karenuhlenbeck andthecalculusofvariationssupported variations of is a second order differential...
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Karen Uhlenbeckand the Calculus of Variations
Simon DonaldsonIn this article we discuss the work of Karen Uhlenbeck,mainly from the 1980s, focused on variational problemsin differential geometry.
The calculus of variations goes back to the 18th century.In the simplest setting we have a functional
πΉ(π’) = β«Ξ¦(π’,π’β²)ππ₯,
Simon Donaldson is a permanent member of the Simons Center for Geometryand Physics at Stony Brook University and Professor at Imperial College London.His email address is [email protected] by Notices Associate Editor Chikako Mese.
For permission to reprint this article, please contact:[email protected].
DOI: https://doi.org/10.1090/noti1806
defined on functions π’ of one variable π₯. Then the condi-tion that β± is stationary with respect to compactlysupported variations of π’ is a second order differentialequationβthe EulerβLagrange equation associated to thefunctional. One writes
πΏβ± = β«πΏπ’ π(π’) ππ₯,
where
π(π’) = πΞ¦ππ’ β π
ππ₯πΞ¦ππ’β² . (1)
The EulerβLagrange equation is π(π’) = 0. Similarlyfor vector-valued functions of a variable π₯ β ππ. Depend-ing on the context, the functions would be required to sat-isfy suitable boundary conditions or, as in most of this ar-ticle, might be defined on a compact manifold rather thana domain inππ, and π’might not exactly be a function but
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Figure 1. Finding a critical point with a minimaxsequence.
a more complicated differential geometric object such as amap, metric, or connection. One interprets π(π’), definedas in (1), as the derivative at π’ of the functional β± on asuitable infinite dimensional space π³ and the solutionsof the EulerβLagrange equation are critical points of β±.
A fundamental question is whether one can exploit thevariational structure to establish the existence of solutionsto EulerβLagrange equations. This question came into fo-cus at the beginning of the 20th century. Hilbertβs 22ndproblem from 1904 was:
Has not every regular variational problem a solution, pro-vided certain assumptions regarding the given boundary condi-tions are satisfied?
If the functional β± is bounded below one might hopeto find a solution of the EulerβLagrange equation whichrealises the minimum of β± on π³. More generally, onemight hope that ifπ³ has a complicated topology then thiswill force the existence ofmore critical points. For example,if Ξ is a homotopy class of maps from the π-sphere ππ
to π³ one can hope to find a critical point via a minimaxsequence, minimising over maps π β Ξ the maximum ofβ±(π(π£)) over points π£ β ππ.
In Hilbertβs time the only systematic results were in thecase of dimension π = 1 and for linear problems, suchas the Dirichlet problem for the Laplace equation. The de-velopment of a nonlinear theory in higher dimensions hasbeen the scene for huge advances over the past century andprovides the setting for much of Karen Uhlenbeckβs work.
Harmonic Maps in Dimension 2We begin in dimension 1 where geodesics in a Riemann-ian manifold are classical examples of solutions to a vari-ational problem. Here we take π to be a compact, con-nected, Riemannianmanifold and fix two pointsπ,π inπ.We takeπ³ to be the space of smooth pathsπΎ βΆ [0, 1] β π
with πΎ(0) = π,πΎ(1) = π, and the energy functional
β±(πΎ) = β«1
0|βπΎ|2,
where the norm of the βvelocity vectorβ βπΎ is computedusing the Riemannian metric on π. The EulerβLagrangeequation is the geodesic equation, in local co-ordinates,
πΎβ³π ββ
π,πΞππππΎβ²
ππΎβ²π = 0,
where the βChristoffel symbolsβ Ξπππ are given by well-
known formulae in terms of the metric tensor and itsderivatives. In this case the variational picture works aswell as one could possibly wish. There is a geodesic fromπ to π minimising the energy. More generally one can useminimax arguments and (at least if π and π are taken ingeneral position) the Morse theory asserts that the homol-ogy of the path space π³ can be computed from a chaincomplex with generators corresponding to the geodesicsfrom π to π. This can be used in both directions: factsfrom algebraic topology about the homology of the pathspace give existence results for geodesics, and, conversely,knowledge of the geodesics can feed into algebraic topol-ogy, as in Bottβs proof of his periodicity theorem.
The existence of a minimising geodesic between twopoints can be proved in an elementaryway and the originalapproach of Morse avoided the infinite dimensional pathspace π³, working instead with finite dimensional approx-imations, but the infinite-dimensional picture gives thebest starting point for the discussion to follow. The basicpoint is a compactness property: any sequence πΎ1, πΎ2,β¦ inπ³ with bounded energy has a subsequence which converges inπΆ0 to some continuous path from π to π. In fact for a pathπΎ β π³ and 0 β€ π‘1 < π‘2 β€ 1 we have
π(πΎ(π‘1), πΎ(π‘2)) β€ β«π‘2
π‘1|βπΎ| β€ β±(πΎ)1/2 |π‘1 β π‘2|1/2,
where the last step uses the CauchyβSchwartz inequality.Thus a bound on the energy gives a 1
2 -HΓΆlder bound onπΎ and the compactness property follows from the Ascoli-Arzela theorem.
In the same vein as the compactness principle, one canextend the energy functional β± to a completion π³ of π³which is an infinite dimensional Hilbert manifold, and el-ements of π³ are still continuous (in fact 1
2 -HΓΆlder con-tinuous) paths in π. In this abstract setting, Palais andSmale introduced a general βCondition Cβ for functionalson Hilbert manifolds, which yields a straightforward vari-ational theory. (This was extended to Banach manifoldsin early work of Uhlenbeck [24].) The drawback is that,beyond the geodesic equations, most problems of interestin differential geometry do not satisfy this PalaisβSmalecondition, as illustrated by the case of harmonic maps.
