ, θ) =

1

4ε2 sinϕ,

with D = (ϕ, θ) ∈ R2 : −π/2 < ϕ < π/2, 0 ≤ θ < 2π. Let us computethe perimeter measure dµ on Sε as given in equations (5.5) and (5.6). Byexplicite computations we obtain

fϕ × fθ =

(−ε

3

4cos3/2 ϕ cos θ,−ε

3

4cos3/2 ϕ sin θ,−ε

2

2sinϕ

),

thus|A~n| = |A(fϕ × fθ)| =

1

4ε3√

cosϕ.


We therefore may conclude that

PH(Bε,Ω) =

ˆSε∩Ω

dµ

=

ˆ(ϕ,θ)∈D:(ε

√cosϕ exp(iθ), 1

4ε2 sinϕ)∈Ω

1

4ε3√

cosϕdϕdθ. (5.10)

We want to explicite (5.9) with f ∈ C∞0 (H), g = ||x||−2H and D = Dε =

Bcε ∩ spt(f). To simplify the notation in what follows we write N(x) = ||x||H

for the Heisenberg norm. Then |∇0N(x)| = |z|/N(x), where x = (z, x3). By(4.7), νH = ∇0u/|∇0u| = −∇0N/|∇0N |, then

〈∇0g, νH〉1 =

⟨∇0g,−

∇0N

|∇0N |

⟩1

= 2N−3|∇0N |. (5.11)

Moreover, by a long but easy computation, one can show that

∆g = 0 in H \ 0. (5.12)

Thus, by equations (5.11) and (5.12), (5.9) becomes

−ˆDε

N−2∆f =

ˆSε

2fN−3|∇0N |+N−2

⟨∇0f,

∇0N

|∇0N |

⟩1

dµ. (5.13)

Lemma 5.12. ˆSε

N−2

⟨∇0f,

∇0N

|∇0N |

⟩1

dµ = o(ε).

Proof. Using (5.10) in the desired integral givesˆ π/2

−π/2

ˆ 2π

0

ε2⟨∇0f,

∇0N

|∇0N |

⟩1

∣∣∣∣x=(ε

√cosϕ exp(iθ), 1

4ε2 sinϕ)

ε3

4

√cosϕdθdϕ.

Another use of (5.10) givesˆSε

2N−3|∇0N |dµ = 8π.

In the first term on the right-hand side of (5.13), we add and substract f(0)and observe that the termˆ

Sε

2(f − f(0))N−3|∇0N | = o(1)


by uniform continuity of f in Bε. In conclusion, passing to the limit as ε→ 0in (5.13), we deduce the identity

−ˆHN−2∆f = 8πf(0),

valid for all f ∈ C∞0 (H). An application of the horizontal integration byparts formulas

´H(Xif)g = −

´H f(Xig), for i = 1, 2, converts this into the

following singular integral representation formula:

f(0) =1

8π

ˆH〈∇0f(y),∇0(N−2)(y)〉1dy

= − 1

4π

ˆH〈∇0f(y),∇0N(y)〉1N(y)−3dy.

(5.14)

Another form for (5.14) can be obtained by inserting the test function fx(y) =f(xy−1) and transforming the integrand via the isometry y 7→ xy−1:

f(x) =1

8π

ˆH〈∇0f(y),∇0(N−2)(y−1x)〉1dy

= − 1

4π

ˆH〈∇0f(y),∇0N(y−1x)〉1N(y−1x)−3dy.

(5.15)

Now we are ready to prove the weak-type Sobolev-Gagliardo-Nirenberginequality.

Proof of Proposition 5.11. Our starting point is the representation formula(5.15). The estimate |∇0N | ≤ 1 yields

|f(x)| ≤ 1

4π

ˆH|∇0f(y)|(dH(x, y))−3dy

≤ C

ˆH|∇0f(y)|d(x, y)−3dy. (5.16)

Here the last inequality holds by the equivalence of the Carnot-Carathéodorymetric d and dH (see equation (3.23)). In view of (5.16), we see that in orderto show the theorem we need only to prove that the fractional integrationoperator

g → I1g(x) =

ˆH|g(y)|d(x, y)−3dy,

satisfies a weak-type (1, 4/3) estimate, i.e., there exists a positive constantC such that

|x ∈ B(0, r) : |I1g(x)| > λ| ≤ Cλ−43 ||g||

43

L1(H) (5.17)


for all g ∈ L1(H) with compact support in B(0, r). Notice that I1g = 0outside B(0, r). Let ε > 0 be a sufficiently small constant (to be chosenlater) which depends on λ and ||g||L1(H) and write

I1g = I11g + I2

1g,

whereI1

1g(x) =

ˆB(x,ε)

|g(y)|d(x, y)−3dy

andI2

1g(x) =

ˆB(0,r)\B(x,ε)

|g(y)|d(x, y)−3dy.

By a dyadic decomposition argument and by the definition of the Hardy-Littlewood maximal operator, we get the existence of a constant C1 > 0 suchthat

I11 (g) ≤

∞∑k=0

(2−k−1ε)−3

ˆB(x,2−kε)\B(x,2−k−1ε)

|g(y)|dy

≤ C1

∞∑k=0

2−kε

(1

|B(x, 2−kε)|

ˆB(x,2−kε)

|g(y)|dy)

≤ 2C1εMg(x).

(5.18)

We also have the trivial estimate

I21g(x) ≤ ε−3||g||L1(H). (5.19)

Next, using (5.8) and (5.18), we have

|I1g ≥ λ| ≤∣∣∣∣I1

1g ≥λ

2

∣∣∣∣+

∣∣∣∣I21g ≥

λ

2

∣∣∣∣≤

∣∣∣∣Mg >λ

4C1ε

∣∣∣∣+

∣∣∣∣I21g ≥

λ

2

∣∣∣∣≤ 4C1ε

λ||g||L1(H) +

∣∣∣∣I21g ≥

λ

2

∣∣∣∣ . (5.20)

Now, if λ ≤ 2r−3||g||L1(H), then λ−4/3||g||4/3L1(H) ≥ 2−4/3r4, that is

|B(0, r)| ≤ Cλ−4/3||g||4/3L1(H)

and we obtain the desired estimate (5.17):

|x ∈ B(0, r) : |I1g(x)| > λ| ≤ |B(0, r)| ≤ Cλ−4/3||g||4/3L1(H).


Hence, to estimate the second term on the right-hand side of (5.20) we mayassume

λ > 2r−3||g||L1(H). (5.21)

Choose ε = (λ−12||g||L1(H))1/3 < r. By (5.19), I2

1g(x) < λ/2 whence |I21 >

λ/2| = 0 and

|I1g ≥ λ| ≤ 4C1

λ||g||L1(H)(λ

−12||g||L1(H))1/3 ≤ Cλ−4/3||g||4/3L1(H)

which concludes the proof.

Making use of the weak-type inequality, we prove the correspondingstrong-type inequality.

Proposition 5.13. There exists a constant C1(H) <∞ so that

||f ||4/3 ≤ C1(H)||∇0f ||1

for all f ∈ S1,1(H).

Proof. By the approximation lemma 5.4, it suffices to prove the estimate forsmooth, compactly supported functions f on H. Moreover, if we prove theproposition for all nonnegative functions, we easily conclude the proof forevery f . Indeed, f = f+ − f−, where f+ and f− denote the positive andnegative part of f respectively. Then

||f || 43≤ ||f+|| 4

3+ ||f−|| 4

3≤ C1(H)(||∇0f+||1 + ||∇0f−||1)

= C1(H)

ˆH|∇0f+|+ |∇0f−|dy = C1(H)||∇0f ||1.

Hence, let f be a nonnegative, smooth, compactly supported function,choose R > 0 so that B(0, R) contains the support of f , and write

Aj = x ∈ B(0, R) : 2j < f(x) ≤ 2j+1, j ∈ Z.

Let fj = max0,minf − 2j, 2j, i.e.

fj(x) =

2j if x ∈

⋃i>j Ai

f − 2j if x ∈ Aj0 otherwise.

We observe that ∇0fj is supported on Aj and that if x ∈ Aj+1 then fj(x) =2j > 2j−1. Since fj is Lipschitz we may apply the weak-type estimate in


Proposition 5.11 and

|Aj+1| ≤ |fj > 2j−1|≤ |I1(|∇0fj|) > C−12j−1|

≤ C

(2−j

Âj

|∇0fj|

)4/3

= C

(2−j

Âj

|∇0f |

)4/3

.