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The harmonic map equations were first studied system-atically by Eells and Sampson [5]. We now take π,πto be a pair of Riemannian manifolds (say compact) andπ³ = Maps(π,π) the space of smooth maps. The energyof a map π’ βΆ π β π is given by the same formula
β±(π’) = β«π|βπ’|2,
where at each point π₯ β π the quantity |βπ’| is the stan-dard norm defined by the metrics onπππ₯ andπππ’(π₯). Inlocal co-ordinates the Euler Lagrange equations have theform
Ξππ’π ββππ
Ξπππβπ’πβπ’π = 0, (2)
where Ξπ is the Laplacian on π. This is a quasi-linearelliptic system, with a nonlinear term which is quadraticin first derivatives. The equation is the natural commongeneralisation of the geodesic equation inπ and the linearLaplace equation on π.
The key point now is that when dim π > 1 the energyfunctional does not have the same compactness property.This is bound up with Sobolev inequalities and, most funda-mentally, with the scaling behaviour of the functional. Toexplain, in part, the latter consider varying the metric ππon π by a conformal factor π. So π is a strictly positivefunction on π and we have a new metric Μππ = π2ππ.Then one finds that the energy ΜπΉ defined by this new met-ric is
β±Μ(π’) = β«ππ2βπ|βπ’|2,
where π = dimπ. In particular if π = 2 we have β±Μ = β±.Now take π = π2 with its standard round metric and π βΆπ2 β π2 a MΓΆbius map. This is a conformal map and itfollows from the above that for any π’ βΆ π2 β π we haveβ±(π’βπ) = β±(π’). Since the space of MΓΆbius maps is notcompact we can construct a sequence of maps π’βππ withthe same energy but with no convergent subsequence.
We now recall the Sobolev inequalities. Let π be asmooth real valued function on ππ, supported in the unitball. We take polar co-ordinates (π, π) in ππ, with π βππβ1. For any fixed π we have
π(0) = β«1
π=0
ππππππ.
So, integrating over the sphere,
π(0) = 1ππ
β«ππβ1
β«1
π=0
ππππππππ,
where ππ is the volume of ππβ1. Since the Euclidean vol-ume form is πππ₯ = ππβ1ππππ we can write this as
π(0) = 1ππ
β«π΅π
|π₯|1βπ ππππ πππ₯.
The function π₯ β¦ |π₯|1βπ is in πΏπ over the ball for anyπ < π/πβ1. Let π be the conjugate exponent, with πβ1+πβ1 = 1, so π > π. Then HΓΆlderβs inequality gives
|π(0)| β€ πΆπββπβπΏπ
where πΆπ is πβ1π times the πΏπ(π΅π) norm of π₯ β¦ |π₯|1βπ.
The upshot is that for π > π there is a continuous em-bedding of the Sobolev space πΏπ
1βobtained by completingin the norm ββπβπΏπβinto the continuous functions onthe ball. In a similar fashion, if π < π there is a contin-uous embedding πΏπ
1 β πΏπ for the exponent range π β€ππ/(π β π), which is bound up with the isoperimetricinequality in ππ. The arithmetic relating the exponentsand the dimension π reflects the scaling behaviour of thenorms. If we define ππ(π₯) = π(ππ₯), for π β₯ 1, then
βππβπΆ0 = βπβπΆ0 ,βππβπΏπ = πβπ/πβπβπΏπ ,βππβπΏπ
1= π1βπ/πβπβπΏπ
1.
It follows immediately that there can be no continuousembedding πΏπ
1 β πΆ0 for π < π or πΏπ1 β πΏπ for π >
ππ/(π β π).The salient part of this discussion for the harmonicmap
theory is that the embedding πΏπ1 β πΆ0 fails at the criti-
cal exponent π = π. (To see this, consider the functionlog log πβ1.) Taking π = 2 this means that the energy ofa map from a 2-manifold does not control the continuityof the map and the whole picture in the 1-dimensionalcase breaks down. This was the fundamental difficulty ad-dressed in the landmark paper [12] of Sacks and Uhlen-beckwhich showed that, with a deeper analysis, variationalarguments can still be used to give general existence re-sults.
Rather thanworking directlywithminimising sequences,Sacks and Uhlenbeck introduced perturbed functionals onπ³ = Maps(π,π) (with π a compact 2-manifold):
β±πΌ(π’) = β«π(1 + |βπ’|2)πΌ.
For πΌ > 1 we are in the good Sobolev range, just as in thegeodesic problem. Fix a connected component π³0 of π³(i.e. a homotopy class of maps from π to π). For πΌ > 1there is a smooth map π’πΌ realising the minimum of β±πΌon π³0. This map π’πΌ satisfies the corresponding EulerβLagrange equation, which is an elliptic PDE given by a vari-ant of (2). The strategy is to study the convergence ofπ’πΌ asπΌ tends to 1. The main result can be outlined as follows.To simplify notation, we understand that πΌ runs over asuitable sequence decreasing to 1.
β’ There is a finite set π β π such that the π’πΌ con-verge in πΆβ over π\π.
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Figure 2. Schematic representation of βbubbling.β
β’ The limit π’ of the maps π’πΌ extends to a smoothharmonic map from π to π (which could be aconstant map).
β’ If π₯ is a point inπ such that theπ’πΌ do not convergeto π’ over a neighbourhood of π₯ then there is anon-trivial harmonic map π£ βΆ π2 β π such thata suitable sequence of rescalings of the π’πΌ near π₯converge to π£.
In brief, the only way that the sequence π’πΌ may fail toconverge is by forming βbubbles,β in which small discs inπ are blown up into harmonic spheres inπ. We illustratethe meaning of this bubbling through the example of ra-tional maps of the 2-sphere. (See also the expository article[11].) For distinct points π§1,β¦π§π in π and non-zero co-efficients ππ consider the map
π’(π§) =πβπ=1
πππ§ β π§π
,
which extends to a degree π holomorphic map π’ βΆ π2 βπ2 with π’(β) = 0. These are in fact harmonic maps, withthe same energy 8ππ. Take π§1 = 0,π1 = π. If we makeπ tend to 0, with the other ππ fixed, then away from 0 themaps converge to the degree (πβ1)mapβπ
2 ππ(π§βπ§π)β1.On the other hand if we rescale about 0 by setting
οΏ½ΜοΏ½(π§) = π’(ππ§) = 1π§ +
πβπ=2
ππππ§ β π§π
the rescaled maps converge (on compact subsets of π) tothe degree 1 map
π£(π§) = 1π§ β π
with π = βπ2 ππ/π§π.
A key step in the Sacks andUhlenbeck analysis is a βsmallenergyβ statement (related to earlier results ofMorrey). Thissays that there is some π > 0 such that if the energy of amap π’πΌ on a small disc π· β π is less than π then there
are uniform estimates of all derivatives of π’πΌ over the half-sized disc. The convergence result then follows from a cov-ering argument. Roughly speaking, if the energy of themap on π is at most πΈ then there can be at most a fixednumber πΈ/π of small discs on which the map is not con-trolled. The crucial point is that π does not depend on thesize of the disc, due to the scale invariance of the energy.To sketch the proof of the small energy result, consider asimpler model equation
Ξπ = |βπ|2, (3)
for a function π on the unit disc in π. Linear elliptic the-ory, applied to the Laplace operator, gives estimates of theschematic form
ββπβπΏπ1β€ πΆβΞπβπΏπ + LOT,
where LOT stands for βlower order termsβ in which (forthis sketch) we include the fact that one will have to restrictto an interior region. Take for example π = 4/3. Thensubstituting into the equation (3) we have
ββπβπΏ4/31
β€ πΆβ|βπ|2βπΏ4/3 + LOT β€ πΆββπβ2πΏ8/3 + LOT.
Now in dimension2wehave a Sobolev embeddingπΏ4/31 β
πΏ4 which yields
ββπβπΏ4 β€ πΆββπβ2πΏ8/3 + LOT.
On the other hand, HΓΆlders inequality gives the interpola-tion
ββπβπΏ8/3 β€ ββπβ1/2πΏ2 ββπβ1/2
πΏ4 .So, putting everything together, one has
ββπβπΏ4 β€ πΆββπβπΏ4ββπβπΏ2 + LOT.If ββπβπΏ2 β€ 1/2πΆ we can re-arrange this to get
ββπβπΏ4 β€ LOT.In other words, in the small energy regime (with βπ =1/2πΆ) we can bootstrap using the equation to gain an es-timate on a slightly stronger norm (πΏ4 rather than πΏ2) andone continues in similar fashion to get interior estimateson all higher derivatives.
This breakthrough work of Sacks and Uhlenbeck ties inwith many other developments from the same era, someof which we discuss in the next section and some of whichwe mention briefly here.
β’ Inminimal submanifold theory: whenπ is a2-spherethe image of a harmonic map is a minimal surfaceinπ (ormore precisely a branched immersed sub-manifold). In this way, Sacks and Uhlenbeck ob-tained an important existence result for minimalsurfaces.
β’ In symplectic topology the pseudoholomorphiccurves, introduced by Gromov in 1986, are exam-ples of harmonic maps and a variant of the Sacksβ
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Uhlenbeck theory is the foundation for all the en-suing developments (see, for example, [7]).
β’ In PDE theory other βcritical exponentβ variationalproblems, in which similar bubbling phenomenaarise, were studied intensively (see for examplethe work of Brezis and Nirenberg [4]).
β’ In Riemannian geometry the Yamabe problem offinding a metric of constant scalar curvature in agiven conformal class (on a manifold of dimen-sion 3 or more) is a critical exponent variationalproblem for the Einstein-Hilbert functional (theintegral of the scalar curvature), restricted to met-rics of volume 1. Schoen proved the existence ofa minimiser, completing the solution of the Yam-abe problem, using a deep analysis to rule out therelevant bubbling [14].
A beautiful application of the SacksβUhlenbeck theorywas obtained in 1988 byMicallef andMoore [8]. The argu-ment is in the spirit of classical applications of geodesicsin Riemannian geometry. Micallef and Moore considereda curvature condition on a compact Riemannian manifoldπ (of dimension at least 4) of having βpositive curvatureon isotropic 2-planes.β They proved that if π satisfies thiscondition and is simply connected then it is a homotopysphere (and thus, by the solution of the PoincarΓ© conjec-ture, is homeomorphic to a sphere). The basic point isthat a non-trivial homotopy class in ππ(π) gives a non-trivial element of ππβ2(π³), where π³ = Maps(π2,π),which gives a starting point for a minimax argument. Ifπ is not a homotopy sphere then by standard algebraictopology there is some π with 2 β€ π β€ 1
2dimπ suchthat ππ(π) β 0, which implies that ππβ2(π³) is non-trivial. By developingmini-max arguments with the SacksβUhlenbeck theory, using the perturbed energy functional,Micallef and Moore were able to show that this leads to anon-trivial harmonic map π’ βΆ π2 β π of index at mostπ β 2. (Here the index is the dimension of the space onwhich the second variation is strictly negative.) On theother hand the LeviβCivita connection ofπ defines a holo-morphic structure on the pull-back π’β(ππ β π) of thecomplexified tangent bundle. By combining results aboutholomorphic bundles over π2 and aWeitzenbock formula,in which the curvature tensor of π enters, they show thatthe index must be at least 1
2dimπ β 32 and thus derive a
contradiction.If the sectional curvature of π is β1
4 -pinchedβ (i.e. lies
between 14 and 1 everywhere) then π has positive curva-
ture on isotropic 2-planes. Thus the Micallef and Mooreresult implies the classical sphere theorem of Berger andKlingenberg, whose proof was quite different. In turn,much more recently, Brendle and Schoen [3] proved that a
(simply connected) manifold satisfying this isotropic cur-vature condition is in fact diffeomorphic to a sphere. Theirproof was again quite different, using Ricci flow.