Thus

ˆH|f |4/3 =

∑j∈Z

Âj

|f |4/3 ≤∑j∈Z

(2j+1)4/3|Aj|

≤ C∑j∈Z

(Âj

|∇0f |

)4/3

≤ C

(∑j∈Z

Âj

|∇0f |

)4/3

= C

(ˆH|∇0f |

)4/3

,

which completes the proof.

The just proved Sobolev embedding theorem 5.13 holds for functions de-fined on all of H. However, for future applications, we are interested in asimilar relation which can be applied to functions defined only on a ballB ⊂ H. It is easy to see that the desidered estimate cannot be exactly thatof theorem 5.13; infact, consider for example a constant function f differentfrom zero, then the inequality ||f ||4/3,B ≤ C||∇0f ||1,B doesn’t hold for anyconstant C. An appropriate analog to theorem 5.13 is the following Poincaréinequality

(1

|C−1B|

ˆC−1B

|f(x)− fB|43dx

) 34

≤ Cr

(1

|B|

ˆB

|∇0f(x)|dx)

(5.22)

which holds for functions f ∈ S1,1(B) defined on a ball B with radius r,where C is independent of both f and B. Here fB = 1

|B|

´Bf denotes the

average of f on B, and C−1B is the ball concentric with B whose radius isC−1 times that of B.

Applying Hölder’s inequality to both sides of (5.22), we obtain the fol-


lowing family of Poincaré-type inequalities:

1

|C−1B|

ˆC−1B

|f(x)− fB|dx ≤(

1

|C−1B|

ˆC−1B

|f(x)− fB|4/3dx)3/4

≤ Cr

(1

|B|

ˆB

|∇0f(x)|dx)

≤ Cr1

|B||B|1−1/p

(ˆB

|∇0f(x)|pdx)1/p

= Cr

(1

|B|

ˆB

|∇0f(x)|pdx)1/p

,

for each 1 ≤ p < ∞. Summing up, we obtain the so-called (1, p)-Poincaréinequality :

1

|C−1B|

ˆC−1B

|f(x)− fB|dx ≤ Cr

(1

|B|

ˆB

|∇0f(x)|pdx)1/p

. (5.23)

5.2.2 Compactness of the embedding BV ⊂ L1 on Johndomains

In this section we focus our attention to functions with bounded variationdefined on a John domain.

Definition 5.14. A bounded, connected open set Ω in a metric space (X, d)is a John domain if there exists a point x0 ∈ Ω (the center of the domain)and a constant δ > 0 such that for any point x ∈ Ω there exists an arc lengthparameterized rectifiable path γ : [0, L] → Ω so that γ(0) = x, γ(L) = x0

and d(γ(t), X \ Ω) ≥ δt.

We are interested in John domains because we can state some theoremson them, which are crucial for proving the existence of minimizers for theisoperimetric inequality in H. In Euclidean spaces every Lipschitz domainis a John domain. In the Heisenberg group it is more difficult to verify theJohn property.

Lemma 5.15. For each x ∈ H and R > 0, both Carnot-Carathéodory ballB(x,R) and gauge ball BH(y,R) = x ∈ H : dH(x, y) < R are John do-mains.

Theorem 5.16 (Garofalo-Nhieu). Let Ω ⊂ H be a John domain. The spaceBV (Ω) is compactly embedded in L1(Ω).


To prove Theorem 5.16 we need the following two lemmas.

Lemma 5.17. Let Ω ⊂ H be a John domain. Then there exists a constantC1,Ω > 0 so that(

1

|Ω|

ˆΩ

|u− uΩ|4/3)3/4

≤ C1,Ωdiam(Ω)

|Ω|VarH(u,Ω)

for all u ∈ BV (Ω). Here we have denoted by uΩ the average of the functionu over the domain Ω.

The second lemma is a relative isoperimetric inequality for John domains(similar to that one given in the euclidean case). We state it only in the caseof metric balls.

Lemma 5.18. Let B ⊂ H be a metric ball. Then there exists a constantC > 0 so that

min(|A|, |B \ A|)3/4 ≤ CPH(A,B)

for any measurable set A ⊂ B.

Proof of Theorem 5.16. We have to show that BV (Ω) ⊂ L1(Ω) is compactfor each John domain Ω. By the characterization of compact sets in a metricspace, a set in a metric space is compact if and only if it is closed and totallybounded. Therefore, in view of the approximation Lemma 5.5 it is sufficientto prove that the set S = u ∈ BV (Ω) ∩ C∞(Ω) : ||u||BV ≤ 1 is totallybounded. In other words, we have to prove that for each ε > 0 there existfunctions u1, . . . , uM ∈ S such that S is covered by the balls Bk with centerin uk and radius ε, where k = 1, . . . ,M .

Let ε > 0 and let Ωε be an open set which approximates Ω in the sensethat Ωε ⊂ Ω and

CPdiam(Ω)

|Ω|1/4|Ω \ Ωε|1/4 +

|Ω \ Ωε||Ω|

<ε

6, (5.24)

where CP is a constant as in Lemma 5.17. For any u ∈ S, using Lemma 5.17and Hölder’s inequality, we have the estimateˆ

Ω\Ωε|u| ≤

ˆΩ\Ωε|u− uΩ|+

ˆ|Ω\Ωε|

|uΩ|

≤(ˆ

Ω\Ωε1dx

)1/4(ˆΩ

|u− uΩ|4/3)3/4

+ |Ω \ Ωε||uΩ|

≤(ˆ

Ω

|u− uΩ|4/3)3/4

|Ω \ Ωε|1/4 + |Ω \ Ωε||uΩ|

≤ CPdiam(Ω)

|Ω|1/4|Ω \ Ωε|

14 VarH(u,Ω) +

|Ω \ Ωε||Ω|

ˆΩ

|u|. (5.25)


Since u ∈ S, (5.24) and (5.25) yieldˆ

Ω\Ωε|u| < ε

6. (5.26)

We prove now that there exists a positive integerM so that for any sufficientlysmall δ > 0 one can construct a family of balls Bj, j = 1, . . . , N with thefollowing properties:

1. the diameter of each Bj is less than or equal to δε,

2. Ωε ⊂⋃Nj=1Bj ⊂ Ω,

3.∑N

j=1 χCPBj ≤MχΩ.

To show these statements, let r := δε be such that 0 < CP r < dist(Ωε,H\Ω).Let S be a set of points in Ωε such that the pairweise mutual distance betweenthem is at least r and let S be maximal with respect to this property.

Firstly let us prove that S is finite: by definition the family of ballsB(x, r/2) : x ∈ S is disjoint and

⋃x∈S B(x, r/2) ⊂ Ω. Thus, if N = ]S,

N |B(0, 1)|(r

2

)4

=∑x∈S

|B(x,r

2)| ≤ |Ω|

which proves that N is finite. Let us denote the balls in B(x, r) : x ∈ S asB1 = B(x1, r), . . . , BN = B(xN , r). Again, by definition,

Ωε ⊂N⋃j=1

Bj ⊂ Ω.

Now let us prove that there exists M independent of r such that

N∑j=1

χCPBj ≤MχΩ.

To this end, let x ∈ Ω. Eventually by translation, we may suppose thatx corresponds to the origin. Suppose that 0 ∈ CPBj1 ∩ . . . ∩ CPBjM , thenCPBjk ⊂ B(0, 2CP r) for each k = 1, . . . ,M and

M |B(0,r

2)| =

∣∣∣∣∣M⋃k=1

B(xjk ,r

2)

∣∣∣∣∣ ≤∣∣∣∣∣M⋃k=1

Bj

∣∣∣∣∣ ≤∣∣∣∣∣M⋃k=1

CPBj

∣∣∣∣∣ ≤ |B(0, 2CP r)|.


ThusM ≤ |B(0, 2CP r)|

|B(0, r/2)|= 28C4

P ,

and we get (1), (2) and (3).Let δ > 0 so that

CPMδ <1

6. (5.27)

Next we consider any function v ∈ BV (Ω) ∩ C∞(Ω) and, using (5.23), weestimate

ˆΩε

|v| ≤N∑j=1

ˆBj

|v|dx ≤N∑j=1

ˆBj

|v − vBj |dx+ |Bj||vBj |

≤N∑j=1

(CP δε

ˆCPBj

|∇0v|dx+

∣∣∣∣∣ˆBj

v dx

∣∣∣∣∣)

≤ CPKδεVarH(v,Ω) +N∑j=1

∣∣∣∣∣ˆBj

v dx

∣∣∣∣∣≤ ε

6VarH(v,Ω) +

N∑j=1

∣∣∣∣∣ˆBj

v dx

∣∣∣∣∣ .

(5.28)

At this point we define a linear operator T : BV (Ω)→ RN as follows:

T (u) =

(ˆB1

u dx, . . . ,

ˆBN

u dx

).

Notice that the image of the bounded set S under T is bounded in RN . Thenthere exist functions u1, . . . , uM ∈ S so that for any u ∈ S,

|Tu− Tuj| =N∑k=1

ˆBk

|u− uj|dx <ε

3(5.29)

for some j ∈ 1, . . . ,M. Moreover,

||u− uj||L1(Ω) ≤ˆ

Ωε

|u− uj|dx+

ˆΩ\Ωε|u− uj|dx = I + II. (5.30)

From (5.28) and (5.29) we have

I ≤ ε

6VarH(u− uj) +

N∑k=1

∣∣∣∣ˆBk

(u− uj)dx∣∣∣∣

≤ ε

6(||u||BV (Ω) + ||uj||BV (Ω)) +

ε

3≤ 2ε

3.

(5.31)

5.3 First variation of the perimeter 109

For the second term in (5.30) we use (5.26) to estimate

II ≤ˆ

Ω\Ωε|u|dx+

ˆΩ\Ωε|uj|dx ≤

ε

3. (5.32)

From (5.30), (5.31) and (5.32) we obtain that S is totally bounded.

5.3 First variation of the perimeter

In this section we present two derivations of the first variation formula for theperimeter of surfaces in H. First we consider parametric representations ofthe surface and variations which vanish in a neighborhood of the character-istic set. Next, we present an argument in which variations over full surfaceare allowed.

We start by introducing some useful notation: Let Ω ⊂ H be a boundedregion enclosed by a surface S, and consider variations Ωt and St along agiven vector field Z ∈ C1(H,R3). We say that Ω is perimeter stationary if

d

dtPH(Ωt)

∣∣∣∣t=0

= 0 for all choices Z ∈ C1(H,R3). (5.33)

We say that Ω is volume-preserving perimeter stationary if

d

dtPH(Ωt)

∣∣∣∣t=0

= 0 ∀Z ∈ C1(H,R3) s.t.d

dt|Ωt|∣∣∣∣t=0

0.

Since a flow line of a vector field Z is a path whose velocity field lies in Z,if Z is tangent to S at every point, then variations along Z do not changeS = St, hence for tangential variations condition (5.33) always holds.

5.3.1 Parametric surface and noncharacteristic varia-tions

We compute the first variation of the perimeter wih respect to the Euclideannormal ~n and then with respect to a horizontal frame X1, X2. Throughoutthe section the emphasis is on explicit computations using the underlyingEuclidean structure of R3.

Let Ω ⊂ R2 be a domain, let ε > 0, and let Ξ : Ω × (−ε, ε) → Hrepresent a flow St = Ξ(Ω, t) of noncharacteristic surface patches in H, withΞ(Ω, 0) = S. Denote by A the 2 × 3 matrix of coefficients of the horizontal


frame X1, X2 and recall that the horizontal perimeter of St may be calculatedusing Proposition 5.8. Our goal is to evaluate

d

dtPH(St)|t=0 =

d

dt

ˆSt

|A~n|dσ∣∣∣∣t=0

(5.34)

=d

dt

ˆΩ

|A(Ξu × Ξv)|dudv∣∣∣∣t=0

=

ˆΩ

1

|AV |

(⟨AV,A

dV

dt

⟩+

⟨AV,

(dA

dt

)V

⟩)∣∣∣∣t=0

,

where we have set V = Ξu × Ξv and ~n = V/|V |. Here the second equalityfollows from the change of variable formula. The underlying metric in usehere is the Euclidean inner product and norm.

To simplify the notation we will always ignore the higher order terms inthe Taylor expansion of Ξ, writing

Ξ(u, v, t) = Ξ(u, v, 0) + tdΞ

dt(u, v, 0) =: X(u, v) + tλ~n(u, v, 0).

A simple computation yields V = Xu × Xv + t(Xu × (λ~n)v + (λ~n)u × Xv),and consequently

dV

dt

∣∣∣∣t=0

= λ(Xu × ~nv + ~nu ×Xv) + λvXu × ~n+ λu~n×Xv.

Recall the definition of the shape operator given in Definition 1.24 anddenote by bβα the coefficients of its matrix in the basis given by Xu, Xv. Then

~nα = −buαXu − bvαXv

for every α ∈ u, v. Consequently Xu × ~nv = −bvvXu ×Xv and ~nu ×Xv =−buuXu×Xv. Endow S with the Riemannian metric induced by the Euclideanmetric in C× R. Thus we have

dV

dt

∣∣∣∣t=0

= −λHXu ×Xv + λvXu × ~n+ λu~n×Xv, (5.35)

where H = buu + bvv denotes the mean curvature of S.Since

A|Ξ =

(1 0 −1

2Ξ2

0 1 12Ξ1

)we find

dA

dt

∣∣∣∣Ξ

=

(0 0 −1

2λn2

0 0 12λn1

), with ~n = (n1, n2, n3).


Thus

dA

dtV =

1

2λV3(−n2, n1), where V = (V1, V2, V3). (5.36)

By virtue of (5.34), (5.35) and (5.36) we obtain

d

dtPH(St)|t=0 =

ˆΩ

1

|AV |〈AV,−λHA(Xu ×Xv)

+λvA(Xu × ~n) + λuA(~n×Xv) +1

2λV3(−n2, n1)

∣∣∣∣t=0

⟩dudv. (5.37)

An integration by parts in u and v gives the following expression for the firstvariation of the perimeter:

ˆΩ

λ

[−H|AV | −

(〈AV,A(Xu × ~n)〉

|AV |

)v

−(〈AV,A(Xv × ~n)

|AV |

)u

+1

2

V3

|AV |〈AV, (−n2, n1)〉

]∣∣∣∣t=0

dudv.

(5.38)

In essence, (5.38) is the derivative of the perimeter functional in the direc-tion Y = λ~n. Since variations along purely tangential directions are zero,then normal variations represent the complete “gradient”of the perimeterfunctional.

We now want to generalize the preceding to the case of general pertur-bations of the original surface S. Thus let us take a basis of R3 given bythe standard left invariant basis of the horizontal bundle X1 = (1, 0,−1

2x2)

and X2 = (0, 1, 12x1) together with the vector T = (1

2x2,−1

2x1, 1) (note that

T differs from the standard left invariant vector field X3 = (0, 0, 1)) andconsider a general variation of the form

Ξ = X + t(aX1 + bX2 + cT ) + o(t),

where X = (x1, x2, x3) is a parameterization of S. Our goal is still to evaluate(5.34), which remains unchanged. In order to compute Ξu×Ξv we make hereall the calculations


Xα ×X1 =

(−1

2x2x2,α,

1

2x2x1,α + x3,α,−x2,α

),

Xα ×X2 =

(−x3,α +

1

2x1x2,α,−

1

2x1x1,α, x1,α

),

X1,α ×Xβ =

(1

2x2,αx2,β,−

1

2x2αx1,β, 0

),

X2,α ×Xβ =

(−1

2x1,αx2,β,

1

2x1,αx1,β, 0

),

Xα × T =

(x2,α +

1

2x1x3,α,

1

2x2x3,α − x1,α,−

1

2x1x1,α −

1

2x2x2,α

),

Xβ × Tα =

(1

2x1,αx3,β,

1

2x2,αx3,β,−

1

2x1,αx1,β −

1

2x2,αx2,β

).

Here α, β ∈ u, v and we have written xi,α = ∂xi/∂α, etc. These in turnyield

A(Xα ×X1) = (0, ω(Xα)),

A(Xα ×X2) = (−ω(Xα), 0),

A(X1,α ×Xβ) =1

2x2,αizβ,

A(X2,α ×Xβ) = −1

2x1,αizβ,

A(Xα × T ) =1

4(2ω(Xα)z − (4 + |z|2)izα)

A(Xβ × Tα) =1

4(2ω(Xβ)zα −

1

2(|z|2)αizβ), (5.39)

where we have let z = x1 + ix2 and ω(Xα) = x3,α + 12(x2x1,α − x1x2,α).

Thanks to (5.39) we have

Ad

dt(Ξu × Ξv) = av(0, ω(Xu))− au(0, ω(Xv)) +

1

2ai(x2uzv − x2vzu)

+ bv(−ω(Xu), 0)− bu(−ω(Xv), 0) +1

2bi(x1vzu − x1uzv)

+ c

[1

2ω(Xu)zv −

1

2ω(Xv)zu −

1

8(|z|2)vizu +

1

8(|z|2)uizv

]+ cv

[(1 +

1

4|z|2)izu +

1

2(ω(Xu)z

]− cu

[(1 +

1

4|z|2)izv +

1

2(ω(Xv)z

].