Gauge Theory in Dimension 4From the late 1970s, mathematics was enriched by ques-tions inspired by physics, involving gauge fields and theYang-Mills equations. These developments weremany-faceted and here we will focus on aspects related tovariational theory. In this set-up one considers a fixed Rie-mannian manifold π and a πΊ-bundle π β π where πΊis a compact Lie group. The distinctive feature, comparedto most previous work in differential geometry, is that π isan auxiliary bundle not directly tied to the geometry of π.The basic objects of study are connections on π. In a localtrivialisation π of π a connection π΄ is given by a Lie(πΊ)-valued 1-form π΄π. For simplicity we take πΊ to be a matrixgroup, so π΄π is a matrix of 1-forms. The fundamental in-variant of a connection is its curvature πΉ(π΄) which in thelocal trivialisation is given by the formula
πΉπ = ππ΄π +π΄π β§π΄π.The Yang-Mills functional is
β±(π΄) = β«π|πΉ(π΄)|2,
and the EulerβLagrange equation is πβπ΄πΉ = 0 where πβ
π΄ isan extension of the usual operator πβ from 2-forms to 1-forms, defined using π΄. This Yang-Mills equation is a non-linear generalisation ofMaxwellβs equations of electromag-netism (which one obtains taking πΊ = π(1) and passingto Lorentzian signature).
In the early 1980s, Uhlenbeck proved fundamental an-alytical results which underpin most subsequent work inthis area. Themain case of interest is when themanifoldπhas dimension 4 and the problem is then of critical expo-nent type. In this dimension the Yang-Mills functional isconformally invariant and there are many analogies withthe harmonic maps of surfaces discussed above. A newaspect involves gauge invariance, which does not have ananalogy in the harmonic maps setting. That is, the infinitedimensional group π’ of automorphisms of the bundle πacts on the space π of connections, preserving the Yang-Mills functional, so the natural setting for the variationaltheory is the quotient space π/π’. Locally we are free tochange a trivialisation π0 by the action of aπΊ-valued func-tion π, which will change the local representation of theconnection to
π΄ππ0 = ππ(πβ1) + ππ΄π0πβ1.While this action of the gauge group π’ may seem un-
usual, within the context of PDEs, it represents a funda-mental phenomenon in differential geometry. In study-ing Riemannian metrics, or any other kind of structure, on
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a manifold one has to take account of the action of theinfinite-dimensional group of diffeomorphisms: for exam-ple the round metric on the sphere is only unique up tothis action. Similarly, the explicit local representation of ametric depends on a choice of local co-ordinates. In factdiffeomorphism groups are much more complicated thanthe gauge group π’. In another direction one can have inmind the case of electromagnetism, where the connection1-form π΄π is equivalent to the classical electric and mag-netic potentials on space-time. The π’-action correspondsto the fact that these potentials are not unique.
Two papers of Uhlenbeck [25], [26] addressed both ofthese aspects (critical exponent and gauge choice). The pa-per [25] bears on the choice of an βoptimalβ local trivialisa-tionπ of the bundle over a ball π΅ β π given a connectionπ΄. The criterion that Uhlenbeck considers is the Coulombgauge fixing condition: πβπ΄π = 0, supplemented withthe boundary condition that the pairing of π΄π with thenormal vector vanishes. Taking π = ππ0, for some arbi-trary trivialisation π0, this becomes an equation for the πΊ-valued function π which is a variant of the harmonic mapequation, with Neumann boundary conditions. In fact theequation is the EulerβLagrange equation associated to thefunctional βπ΄πβπΏ2 , on local trivialisations π. The Yang-Mills equations in such a Coulomb gauge form an ellipticsystem. (Following the remarks in the previous paragraph;an analogous discussion for Riemannian metrics involvesharmonic local co-ordinates, in which the Einstein equa-tions, for example, form an elliptic system.)
The result proved by Uhlenbeck in [25] is of βsmall en-ergyβ type. Specialising to dimension 4 for simplicity, sheshows that there is an π > 0 and a constant πΆ such that ifβπΉβπΏ2(π΅) < π there is a Coulomb gauge π over π΅ in which
ββπ΄πβπΏ2 + βπ΄πβπΏ4 β€ πΆβπΉβπΏ2 .
The strategy of proof uses the continuity method, appliedto the family of connections given by restricting to smallerballs with the same centre, and the key point is to obtain apriori estimates in this family. The PDE arguments derivingthese estimates have some similarity with those sketchedin Section βHarmonic maps in dimension 2β above. Animportant subtlety arises from the critical nature of theSobolev exponents involved. Ifπ = ππ0 then anπΏ2 boundon βπ΄π gives an πΏ2 bound on the second derivative of πbut in dimension 4 this is the borderline exponent wherewe do not get control over the continuity of π. That makesthe nonlinear operations such as π β¦ πβ1 problematic.Uhlenbeck overcomes this problem by working with πΏπ
for π > 2 and using a limiting argument.In the companion paper [26], Uhlenbeck proves a
renowned βremoval of singularitiesβ result. The statementis that a solution π΄ of the Yang-Mills equations over the
punctured ball π΅4 \{0} with finite energy (i.e. with curva-ture πΉ(π΄) in πΏ2) extends smoothly over 0 in a suitable lo-cal trivialisation. One important application of this is thatfinite-energy Yang-Mills connections overπ4 extend to theconformal compactification π4. We will only attempt togive the flavour of the proof. Given our finite-energy solu-tion π΄ over the punctured ball let
π(π) = β«|π₯|<π
|πΉ(π΄)|2,
for π < 1. Then the derivative is
ππππ = β«
|π₯|=π|πΉ(π΄)|2.
The strategy is to express π(π) also as a boundary integral,plus lower order terms. To give a hint of this, considerthe case of an abelian groupπΊ = π(1), so the connectionformπ΄π is an ordinary1-form, the curvature is simplyπΉ =ππ΄π, and the Yang-Mills equation is πβπΉ = 0. Fix smallπ < π and work on the annular regionπ where π < |π₯| <π. We can integrate by parts to write
β«π|πΉ|2 = β«
πβ¨ππ΄π, πΉβ© = β«
πβ¨π΄π, πβπΉβ©+β«
πππ΄πβ§βπΉ.