(5.40)


Set F = X1, X2 the horizontal, orthonormal frame and express the hori-zontal normal νH in this frame as the coordinate vector

[νH ]F =A(Xu ×Xv)|A(Xu ×Xv)|

= (ν1H , ν

2H). (5.41)

Since

A(Xu ×Xv) =(x2ux3v − x3ux2v −

1

2x2(x1ux2v − x2ux1v),

x3ux1v − x1ux3v +1

2x1(x1ux2v − x2ux1v)

)=

(− x2vω(Xu) + x2uω(Xv) , x1vω(Xu)− x1uω(Xv)

)= i[ω(Xu)zv − ω(Xv)zu]. (5.42)

using (5.40) we have (after several cancellations)⟨[νH ]F , A

d

dt(Ξu × Ξv)

⟩= a(〈[νH ]F , (0, ω(Xv))〉u − 〈[νH ]F , (0, ω(Xu))〉v

)+ b(〈[νH ]F , (ω(Xu), 0)〉v − 〈[νH ]F , (ω(Xv), 0)〉u

)+ c(〈∂u[νH ]F , (1 +

1

4|z|2)izv +

1

2ω(Xv)z〉

− 〈∂v[νH ]F , (1 +1

4|z|2)izu +

1

2ω(Xu)z〉

). (5.43)

On the other hand, (dA

dt

)Xu ×Xv =

1

2V3(−Y2, Y1),

where we have set V = Xu ×Xv and Y = (Y1, Y2, Y3) ∈ R3 such that

Y1 =d

dtΞ1 = a+

1

2cx2 Y2 =

d

dtΞ2 = b− 1

2cx1.

We have(−Y2, Y1) = a(0, 1) + b(−1, 0) +

1

2cz.

Combining the latter with (5.34) and (5.43) we finally obtain the first varia-tion of the perimeter. With respect to every single direction it is:Variation along X1 (a = 1, b = 0, c = 0):

[ν2Hω(Xv)]u − [ν2

Hω(Xu)]v +1

2V3ν

2H . (5.44)


Variation along X2 (a = 0, b = 1, c = 0):

[ν1Hω(Xu)]v − [ν1

Hω(Xv)]u −1

2V3ν

1H . (5.45)

Variation along X3 (a = 0, b = 0, c = 1):⟨∂u[ν

2H ]F ,

(1 +

1

4|z|2)izv +

1

2ω(Xv)]z

⟩−⟨∂v[ν

2H ]F ,

(1 +

1

4|z|2)izu +

1

2ω(Xu)]z

⟩+

1

4V3 〈[νH ]F , z〉 . (5.46)

At this point we restrict our attention to horizontal variations by settingc = 0. Given a horizontal variation, determined by the flow in directionxX1+yX2, the corresponding variation can be dermined by the scalar productbetween the vectors (x, y) and (a, b), where (a, b) is chosen according to (5.44)and (5.45), i.e., it is given by(

[ν2Hω(Xv)]u − [ν2

Hω(Xu)]v +1

2V3ν

2H ; [ν1

Hω(Xu)]v − [ν1Hω(Xv)]u −

1

2V3ν

1H

).

(5.47)Now, since the Schwarz inequality gives

(x, y) · (a, b) ≤ ||(x, y)|| ||(a, b)||,

the variation corresponding to a flow in the direction aX1 + bX2 correspondsto the maximum variation, among horizontal variations. We summarize thepreceding computations in the following proposition.

Proposition 5.19. The maximum variation of the perimeter among hor-izontal variations and outside the characteristic locus is obtained along thedirection aX1+bX2, where (a, b) is chosen as in (5.47), and is equal to I+II,where

I =

(ω(Xu)v − ω(Xv)u −

1

2V3

)i[νH ]F

andII = ω(Xu)∂v(i[νH ]F)− ω(Xv)∂u(i[νH ]F), (5.48)

here [νH ]F is as in (5.41). Moreover, the component corresponding to I istangent to the surface.

Proof. We are only left with the proof of the last statement. Since [νH ]F isa unit vector, the factors I and II are orthogonal. The vector

−ν2HX1 + ν1

HX2 = AT iA(Xu ×Xv)|A(Xu ×Xv)|


is in the same direction of I, then we conclude by observing that

⟨~n,−ν2

HX1 + ν1HX2

⟩=

⟨(Xu ×Xv)|(Xu ×Xv)|

,AT iA(Xu ×Xv)|A(Xu ×Xv)|

⟩= 0.

Remark 5.20. Observe that if we can choose a parameterization of thesurface so that zu and zv are orthonormal and use the fact that V3 = 〈izu, zv〉,we can rewrite I more explicitly in the following form:

I =

[〈izu, zv〉 −

1

2V3

]i[νH ]F =

1

2V3i[νH ]F . (5.49)

Since we just proved that the component of the variation correspondingto I is horizontal and tangent to the surface, it thus corresponds only to areparameterization of the surface. In the following we ignore it and focus onthe component II.

Proposition 5.21. The non-tangential component of the maximal variationof the perimeter PH occurs along the vector Z = aX1 + bX2, where

(a, b) = −H0|A(Xu ×Xv)|[νH ]F ,

i.e., Z = −H0νH |πH(ν1)|1 with πH denoting the (Euclidean) orthogonal pro-jection on the horizontal bundle.

Proof. To identify II, we consider the Legendrian foliation on the surface.Recall that this foliation is composed of horizontal curves γ lying on thesurface which are flow lines of the horizontal vector field

−i[νH ]F · ∇0 := ν2HX1 − ν1

HX2.

Since we are using a parametric representation of the surface S, we considera curve γ = (u, v) : [0, L]→ R2, such that γ(s) = X (γ(s)) = X (u(s), v(s)).

Note thatd

dsπzγ = zuu

′ + zvv′.

On the other hand, by (5.42), and by the definition of the Legendrian foliationwe also have

d

dsπzγ = −i[νH ]F = − ω(Xv)zu − ω(Xu)zv

|ω(Xv)zu − ω(Xu)zv|.


As a consequence,

u′ = − ω(Xv)|ω(Xv)zu − ω(Xu)zv|

, and v′ =ω(Xu)

|ω(Xv)zu − ω(Xu)zv|.

Now, by virtue of Proposition 4.6 we have that the curvature of πzγ is givenby H0[νH ]F , hence

H0[νH ]F =d2

ds2πzγ =

d

ds(−i[νH ]F(u(s), v(s)))

= −(i[νH ]F)uu′ − (i[νH ]F)vv

′

=(i[νH ]F)uω(Xv)− (i[νH ]F)vω(Xu)

|A(Xu ×Xv)|(5.50)

=−II

|A(Xu ×Xv)|.

Formula (5.50) provides an explicit representation of the horizontal meancurvature for noncharacteristic parametric surface.

Proposition 5.22. Let S ⊂ H be a C2 surface, which is parametrized by themap X : Ω→ H. Then outside the characteristic set Σ(S) one has

H0[νH ]F =ω(Xv)∂u(i[νH ]F)− ω(Xu)∂v(i[νH ]F)

|A(Xu ×Xv)|, (5.51)

where [νH ]F is given by (5.41) and ω = dx3 − 12(x1dx2 − x2dx1).

Remark 5.23. Since tangential variations result in reparametrizations of thesurface, which do not modify the perimeter, it is important that we consideronly the normal component of the maximal horizontal variation identifiedabove, i.e.,⟨

AT(−H0|A(Xu ×Xv)|[νH ]F +

1

2〈izu, zv〉i[νH ]F

), ~n

⟩= −H0

|A(Xu ×Xv)||(Xu ×Xv)|

= −H0|A~n| = −H0|πH(ν1)|1. (5.52)

5.3.2 General variations

In the next proposition we give an extension of the first variation formulawhich can be applied also to variations across the characteristic locus.


Proposition 5.24. Let S ⊂ H be an oriented C2 immersed surface withg1-Riemannian normal ν1 and horizontal normal νH . Suppose that U is a C1

vector field with compact support on S, let φt(p) = expp(tU) and let St be thesurface φt(S). Then

d

dtPH(St)

∣∣∣∣t=0

=

ˆS\Σ(S)

[u(divSνH)− divS(u(νH)tang)] dσ, (5.53)

where u = 〈U, ν1〉1. Moreover, if divSνH ∈ L1(S, dσ), then

d

dtPH(St)

∣∣∣∣t=0

=

ˆS

u(divSνH)dσ −ˆS

divS(u(νH)tang)dσ, (5.54)

where u = 〈U, ν1〉1.