Since πβπΉ = 0 the first term on the right hand side van-ishes. If one can show that the contribution from the innerboundary |π₯| = π tends to 0 with π then one concludesthat
π(π) = β«|π₯|=π
π΄π β§βπΉ.
In the nonabelian case the same discussion applies up tothe addition of lower-order terms, involvingπ΄πβ§π΄π. Thestrategy is then to obtain a differential inequality of theshape
π(π) β€ 14π
ππππ + LOT, (4)
by comparing the boundary terms over the 3-sphere. Thisdifferential inequality integrates to give π(π) β€ πΆπ4 andfrom there it is relatively straightforward to obtain an πΏβ
bound on the curvature and to see that the connection canbe extended over 0. The factor 1
4 in (4) is obtained froman inequality over the 3-sphere. That is, any closed 2-formπ on π3 can be expressed as π = ππ where
βπβ2πΏ2(π3) β€
14βπβ2
πΏ2(π3).The main work in implementing this strategy is to con-struct suitable gauges over annuli in which the lower order,nonlinear terms π΄π β§π΄π are controlled.
These results of Uhlenbeck lead to a Yang-Mills analogueof the SacksβUhlenbeck picture discussed in the previoussection. This was not developed explicitly in Uhlenbeckβs1983 papers [25], [26] but results along those lines wereobtained by her doctoral student S. Sedlacek [16]. Let π
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be the infimum of the Yang-Mills functional on connec-tions on π β π, where π is a compact 4-manifold. Letπ΄π be a minimising sequence. Then there is a (possiblydifferent) πΊ-bundle Μπ β π, a Yang-Mills connection π΄βon Μπ, and a finite set π β π such that, after perhaps pass-ing to a subsequence πβ², theπ΄πβ² converge toπ΄β overπ\π.(More precisely, this convergence is in πΏ2
1,loc and implic-itly involves a sequence of bundle isomorphisms of π andΜπ over π\π.) If π₯ is a point in π such that the π΄πβ² do
not converge to π΄β over a neighbourhood of π₯ then oneobtains a non-trivial solution to the Yang-Mills equationsover π4 by a rescaling procedure similar to that in the har-monic map case. Similar statements apply to sequences ofsolutions to the Yang-Mills equations over π and in par-ticular to sequences of Yang-Mills βinstantons.β These spe-cial solutions solve the first order equation πΉ = Β±βπΉ andare closely analogous to the pseudoholomorphic curves inthe harmonic map setting. Uhlenbeckβs analytical resultsunderpinned the applications of instanton moduli spacesto 4-manifold topology which were developed vigorouslythroughout the 1980s and 1990sβjust as for pseudoholo-morphic curves and symplectic topology. But we will con-centrate here on the variational aspects.
For simplicity fix the group πΊ = ππ(2); the ππ(2)-bundles π over π are classified by an integer π = π2(π)and for each π we have a moduli space β³π (possiblyempty) of instantons (where the sign in πΉ = Β± β πΉ de-pends on the sign of π). Recall that the natural domain forthe Yang-Mills functional is the infinite-dimensional quo-tient space π³π = ππ/π’π of connections modulo equiv-alence. The moduli space β³π is a subset of π³π and (ifnon-empty) realises the absolute minimum of the Yang-Mills functional on π³π. In this general setting one could,optimistically, hope for a variational theory which wouldrelate:
(1) The topology of the ambient space π³π,(2) The topology of β³π,(3) The non-minimal critical points: i.e. the solutions
of the Yang-Mills equation which are not instan-tons.
A serious technical complication here is that the group π’πdoes not usually act freely on ππ, so the quotient spaceis not a manifold. But we will not go into that furtherhere and just say that there are suitable homology groupsπ»π(π³π), which can be studied by standard algebraic topol-ogy techniques and which have a rich and interesting struc-ture.
Much of the work in this area in the late 1980s wasdriven by two specific questions.
β’ The Atiyah-Jones conjecture [1]. They consideredthe manifold π = π4 where (roughly speaking)the spaceπ³π has the homotopy type of the degree
π mapping space Mapsπ(π3, π3), which is in factindependent of π. The conjecture was that the in-clusion β³π β π³π induces an isomorphism onhomology groups π»π for π in a range π β€ π(π),where π(π) tends to infinity with π. One motiva-tion for this idea came from results of Segal in theanalogous case of rational maps [17].
β’ Again focusing on π = π4: are there any non-minimal solutions of the Yang-Mills equations?
A series of papers of Taubes [20], [22] developed a varia-tional approach to the Atiyah-Jones conjecture (andgeneralisations to other 4-manifolds). In [20] Taubesestablished a lower bound on the index of any non-minimal solution over the 4-sphere. If the problem sat-isfied the PalaisβSmale condition this index bound wouldimply the Atiyah-Jones conjecture (with π(π) roughly 2π)but the whole point is that this condition is not satisfied,due to the bubbling phenomenon formini-max sequences.Nevertheless, Taubes was able to obtain many partial re-sults through a detailed analysis of this bubbling. TheAtiyah-Jones conjecture was confirmed in 1993 by Boyer,Hurtubise, Mann, and Milgram [2] but their proof workedwith geometric constructions of the instanton modulispaces, rather than variational arguments.
The second question was answered, using variationalmethods, by Sibner, Sibner, and Uhlenbeck in 1989 [18],showing that indeed such solutions do exist. In their proofthey considered a standardπ1 action onπ4 with fixed pointset a 2-sphere, an π1-equivariant bundle π over π4 andπ1-invariant connections on π. This invariance forces theβbubbling pointsβ arising in variational arguments to lieon the 2-sphere π2 β π4 and there is a dimensional re-duction of the problem to βmonopolesβ in 3-dimensionswhich has independent interest.
A connection over π4 which is invariant under the ac-tion of translations in one direction can be encoded as apair (π΄,π) of a connection π΄ over π3 and an additionalHiggs field π which is a section of the adjoint vector bun-dle adπwhose fibres are copies of Lie(πΊ). The Yang-Millsfunctional induces a Yang-Mills-Higgs functional
β±(π΄,π) = β«π3
|πΉ(π΄)|2 + |βπ΄π|2
on these pairs over π3. One also fixes an asymptotic con-dition that |π| tends to 1 at β in π3. In 3 dimensionswe are below the critical dimension for the functional, butthe noncompactness of π3 prevents a straightforward ver-ification of the PalaisβSmale condition. Nonetheless, in aseries of papers [19], [21] Taubes developed a far-reachingvariational theory in this setting. By a detailed analysis,Taubes showed that, roughly speaking, aminimax sequencecan always be chosen to have energy density concentrated
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in a fixed large ball in π3 and thus obtained the neces-sary convergence results. In particular, using this analysis,Taubes established the existence of non-minimal criticalpoints for the functional β±(π΄,π).
The critical points of the Yang-Mills-Higgs functional onπ3 yield Yang-Mills solutions over π4, but these do nothave finite energy. However the same ideas can be appliedto the π1-action. The quotient of π4 \π2 by the π1-actioncan naturally be identified with the hyperbolic 3-spaceπ»3,and π1-invariant connections correspond to pairs (π΄,π)overπ»3. There is a crucial parameter πΏ in the theory whichfrom one point of view is the weight of the π1 action onthe fibres of π over π2. From another point of view thecurvature of the hyperbolic space, after suitable normali-sation, is βπΏβ2. The fixed set π2 can be identified withthe sphere at infinity of hyperbolic space and bubbling ofconnections over a point in π2 β π4 corresponds, in theYang-Mills-Higgs picture, to some contribution to the en-ergy density of (π΄,π) moving off to the correspondingpoint at infinity.
The key idea of Sibner, Sibner, and Uhlenbeck was tomake the parameter πΏ very large. This means that the cur-vature of the hyperbolic space is very small and, on sets offixed diameter, the hyperbolic space is well-approximatedby π3. Then they show that Taubesβ arguments on π3 goover to this setting and are able to produce the desirednon-minimal solution of the Yang-Mills equations overπ4.Later, imposingmore symmetry, other solutionswere foundusing comparatively elementary arguments [13], but theapproach of Taubes, Sibner, Sibner, and Uhlenbeck is aparadigmof theway that variational arguments can be usedβbeyond PalaisβSmale,β via a delicate analysis of the be-haviour of minimax sequences.
We conclude this section with a short digression fromthe main theme of this article. This brings in other rela-tions between harmonic mappings of surfaces and4-dimensional gauge theory, and touches on another veryimportant line of work by Karen Uhlenbeck, representedby papers such as [27], [28]. In this setting the target spaceπ is a symmetric space and the emphasis is on explicit so-lutions and connectionswith integrable systems. There is ahuge literature on this subject, stretching back to work ofCalabi and Chern in the 1960s, and distantly connectedwith the Weierstrass representation of minimal surfaces inπ3. From around 1980 there were many contributionsfrom theoretical physicists and any kind of proper treat-ment would require a separate article, so we just include afew remarks here.
As we outlined above, the dimension reduction of Yang-Mills theory on π4 obtained by imposingtranslation-invariance in one variable leads to equationsfor a pair (π΄,π) on π3. Now reduce further by imposing
translation-invariance in two directions. More precisely,write π4 = π2
1 Γπ22, fix a simply-connected domain Ξ© β
π21, and consider connections on a bundle over Ξ© Γ π2
2which are invariant under translations in π2
2. These corre-spond to pairs (π΄,Ξ¦)whereπ΄ is a connection on a bundleπ over Ξ© and Ξ¦ can be viewed as a 1-form on Ξ© with val-ues in the bundle adπ. Now π΄+ πΞ¦ is a connection overΞ© for a bundle with structure group the complexificationπΊπ: for example if πΊ = π(π) the complexified group isπΊπ = πΊπΏ(π,π). The Yang-Mills instanton equations onπ4 imply thatπ΄+πΞ¦ is a flat connection. By the fundamen-tal property of curvature, sinceΞ© is simply-connected, thisflat connection can be trivialised. The original data (π΄,Ξ¦)is encoded in the reduction of the trivial πΊπ-bundle to thesubgroup πΊ, which amounts to a map π’ from Ξ© to thenon-compact symmetric space πΊπ/πΊ. For example, whenπΊ = π(π) the extra data needed to recover (π΄,Ξ¦) is a Her-mitian metric on the fibres of the complex vector bundle,and πΊπΏ(π,π)/π(π) is the space of Hermitian metrics onππ. The the remaining part of the instanton equations infour dimensions is precisely the harmonic map equationfor π’. This is one starting point for Hitchinβs theory of βsta-ble pairsβ over compact Riemann surfaces [6].
One is more interested in harmonic maps to compactsymmetric spaces and, asUhlenbeck explained in [28], thiscan be achieved by a modification of the set-up above. Shetakes π4 with an indefinite quadratic form of signature(2, 2) and a splitting π4 = π2
1 Γ π22 into positive and
negative subspaces. Then the invariant instantons corre-spond to harmonicmaps fromΞ© to the compact Lie groupπΊ. Other symmetric spaces can be realised as totally ge-odesic submanifolds in the Lie group, for example com-plex Grassmann manifolds inπ(π), and the theory can bespecialised to suit. This builds a bridge between the βinte-grableβ nature of the 2-dimensional harmonic map equa-tions and the Penrose-Ward twistor description of Yang-Mills instantons over π4, although as we have indicatedabovemuch of thework on the former predates twistor the-ory. In her highly influential paper [28], Uhlenbeck foundan action of the loop group on the space of harmonicmapsfromΞ© toπΊ, introduced an integer invariant βuniton num-ber,β and obtained a complete description of all harmonicmaps from the Riemann sphere to πΊ.