Proof. Let us denote by vtang the tangential component of a vector v and bydσ the Riemannian surface area element on S. Let NH be the g1 projection ofν1 on the horizontal distribution, that is NH =

∑2i=1〈ν1, Xi〉1Xi. Then NH is

the component of ν1 in the direction of νH and |NH | = 〈ν1, νH〉. Recall thatin view of Corollary 5.9, the perimeter measure is dµ = |πH(ν1)|dσ = |NH |dσ.Let dσt be the Riemannian surface area element on the surface St.

Let ν1 be the unitary vector field normal to the surfaces St. Note that,since the differential of φt sends the tangent space to S at p in the tangentspace to φt(S) at φt(p), the vector field ν1 is such that ν1(φt(p)) = dφt(ν1(p))and is therefore a C1 vector field. By taking a partition of the unity we canextend this vector field defined a priori on the surfaces St to the whole H.Let us still denote this extension by ν1.

By Corollary 5.9 and using the Rimannian co-area formula 1.44 we obtain

PH(St) =

ˆSt

dµ =

ˆSt

|NH |dσt

=

ˆS

(|NH | φt)|Jφt |dσ

=

ˆS\Σ(S)

(|NH | φt)|Jφt|dσ

where the last equality follows because the Riemannian surface measure ofΣ(S) is zero. This result is due to Derridj. We just observe here that Σ(S) isnowhere dense, that is it doesn’t admit proper open subsets. In fact, Σ(S) isthe set of points where the tangent space coincides with the horizontal plane.The horizontal plane, however, is not involutive, because it isn’t closed under


the commutator law. Then by Frobenius theorem 1.6, the distribution consti-tuted by the horizontal planes cannot be the tangent space of a submanifold.Since these are already tangent to Σ(S), Σ(S) can’t be a submanifold.

In the next computation, we use the divergence identity (1.24)

d

dt|Jφt|

∣∣∣∣t=0

= divSU (5.55)

and the standard formula

divS(fV ) = fdivSV + Vtang(f) (5.56)

for f : S → R and a C1 vector field V on S, computed in equation (1.20).Here, as before, we will use the decomposition V = Vnorm +Vtang, where Vnorm

and Vtang denote the (Riemannian) normal and the tangential components toS of V respectively. Let us use equations (5.55) and (5.56) to compute thedifferential of PH(St) with respect to t, then

d

dtPH(St)

∣∣∣∣t=0

=

ˆS\Σ(S)

d(|NH | φt)|Jφt|dt

∣∣∣∣t=0

dσ

=

ˆS\Σ(S)

d(|NH | φt)dt

∣∣∣∣t=0

|Jφt ||t=0 + (|NH | φt)|t=0

d|Jφt|dt

∣∣∣∣t=0

dσ

=

ˆS\Σ(S)

〈∇|NH |,dφtdt〉1∣∣∣∣t=0

+ |NH |divSUdσ

=

ˆS\Σ(S)

U(|NH |) + divS(|NH |U)− Utang(|NH |)dσ

=

ˆS\Σ(S)

Unorm(|NH |) + divS(|NH |Unorm) + divS(|NH |Utang)dσ

Since Utang is tangent to the surface, we can apply the divergence theorem1.29 and obtain that

´S\Σ(S)

divS(|NH |Utang))dσ =´S

divS(|NH |Utang))dσ =0, because U is Lipschitz and has compact support in S. Hence

d

dtPH(St)

∣∣∣∣t=0

=

ˆS\Σ(S)

Unorm(|NH |) + divS(|NH |(Unorm))dσ

=

ˆS\Σ(S)

Unorm(|NH |) + |NH |divS(Unorm)dσ (5.57)

Notice that NH = ν1 − 〈ν1, X3〉1X3 and |NH | = 〈NH , NH〉1/|NH | =〈NH , νH〉1. Then, for any C1 vector field V , since the Levi-Civita connectionis compatible with the metric (see equation (1.4)), one has

V (|NH |) = 〈DVNH , νH〉1 + 〈NH , DV νH〉1 = 〈DVNH , νH〉1 .


The last equality follows because |νH | = 1 and then

〈NH , DV νH〉1 = |NH |〈νH , DV νH〉1 =1

2|NH |V 〈νH , νH〉1 = 0. (5.58)

Continuing the calculation, by using the properties of the affine connection,we have

V (|NH |) = 〈DVNH , νH〉1= 〈DV ν1, νH〉1 − 〈ν1, X3〉1 〈DVX3, νH〉1 − V (〈ν1, X3〉1) 〈X3, νH〉1= 〈DV ν1, νH〉1 − 〈ν1, X3〉1 〈DVX3, νH〉1 , (5.59)

where the last equality holds because X3 is orthogonal to νH .We now make use of the following formula, that we will prove in Lemma

5.27:DUnormν1 = −∇Su, (5.60)

where ∇S is the gradient on S.Let us write ν1 = aX1 +bX2 +cX3, then by Definition 1.10 and equations

(3.46) we have

〈Dν1X3, X3〉1 = a〈∇X1X3, X3〉+ b〈∇X2X3, X3〉1 + c〈∇X3X3, X3〉1 = 0

and

〈∇ν1X3, ν1〉1 = a〈∇X1X3, bX2〉1 + b〈∇X2X3, aX1〉1 = ab− ba = 0.

From these computations it follows that

〈DUnormX3, νH〉1 = 0. (5.61)

Using Equations (5.59), (5.60) and (5.61) we obtain that

Unorm(|NH |) = 〈DUnormν1, νH〉1 − 〈ν1, X3〉1 〈DUnormX3, νH〉1 = −〈∇Su, νH〉1 .

Thus the integrand in the last line of (5.57) is

Unorm(|NH |) + |NH |divS(Unorm)

= 〈−∇Su, νH〉1 + |NH |divS(uν1)

= −(νH)tang(u) + u|NH |divSν1

= −divS(u(νH)tang) + udivS((νH)tang) + u(divS(|NH |ν1)− (ν1)tang(|NH |))= −divS(u(νH)tang) + udivSνH ,

(5.62)


where the last equality holds because (νH)norm = 〈νH , ν1〉1ν1 = |NH |ν1.Since Unorm(|NH |)+|NH |divS(Unorm) is compactly supported and bounded

on S (because it is continuous on a compact), it is in L1(S, dσ) and we obtainequation (5.53). Under the additional assumption divSνH ∈ L1(S, dσ), bylinearity also divS(u(νH)tang) ∈ L1(S, dσ). Then we obtain (5.54) and theproof is concluded.

Remark 5.25. The hypothesis of Theorem 5.24 can be slightly weakened:we may consider compactly supported, bounded vector fields U on S suchthat |NH |divS(Unorm) ∈ L1(S) (so the final implication still holds), |NH |Utang

is Lipschitz continuous (so that we can apply the divergence theorem) anddivSνH ∈ L1(S, udσ).

This observation is important as it allows us to consider horizontal vari-ations U = aνH , with a ∈ C1

0(S), which correspond to udσ = adµ; in thiscase all the assumptions hold except the last one, which, by equation (4.9), isessentially reduced to the requirement H0 ∈ L1(S, dµ). In view of (4.14), thelatter condition is always true for C2 surfaces. We will use such variations inProposition 6.7.

Remark 5.26. If U is compactly supported in S \ Σ(S), by an applicationof the divergence theorem, equation (5.53) reduces to

d

dtPH(St)

∣∣∣∣t=0

=

ˆS

u(divSνH)dσ

with u = 〈U, ν1〉1.

With the following Lemma, which was pointed out to me by Prof. Ritoré,we prove formula (5.60) in a more general setting.

Lemma 5.27. Let M be a Riemannian manifold, S ⊂M a hypersurface inM such that it is well defined its unit normal N , U a smooth vector fieldwith compact support in M and φtt∈R the associated flow. Let us denoteby A(v) = −DvN for v ∈ TpS the Weingarten operator. Then

DUN = −∇Su− A(UT ), (5.63)

where (by abuse of notation) N is the unit normal to the hypersurface φt(S)(an extension of the original one) and u := 〈U,N〉|S.

Using the decomposition DUN = DUTN +DU⊥N , the fact that DUTN =−A(UT ) and (5.63) we obtain (5.60).


Proof. Equality (5.63) holds trivially in the interior of U = 0 and, bycontinuity of the formula, extends to the boundary of this set. So we onlyneed to establish (5.63) at a point p ∈ S where Up 6= 0.

We note first that 〈DUpN,Np〉 = 0 since |N | = 1. Pick some v ∈ TpS andextend it to a vector field Z in a neghborhood of p which is invariant by theflow of U , that is Z(φt(q)) = dφt(Zq) for each q in the neighborhood of p. Bythe geometrical interpretation of the Lie bracket (see Proposition 1.7) thisis equivalent to [U,Z] = 0. Moreover, the differential dφt sends the tangentspace to S at p in the tangent space to φt(S) at φt(p). Then the restrictionof Z to the integral curve t 7→ φt(p), i.e. Z(φt(p)) = dφt(v), is tangent toφt(S) and hence 〈N,Z〉 = 0. We have

〈DUpN, v〉 = 〈DUpN,Zp〉 = Up〈N,Z〉 − 〈Np, DUpZ〉= −〈Np, DZpU〉 = −Zp〈N,U〉+ 〈DZpN,Up〉= −〈∇Su, Zp〉 − 〈A(Zp), U

Tp , 〉

= −〈∇Su, v〉 − 〈A(UT ), v〉,

where in the last equality we have used that A is self-adjoint in TpS. So(5.63) is proven.

Chapter 6

The Isoperimetric problem in theSub-Riemannian Heisenberggroup

The solution of the isoperimetric inequality in H is still an open questionin mathematics: recent studies have picked out the “best” constant for theisoperimetric inequality and have proved the existence of minimizers, nev-ertheless, up to now, none has yet characterized the sets which realize theminimum for the inequality in the largest possible class of Cacciopoli sets.

This chapter is the core of this survey. We introduce the isoperimetricinequality in H in the setting of C1 sets and give a proof which make use ofthe Sobolev inequality (5.7). Next we define the isoperimetric profile of Hand present Pansu’s 1982 conjecture. Finally we show a proof of the existenceof an isoperimetric profile and descrive some of the existing literature on theisoperimetric problem.

6.1 The Isoperimetric Inequality in HThe isoperimetric inequality in H with respect to the horizontal perimeterhas first been proved by Pansu. We begin by stating it in the setting of C1

sets.

Theorem 6.1 (Pansu’s isoperimetric theorem in H). There exists a constantC > 0 so that

|E|34 ≤ CPH(E) (6.1)

for any bounded open set E ⊂ H with C1 boundary.

123

124 The Isoperimetric problem in the Sub-Riemannian Heisenberg group

Let us focus our attention on the exponent in (6.1). Comparing it withequation (2.22) we note that the right dimension that is involved in theisoperimetric problem is the homogeneous dimension, which in the Heisen-berg group is equal to 4. In the Euclidean case we didn’t noticed it, sincethe topological and homogeneous dimension are the same in this case.

Now we present a proof of Theorem 6.1 which isn’t the original one givenby Pansu, but is based on the geometric Sobolev embedding S1,1 ⊂ L4/3

viewed in Section 5.2.

Proof. Let us recall the geometric inequality (5.7): there exists a constantC1(H) <∞ such that

||f || 43≤ C1(H)||∇0f ||1

for each f ∈ S1,1(H). If we show the equivalence between this inequality and(6.1), the theorem will be proved.

Let E ⊂ H be a bounded, open, C1 set, and let R > 0 be such thatthe closure E of E is contained in B(0, R). Choose δ > 0 such that 2δ <dist(E, ∂B(0, R)), where dist(·, E) denotes the Euclidean distance from E.Define the function gδ(x) = 1− dist(x,E)

δand let fδ(x) be its positive part, i.e.

fδ(x) =

1 if x ∈ E1− dist(x,E)

δif dist(x, E) < δ

0 otherwise.

Notice that fδ is an (Euclidean) Lipschitz function, indeed let Aδ be theintersection of B(0, R) with an (Euclidean) tubular neighborhood of E ofradius δ. Then it is sufficient to prove Lipschitz condition only for x, y ∈ Aδ.In this case:

|fδ(x)− fδ(y)| =

∣∣∣∣1− dist(x, E)

δ−(

1− dist(y, E)

δ

)∣∣∣∣=

1

δ

∣∣dist(x, E)− dist(y, E)∣∣ ≤ 1

δ|x− y|,

that is Lipschitz condition for fδ with Lipschitz constant 1/δ.Applying the weak-type Sobolev-Gagliardo-Nirenberg inequality (Propo-

sition 5.11) to fδ we obtain

|E|34 ≤ |x ∈ B(0, R) : fδ(x) > t|

34

≤ C

t

ˆB(0,R)

|∇0fδ(y)|dy

≤ C

t

ˆAδ

∣∣∣∣∇0

(dist(y, E)

δ

)∣∣∣∣ dy =C

δt

ˆB(0,R)

|∇0dist(y, E)|dy (6.2)

6.2 The Isoperimetric Profile of H and Pansu’s conjecture 125

for each t < 1. Letting t→ 1, by an application of the co-area formula 1.44,we obtain

|E|34 ≤ C

δ

ˆ δ

0

ˆy∈B(0,R):dist(y,E)=s

|∇0dist(·, E)||∇dist(·, E)|

dHn−1ds,

where dHn−1 denotes the (n−1)-dimensional Hausdorff measure with respectto the backgroud Euclidean metric. The proof is concluded once we let δ → 0and apply Corollary 5.9.

6.2 The Isoperimetric Profile of H and Pansu’sconjecture

We introduce in this section the “isoperimetric profile” of H. This is a familyof sets which realize the equality in the isoperimetric inequality. To presentit we need to define the best isoperimetric constant of the Heisenberg group,which is the best constant Ciso(H) for which the isoperimetric inequality

min|Ω|3/4, |H \ Ω|3/4 ≤ Ciso(H)PH(Ω) (6.3)

holds for each Cacciopoli set in H. We can define it as

Ciso(H) = supΩ

min|Ω|3/4, |H \ Ω|3/4PH(Ω)

, (6.4)

where the supremmum is taken on all Cacciopoli subsets of the Heisenberggroup; in a dual manner we may present it as(Ciso(H)

)−1

=infPH(E) : E ⊂ H is a bounded Cacciopoli set and |E| = 1.(6.5)

Definition 6.2. An isoperimetric profile for H is a family of bounded Cac-ciopoli sets Ωprofile = Ωprofile(V ), with V > 0, such that |Ωprofile(V )| = Vand

|Ωprofile|3/4 = Ciso(H)PH(Ωprofile).

The invariance and scaling properties of the Haar measure and perimetermeasure clearly imply that the sets comprising the isoperimetric profile areclosed under the operations of left translation and group dilation. Pansuconjectured that any set in the isoperimetric profile of H is, up to translationand dilation, a bubble set B(0, R). In Section 3.3 we have studied that B(0, R)is obtained by rotating around the x3-axis a geodesic joining two points at


height ±πR2/2. More precisely, these cylindrically simmetric surfaces haveprofile curve

x3 = fR(r) = ±1

4

(r√R2 − r2 +R2 arccos

r

R

).

Setting u(x) = fR(|z|) − x3, where x = (z, x3), we compute the volume ofB(0, R) in cylindrical coordinates as

|B(0, R)| = 2π

ˆ R

0

r2f(r)dr =3

16π2R4 (6.6)

andPH(B(0, R)) =

ˆ∂B(0,R)

|∇0u||∇u|

dσ = 2

ˆ∂B(0,R)+

|∇0u||∇u|

dσ

where ∂B(0, R)+ denotes the half part of the surface over the C plane, cor-responding to x3 > 0. Applying the area formula 1.42 we then obtain

PH(B(0, R)) = 2

ˆB(0,R)

|∇0u| = 4π

ˆ R

0

r

√f ′(r)2 +

r2

4dr =

1

2π2R3,

where B(0, R) is the ball with center in 0 and radius R in the C-plane andthe second equality follows calculating |∇0u| =

√f ′(r)2 + r2/4.

We can therefore write explicitely the value of the Heisenberg isoperimet-ric constant and the isoperimetric profile, conjectured by Pansu:

Conjecture 6.3 (Pansu).

Ciso(H) =|B(0, R)|3/4

PH(B(0, R))=

33/4

4√π

(6.7)

for any R, and the equality in (6.3) is obtained if and only if Ω is a bubbleset.

6.3 Existence of minimizersIn this section we establish the existence of an isoperimetric profile.

Theorem 6.4 (Leonardi-Rigot). The Heisenberg group H admits an isoperi-metric profile. More precisely, for any V > 0, there exists a bounded setΩ ⊂ H with finite perimeter so that |Ω| = V and

|Ω|3/4 = Ciso(H)PH(Ω).