Higher DimensionsIn a variational theory with a critical dimension π certaincharacteristic features appear when studying questions indimensions greater than π. In the harmonic mapping the-ory, for maps π’ βΆ π β π, the dimension in question isπ = dim π and, as we saw above, the critical dimensionis π = 2. A breakthrough in the higher dimensional the-ory was obtained by Schoen and Uhlenbeck in [12]. Sup-pose for simplicity that π is isometrically embedded in
310 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 3
some Euclidean space ππ and define πΏ21(π,π) to be the
set of πΏ21 functions on π with values in the vector space
ππ which map to π almost everywhere on π. The en-ergy functionalβ± is defined on πΏ2
1(π,π) and Schoen andUhlenbeck considered an energy minimising map π’ βπΏ21(π,π). The main points of the theory are:
β’ π’ is smooth outside a singular set Ξ£ β π whichhas Hausdorff dimension at most πβ 3;
β’ at each point π₯ in the singular set Ξ£ there is a tan-gent map to π’.
The second item means that there is a sequence of realnumbers ππ β 0 such that the rescaled maps
π’π(π) = π’(expπ₯(πππ))converge to a map π£ βΆ ππ β π which is radially invariant,and hence corresponds to a map from the the sphere ππβ1
toπ. (Here expπ₯ is the Riemannian exponential map andwe have chosen a frame to identify πππ₯ with ππ.)
To relate this to the case π = 2 discussed above, thegeneral picture is that a β±-minimising sequence in Maps(π,π) can be taken to converge outside a bubbling set ofdimension at most πβ 2 and the limit extends smoothlyover the (πβ2)-dimensional part of the bubbling set. Thenew feature in higher dimensions is that the limit can havea singular set of codimension 3 or more.
Two fundamental facts which underpin these results areenergy monotonicity and π-regularity. To explain the first,consider a smooth harmonic map π βΆ π΅π β π, whereπ΅π is the unit ball in ππ. For π < 1 set
πΈ(π) = 1ππβ2 β«
|π₯|<π|βπ|2.
Then one has an identity, for π1 < π2:
πΈ(π2) β πΈ(π1) = 2β«π1<|π₯|<π2
|π₯|2βπ|βππ|2, (5)
where βπ is the radial component of the derivative. Inparticular, πΈ is an increasing function of π. The point ofthis is that πΈ(π) is a scale-invariant quantity. If we de-fine ππ(π₯) = π(ππ₯) then πΈ(π) is the energy of the mapππ on the unit ball. The monotonicity property meansthat π βlooks betterβ on a small scale, in the sense of thisrescaled energy. The identity (5) follows from a very gen-eral argument, applying the stationary condition to the in-finitesimal variation of π given by radial dilation. (Oneway of expressing this is through the theory of the stress-energy tensor.) Note that equality πΈ(π2) = πΈ(π1) holdsif and only if π is radially-invariant in the correspondingannulus. This is what ultimately leads to the existence ofradially-invariant tangent maps.
The monotonicity identity is a feature of maps fromππ,but a similar result holds for small balls in a general Rie-mannian π-manifold π. For π₯ β π and small π > 0 we
define
πΈπ₯(π) =1
ππβ2 β«π΅π₯(π)
|βπ|2,
where π΅π₯(π) is the π-ball about π₯. Then if π is a smoothharmonic map and π₯ is fixed the function πΈπ₯(π) is increas-ing in π, up to harmless lower-order terms.
The π-regularity theoremof Schoen andUhlenbeck statesthat there is an π > 0 such that if π’ is an energy minimiserthen π’ is smooth in a neighbourhood of π₯ if and only ifπΈπ₯(π) < π for some π. An easier, related result is that ifπ’ is known to be smooth then once πΈπ₯(π) < π one has apriori estimates (depending on π) on all derivatives in theinterior ball π΅π₯(π/2). The extension to general minimis-ingmaps is one of themain technical difficulties overcomeby Schoen and Uhlenbeck.
We turn now to corresponding developments in gaugetheory, where the critical dimension π is 4. A prominentachievement of Uhlenbeck in this direction is her workwith Yau on the existence of Hermitian-Yang-Mills connec-tions [29]. The setting here involves a rank π holomorphicvector bundle πΈ over a compact complex manifoldπwitha KΓ€hler metric. Any choice of Hermitian metric β on thefibres ofπΈ defines a principleπ(π) bundle of orthonormalframes in πΈ and a basic lemma in complex differential ge-ometry asserts that there is a preferred connection on thisbundle, compatible with the holomorphic structure. Thecurvature πΉ = πΉ(β) of this connection is a bundle-valued2-form of type (1, 1)with respect to the complex structure,andwewriteΞπΉ for the inner product with the (1, 1) formdefined by the KΓ€hler metric. Then ΞπΉ is a section of thebundle of endomorphisms of πΈ. The Hermitian-Yang-Millsequation is a constant multiple of the identity:
ΞπΉ = π 1πΈ
(where the constant π is determined by topology). As thename suggests, these are special solutions of the Yang-Millsequations. The result proved by Uhlenbeck and Yau isthat a βstableβ holomorphic vector bundle admits such aHermitian-Yang-Mills connection. Here stability is a nu-merical condition on holomorphic sub-bundles, or moregenerally sub-sheaves, of πΈ which was introduced by alge-braic geometers studying moduli theory of holomorphicbundles. The result of Uhlenbeck and Yau confirmed con-jectures made a few years before by Kobayashi andHitchin.These extend older results of Narasimhan and Seshadri, forbundles over Riemann surfaces, and fit into a large devel-opment over the past 40 years, connecting various stabilityconditions in algebraic geometry with differential geome-try. We will not say more about this background here butfocus on the proof of Uhlenbeck and Yau.
The problem is to solve the equation ΞπΉ(β) = π 1πΈfor a Hermitian metric β on πΈ. This boils down to a sec-ond order, nonlinear, partial differential equation for β.
MARCH 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 311
While this problemdoes not fit directly into the variationalframework we have emphasised in this article, the samecompactness considerations apply. Uhlenbeck and Yauuse a continuity method, extending to a 1-parameter fam-ily of equations for π‘ β [0, 1] which we write schemati-cally as ΞπΉ(βπ‘) = πΎπ‘, where πΎπ‘ is prescribed and πΎ1 =π 1πΈ. They set this up so that there is a solution β0 forπ‘ = 0 and the set π β [0, 1] for which a solution βπ‘ ex-ists is open, by an application of the implicit function the-orem. The essential problem is to prove that if πΈ is a stableholomorphic bundle then π is closed, hence equal to thewhole of [0, 1] and in particular there is a Hermitian-Yang-Mills connection β1.