6.3 Existence of minimizers 127

Proof. The proof of this theorem follows classical lines: one considers a se-quence of sets Ωi ⊂ H with |Ωi| = 1 whose isoperimetric ratios

Ci =|Ωi|3/4

PH(Ωi)=

1

PH(Ωi)

converge to Ciso(H) as i→∞ and examines the existence and properties ofa subconvergent limit. It sufficies to show that

1. the sequence (Ωi) subconverges to a set Ω∞, and

2. |Ω∞| = 1.

Step (1) requires the following compactness theorem.

Theorem 6.5 (Garofalo-Nhieu). Let (Ωi) be a sequence of measurable setsso that

supiPH(Ωi) <∞.

Then (Ωi) subconverges in L1loc(H) to a measurable set Ω∞ with finite perime-

ter.

Proof. In view of Lemma 5.15, Ω = B(x,R) are John domains and we can ap-ply Theorem 5.16, to conclude that BV (Ω) is compactly embedded in L1(Ω).If (Ωi) is a sequence of Caccioppoli sets as in the statement of Theorem 6.5,then the functions fi = χΩi∩Ω lie in BV (Ω) and

||fi||BV (Ω) = ||fi||L1(Ω) + VarH(fi,Ω)

≤ |B(x,R)|+ PH(Ωi,Ω) ≤M <∞,

where M is a suitable constant. In view of Theorem 5.16 there exists asubsequence of fii that converges in L1(Ω) to an f∞ ∈ BV (Ω). Then,considering pointwise convergence, f∞ = χΩ∞ for some set Ω∞ ⊂ Ω. Thelower-semicontinuity of the perimeter functional with respect to L1

loc conver-gence guaranties that PH(Ω∞) ≤ M . This completes the proof of Theorem6.5.

Let us prove step (2): we have to show that the limit set Ω∞ has volumeequal to 1. A priori, in fact, the sets Ωi may become very thin, spread out,and in the limit lose volume at infinity. But this is not the case, because, aswe are going to see, a fixed amount of volume must lie within a ball of radius1. We need the following Lemma.


Lemma 6.6. Let ωH = |B(0, 1)|, and let A be a bounded set with finiteperimeter. Assume that m ∈ (0, ωH/2) is such that |A ∩B(x, 1)| < m for allx ∈ H. Then there exists a constant c > 0 so that

c

(|A|

PH(A)

)4

≤ m. (6.8)

Proof. Consider the points x ∈ H such that |A ∩ B(x, 12)| > 0. Let S be a

set of such points such that the pairwise mutual distance between them isat least 1/2 and let S be maximal with this property. By definition of S theunion of balls B(x, 1/4) with x ∈ S is a disjoint union and is contained inthe 1/2-neighborhoods of A, A1/2, then, since |A1/2| <∞,∣∣∣∣∣⋃

x∈S

B

(x,

1

4

)∣∣∣∣∣ =∑x∈S

∣∣∣∣B(x, 1

4

)∣∣∣∣ ≤ |A 12|.

Hence S is a finite set and we denote by N its cardinality.By the maximality of S, A ⊂ ∩x∈SB(x, 1); in fact, by contradiction,

if there exists y ∈ A \ ∩x∈SB(x, 1), then S ∪ y were a set which properlycontains S and with the same properties of S; this contradicts the maximalityof S. Hence

|A| ≤∑x∈S

|A ∩B(x, 1)|

=∑x∈S

|A ∩B(x, 1)|1/4|A ∩B(x, 1)|3/4

≤ m1/4∑x∈S

|A ∩B(x, 1)|3/4.

By the relative isoperimetric inequality for balls (Lemma 5.18)

|A ∩B(x, 1)|3/4 < CPH(A,B(x, 1)),

(note that min|A ∩B(x, 1)|, |B(x, 1) \ A| = |A ∩B(x, 1)| < m). Hence

|A| ≤ Cm1/4∑x∈S

PH(A,B(x, 1))

≤ Cm1/4∑x∈S

PH(A) = CNm1/4PH(A).

6.3 Existence of minimizers 129

Using this lemma, we achieve the proof of Theorem 6.4. Recall that(Ciso(H)

)−1

=infPH(E) : E ⊂ H is a bounded Cacciopoli set and |E| = 1

and consider a minimizing sequence Ωi satisfying |Ωi| = 1 and

PH(Ωi) ≤ Ciso(H)−1

(1 +

1

i

). (6.9)

By Lemma 6.6 we say that there exist xi ∈ Ωi for i ∈ N, such that

|Ωi ∩B(xi, 1)| ≥ m0 (6.10)

for some absolute constant m0 > 0. Indeed, if no such constant existed, thenfor each k ∈ N, there exists ik such that |Ωik∩B(x, 1)| < 1/k for each x ∈ Ωik

and therefore for each y ∈ H. So, by Lemma 6.6 there exists c > 0 such that

c

(|Ωik |

PH(Ωik)

)4

≤ 1/k.

Hence1

k1/4≥ |Ωik |PH(Ωik)

≥ |Ωik |

C−1iso (H)

(1 + 1

ik

) = Ciso(H)ik

ik + 1≥ Ciso(H)

2.

Letting k →∞ we reach the contradiction Ciso(H) = 0.Return to equation (6.10); right translating each Ωi by xi, we may assume

that Ωi contains the origin and that

|Ωi ∩B(0, 1)| ≥ m0. (6.11)

Theorem 6.5 assures the existence of Ω∞ with finite perimeter such that Ωi

subconverges to Ω∞ in L1loc(H). Lower semi-continuity of the perimeter and

inequality (6.9) yield

PH(Ω∞) ≤ lim infi→∞

PH(Ωi) ≤ lim infi→∞

Ciso(H)−1

(1 +

1

i

)= Ciso(H)−1.

The choice of m0 implies m0 ≤ limi→∞ |Ωi ∩ B(0, 1)| = |Ω∞ ∩ B(0, 1)| andlower semi-continuity gives

|Ω∞| ≤ lim infi→∞

|Ωi| ≤ 1. (6.12)

Taking all of this together yields

m0 ≤ |Ω∞| ≤ 1.

We complete the proof by showing that Ω∞ is essentially bounded, i.e. byproving that


(3) if r ≥ 2 is such that |Ω∞ ∩B(0, r)| < 1 then r ≤ R0 <∞.

Indeed, if we prove (3), then, by (6.12) and since (Ωi) subconvergies to Ω∞in L1

loc, we have1 ≥ |Ω∞| ≥ |Ω∞ ∩B(0, R0)| = 1

and step (2) is proven.To prove (3), let us introduce mi(ρ) = |Ωi ∩ B(0, ρ)| and let m∞(ρ) =

|Ω∞ ∩B(0, R)|. By assumption, m∞(r) < 1. Using the relation between the(local) perimeter and the rate of change of the (local) volume of a set (seeTheorem 2.12)

mi(ρ)3/4 ≤ Ciso(H)PH(Ωi ∩B(0, ρ)) ≤ Ciso(H)(PH(Ωi, B(0, ρ)) +m′i(ρ))

and

(1−mi(ρ))3/4 ≤ Ciso(H)PH(Ωi ∩B(0, ρ)c)

≤ Ciso(H)(PH(Ωi,H \B(0, ρ)) +m′i(ρ)

)for almost every ρ > 0. Using (6.9) we deduce

mi(ρ)3/4 + (1−mi(ρ))3/4 ≤ Ciso(H) (PH(Ωi) + 2m′i(ρ))

≤ 1 +1

i+ 2Ciso(H)m′i(ρ)

(6.13)

for almost every ρ > 0.Let us analyze the general function Φ(x) = x3/4 + (1 − x)3/4 − 1. Let

ε = minm0,1−m∞(r)

2. Since m∞(r) < 1 by assumption, then

m∞(r) <m∞(r) + 1

2.

Hence, since |Ωi| locally convergies to |Ω∞|, for sufficiently large i we have

mi(r) = |Ωi ∩B(0, r)| ≤ m∞(r) + 1

2≤ 1− ε.

Moreover, by choosing a greater i if necessary, we can suppose

Φ(x) ≥ 1

ifor all x ∈ [ε, 1− ε].

Sinceε ≤ m0 ≤ mi(1) ≤ mi(ρ) ≤ mi(r) ≤ 1− ε

6.4 A C2 isoperimetric profile has constant horizontal mean curvature 131

for all such i and all 1 ≤ ρ ≤ r (see (6.11)), we may rewrite the differentialinequality (6.13) in the form

1 ≤ Cm′i(ρ)

Φ(mi(ρ))− 1i

, a.e. ρ ∈ [1, r].

Integrating from r/2 to r gives

r

2≤ C

ˆ r

r2

m′i(ρ)dρ

Φ(mi(ρ))− 1i

= C

ˆ mi(r)

mi(r/2)

dx

Φ(x)− 1i

≤ C

ˆ 1−ε

ε

dx

Φ(x)− 1i

−→ C

ˆ 1

0

dx

Φ(x)<∞

as i→∞. Then r ≤ 2C´ 1

0Φ(x)−1dx =: R0 as desired.

6.4 A C2 isoperimetric profile has constant hor-izontal mean curvature

In this section we prove that if the isoperimetric profile of H is C2 smooththen it necessarily has constant horizontal mean curvature away from thecharacteristic set. The proof is based on the first variation of the perimeterseen is Section 5.3, with an additional volume constraint. We refer to thepaper [17].

Proposition 6.7. Let Ω ⊂ H be a bounded open set enclosed by an orientedC2 immersed surface S with g1-Riemannian normal ν1. Denote by dµ theperimeter measure defined in Corollary 5.9, and by H0 the horizontal meancurvature of S. If S is volume-preserving and perimeter stationary, then ithas constant horintal mean curvature in S \ Σ(S) equal to

H0 =

´SH0dµ

PH(S). (6.14)

Proof. We use the notation introduced in Section 5.3.2 and consider varia-tions Ωt = φt(Ω) along a vector field U defined on S. The hyphotesis that Sis volume-preserving and perimeter stationary means that

d

dtPH(St)

∣∣∣∣t=0

=

ˆS

u(divSνH)dσ −ˆS

divS(u(νH)tang)dσ = 0 (6.15)

for each U such thatd

dt|Ωt|∣∣∣∣t=0

=

ˆΩ

d

dt|Jφt|

∣∣∣∣t=0

dV =

ˆΩ

divSUdV = −ˆS

〈U, ν1〉dσ = 0, (6.16)


where we have used formula (5.55) for the calculation of d|Jφt |/dt and thedivergence theorem.

Define H0 :=´SH0dµ/PH(S). By the fundamental lemma of the calculus

of variations, we complete the proof if we show thatˆS\Σ(S)

(H0 − H0)adµ = 0 for each a ∈ C∞0 (S \ Σ(S)). (6.17)

To each function a ∈ C∞0 (S \ Σ(S)) is associated a vector field U withcompact support in S \Σ(S), by defining U = aνH . Then by equations (6.15)and (6.16) and using Remark 5.26, we have

d

dtPH(St)

∣∣∣∣t=0

=

ˆS

u(divSνH)dσ = 0

wheneverd

dt|Ωt|∣∣∣∣t=0

= −ˆS

〈aνH , ν1〉dσ = −ˆS

adµ = 0.

We observe that H0 = −divSνH . Indeed let e1, e2 be an orthonormal base ofTpS such that e1, e2, ν1 is positive oriented. Calculating the divergence withrespect to such basis, since ν1 = |NH |νH + 〈ν1, X3〉X3, we obtain

H0 = −divνH = −divSνH − 〈Dν1νH , ν1〉 = −divSνH .

By this observation, we have that if a ∈ C∞0 (S\Σ(S)) is such that´Sadµ = 0

thenˆS\Σ(S)

(H0 − H0)adµ = −ˆS\Σ(S)

divSνH a dµ−ˆS\Σ(S)

H0 a dµ = 0

and (6.17) is verified in this case.Take now a general function a ∈ C∞0 (S \Σ(S)) and the associated vector

field U = aνH . Let Snn∈N be approximating open sets such that Sn ⊂Sn+1 ⊂ S\Σ(S) and

⋃n Sn = S\Σ(S) and take functions ϕn ∈ C∞0 (S\Σ(S))

such that ϕn = 1 on Sn. Let n be such that spt(a) ⊂ Sn and define

a := a− ϕn

´S\Σ(S)

adµ

PH(S).

In this way we have´Sadµ = 0, so that (6.17) holds for a. Hence

ˆS\Σ(S)

(H0 − H0)adµ =

ˆS\Σ(S)

(H0 − H0)ϕn

´S\Σ(S)

adµ

PH(S)dµ

=

´S\Σ(S)

adµ

PH(S)

ˆS\Σ(S)

(H0 − H0)ϕndµ

which tends to 0 as n→∞. This completes the proof of (6.17).

6.5 Recent results 133

6.5 Recent resultsWe collect in this section some of the recent results towards the solutionof the isoperimetric problem in H. They are partial solutions, which provePansu’s conjecture in some restricted class of sets and give some evidence tosupport Pansu’s thesis. We don’t give the proofs here, which can be foundin the respective papers or, sketched, in [6]. Our goal is to cite the recentdevelopments in the area, without pretend to be exhaustive, but with intentto put the basis for a future deeper study.

6.5.1 Existence and characterization of minimizers withadditional symmetries

Danielli, Garofalo and Nhieu have given in [8] a solution of the isoperimetricproblem in a special class of domains, having suitable symmetry and regu-larity properties.

Let us introduce the half-spaces H+ = (z, x3) ∈ H : x3 > 0 and H− =(z, x3) : x3 < 0 and consider the collection

E = E ⊂ H : E satisfies properties (i) and (ii)

where

(i) |E ∩H+| = |E ∩H−|, and

(ii) there exist R > 0, and functions u, v : BR → [0,∞), with u, v ∈C1(BR) ∩ C(BR), u = v = 0 on ∂BR, and such that

∂E ∩H+ = (z, x3) ∈ H+ : |z| < R, x3 = u(z)and ∂E ∩H− = (z, x3) ∈ H− : |z| < R, x3 = −v(z).

Here BR = B((0, 0), R) denotes the ball of radius R in R2.Note that the functions u and v which describe the upper and lower

portions of a set E ∈ E may be different. Besides C1 smoothness, and thefact that their common domain is a metric ball, no additional assumptionsare made on them.

Theorem 6.8 (Danielli-Garofalo-Nhieu). Let V > 0, and define R > 0 sothat V = |B(0, R)| (see formula (6.6)). Then the variational problem

minE∈E:|E|=V

PH(E)

has a unique solution in E given by the bubble set B(0, R).


6.5.2 The C2 isoperimetric profile in HRitoré and Rosales proved in [17] that the bubble sets are the isoperimetricminimizers among the sets with C2 boundaries.

Theorem 6.9 (Ritoré-Rosales). If Ω is an isoperimetric region in H whichis bounded by a C2 smooth surface S, then S is congruent (i.e. equivalent bythe composition of a Heisenberg isometry and dilation) to the boundary of abubble set.

6.5.3 The convex isoperimetric profile of HMonti and Rickly have proved in [15] Pansu’s conjecture among Euclideanconvex sets.

Theorem 6.10 (Monti-Rickly). If Ω is an (Euclidean) convex region in Hwhich is an isoperimetric set, then Ω is congruent to a bubble set.

Note that from the point of view of the regularity the assumption inTheorem 6.10 is stricktly weaker than that in Theorem 6.9, since it is requiredonly that the sets are convex, without assuming any smoothness properties.

6.5.4 The isoperimetric profile in a reduced symmetrysetting

We cite a very recent result due to Ritoré, that can be found in [16]. Wedenote by D a closed disk in the C-plane centered in the origin and by C therelative vertical cylinder over D in H.

Theorem 6.11 (Ritoré). Let E ⊂ H be a bounded set with finite perimetersuch that

D ⊂ E ⊂ C.

Then its perimeter is greater or equal to that of the bubble set with the samevolume as E. The equality holds only for bubble sets.

Notice that this new result generalizes the preceding Theorem 6.8, indeedany set E, for which properties (i) and (ii) of section 6.5.1 hold, satisfies thehyphothesis of Theorem 6.11 with D = BR, but any smoothness assumptionis required now.

Conclusions

In this survey we have collected, in a synthetical and possibly ordered way,the existent literature on the isoperimetric problem in H. At the beginningwe have approached the question in the Euclidean space, with the intent togive a motivation for the successive generalization and to introduce someof the classical notions, which we have transferred then on the Heisenberggroup.

The most interesting study is developed in the second part of the thesis.Here we have introduced the first Heisenberg group and the geometric mea-sure theory on it, reaching the limit of what is known about the isoperimetricproblem in H.

Nowadays we have a lot of solutions of the problem in several specialcases. What we need now is something like the contribution realized by DeGiorgi in 1954 in the Euclidean space: a kind of revolution which will allowus to go back from the most general case to those already known.

135

I declair herewith that I have written this work indipendently and byusing the indicated resources.

Trento, July 7, 2011.

Elisa Paoli

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