The paper of Uhlenbeck and Yau gave two independenttreatments of the core problem, one emphasising complexanalysis and the other gauge theory. We will concentratehere on the latter. For a sequence π‘(π) β π we have con-nections π΄π defined by the hermitian metrics βπ‘(π) and thequestion is whether one can take a limit of the π΄π. The de-formation of the equations by the term πΎπ‘ is rather harm-less here so the situation is essentially the same as if theπ΄π were Yang-Mills connections. In addition, an integralidentity usingChern-Weil theory shows that the Yang-Millsenergy βπΉ(π΄π)β2
πΏ2 is bounded. Then Uhlenbeck and Yauintroduced a small energy result, for connections over aball π΅π₯(π) β π. Since the critical dimension π is 4, therelevant normalised energy in this Yang-Mills setting is
πΈπ₯(π) =1
ππβ4 β«π΅π₯(π)
|πΉ|2,
where π is the real dimension of π. If πΈπ₯(π) is below asuitable threshold there are interior bounds on all deriva-tives of the connection, in a suitable gauge. Then the globalenergy bound implies that after perhaps taking a subse-quence, the π΄π converge outside a closed set π β π ofHausdorff codimension at least 4. Uhlenbeck and Yaushow that if the metrics βπ‘(π) do not converge then a suit-able rescaled limit produces a holomorphic subbundle ofπΈ over π\π. A key technical step is to show that thissubundle corresponds locally to a meromorphic map toa Grassmann manifold, which implies that the subbundleextends as a coherent sheaf over all of π. The differen-tial geometric representation of the first Chern class of thissubsheaf, via curvature, shows that it violates the stabilityhypothesis.
The higher-dimensional discussion in Yang-Mills the-ory follows the pattern of that for harmonic maps above.The corresponding monotonicity formula was proved byPrice [10] and a treatment of the small energy result wasgiven by Nakajima [9]. Some years later, the theory wasdeveloped much further by Tian [23], including the exis-tence of βtangent conesβ at singular points.
This whole circle of ideas and techniques involving thedimension of singular sets, monotonicity, βsmall energyβresults, tangent cones, etc. has had a wide-ranging im-pact in many branches of differential geometry over thepast few decades and forms the focus of much current re-search activity. Apart from the cases of harmonicmaps andYang-Mills fields discussed above, prominent examples areminimal submanifold theory, where many of the ideas ap-peared first, and the convergence theory of Riemannianmetrics with Ricci curvature bounds.
References
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[2] Boyer C, Hurtubise J, Mann B, Milgram R, The topologyof instanton moduli spaces. I. The Atiyah-Jones conjectureAnn. of Math. 137 (1993) 561-609. MR1217348
[3] Brendle S, Schoen R, Manifolds with 1/4-pinched curva-ture are space forms J. Amer. Math. Soc. 22 (2009) 287-307.MR2449060
[4] BrΓ©zis H, Nirenberg L, Positive solutions of nonlinearelliptic equations involving critical Sobolev exponents,Comm. Pure Appl. Math. 36 (1983) 437-477 MR709644
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[6] Hitchin N, The self-duality equations on a Riemann sur-face Proc. Lond. Math. Soc. 55 (1987) 59-126 MR887284
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[8] Micallef M, Moore J, Minimal two-spheres and thetopology of manifolds with positive curvature on totallyisotropic two-planes Annals of Math. 127 (1988) 199-227MR924677
[9] Nakajima H, Compactness of the moduli space of Yang-Mills connections in higher dimensions J. Math. Soc. Japan40 (1988) 383-392 MR945342
[10] Price P, A monotonicity formula for Yang-Mills fieldsManuscripta Math. 43 (1983) 131-166 MR707042
[11] Parker T, What is.... a bubble tree? Notices of the Amer.Math. Soc. Volume 50 (2003) 666-667
[12] Sacks J, Uhlenbeck K, The existence of minimal immer-sions of 2-spheres Annals of Math. Vol 113 (1981) 1-24MR604040
[13] Sadun L, Segert J, Non-self-dual Yang-Mills connectionswith nonzero Chern number Bull. Amer. Math. Soc. 24(1991) 163-170 MR1067574
[14] Schoen R, Conformal deformation of a Riemannianmetric to constant scalar curvature J. Differential Geom. 20(1984) 479-495 MR788292
[15] Schoen R, Uhlenbeck K, A regularity theory for har-monic maps J. Differential Geometry 17 (1982) 307-335MR664498
[16] Sedlacek S, A direct method for minimizing the Yang-Mills functional over 4-manifolds Comm. Math. Phys. 86(1982) 515-527 MR679200
[17] Segal G, The topology of spaces of rational functionsActa Math. 143 (1979) 39-72. MR533892
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[18] Sibner L, Sibner R, Uhlenbeck K, Solutions to Yang-Millsequations that are not self-dual Proc. Natl. Acad. Sci. USA86 (1989) 8610-8613 MR1023811
[19] Taubes C, The existence of a nonminimal solution to theSU(2) Yang-Mills-Higgs equations on π3 I. Comm. Math.Phys. 86 (1982) 257-298 MR676188
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[22] Taubes C, The stable topology of self-dualmoduli spacesJ. Differential Geom. 29 (1989) 163-230 MR978084
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[28] Uhlenbeck K, On the connection between harmonicmaps and the self-dual Yang-Mills and the sine-Gordonequations J. Geometry and Physics 8 (1992) 283-316MR1165884
[29] Uhlenbeck K, Yau ST, On the existence of Hermitian-Yang-Mills connections in stable vector bundles Commun.Pure Appl. Math. Vol XXXIX (1986) S257-S293 MR861491
Credits
Photo of Karen Uhlenbeck is by Andrea Kane, Β©IAS.Author photo is courtesy of the author.
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