Celebratio Mathematica — Uhlenbeck

Karen Uhlenbeck's contributions to gauge theoretic analysis

by Clifford Henry Taubes

Karen Uhlenbeck’s mathematical contributions and mathematical work in the fields of geometry and analysis laid the foundation for a broad swath of continuing research in differential geometry over the past four decades. Think of her work with Jonathan Sacks on minimal surfaces, her work with Richard Schoen on harmonic maps, and her work with S-T. Yau on Hermitian Yang–Mills connections on stable, holomorphic bundles. And then there is her foundational work on gauge theory. But, the full scope of Karen Uhlenbeck’s contributions is not the subject of this essay; this essay is only about Karen Uhlenbeck’s brilliant gauge theory analysis. (Karen Uhlenbeck’s ICM talks in 1983 and 1990 and her Commentary on “analysis in the large” (for the centennial of the AMS) summarize much of her early work. However, these papers seriously down-play her influence — she is exceptionally modest. And, of course, they say nothing about the remarkable subsequent developments and applications. The ICM talks, her AMS commentary, and selected papers by Karen Uhlenbeck on minimal surfaces, harmonic maps, Hermitian Yang–Mills connections and gauge theory are listed at the end of this essay.)

Uhlenbeck’s papers on connections with bounded, integral curvature norms (see [4], [17], [7], [8]) form the foundational analysis behind all of the subsequent applications of gauge theory to topology and geometry. Here is some history: Simon Donaldson had the brilliant idea to consider certain canonical families of equivalence classes (equivalence by bundle automorphism) of connections on a given manifold as a homology cycle in the space of equivalence classes of all connections (see [e1], [e2], [e4] and then [e5]). By virtue of the nature of the family (the connections are solutions to a canonical partial differential equation with Fredholm linearization), the pairing of this cycle with homology classes would give an invariant of the smooth structure of the ambient manifold if the cycle is effectively compact (compact, or compactifiable in a nice way). In hindsight, compactness is the truly difficult analysis issue. Uhlenbeck’s analysis is needed to prove that the cycle does indeed have a canonical compactification. Donaldson brilliantly saw how to exploit her theorem to develop his gauge theoretic 4-manifold invariants. Likewise, her analysis lies behind Andreas Floer’s development of Floer homology for gauge theory [e3] (see also [e6]) because that also requires her compactness theorem to prove that the Floer differential has square zero. All subsequent work on gauge theoretic invariants in low dimensional topology can trace ancestry back to Uhlenbeck’s work. (It is still exciting; see the ICM 2018 plenary talk by Peter Kronheimer and Tom Mrowka.)

By way of background for this compactness issue: The Hodge dual operator from a metric on an oriented, Riemannian 4-manifold maps 2-forms to 2-forms with square 1. Its $+ 1$ and $- 1$ eigenspaces are called (respectively) the bundles of self-dual and anti-self dual 2-forms. Each is a rank 3, real vector bundle. Meanwhile, the curvature of a given connection on some principal bundle over the manifold is a 2-form with values in an associated vector bundle (the fiber is the Lie algebra of the group), and so it has a self-dual part and an anti-self dual part. Donaldson’s space of connections are those with vanishing self-dual curvature; they are connections with anti-self dual curvature.

This space of connections with anti-self dual curvature can not be compact by virtue of the fact that the action by pull-back of the group of automorphisms of the principal bundle (which is infinite dimensional) preserves anti-self duality. But, the space of orbits of connections with anti-self dual curvature under this group action has a chance to be compact because it is locally compact. In fact, this orbit space is locally homeomorphic to the orbit space (via a finite group) of the zero locus of a smooth map between finite dimensional Euclidean spaces. To elaborate: Suppose that $A_{0}$ is a connection on the principal bundle. Any nearby connection (in the $C^{\infty}$ topology) can be written as $A_{0} + a$ where $a$ is a 1-form in that associated bundle with Lie algebra fiber. If $A_{0}$ has anti-self dual curvature and likewise $A_{0} + a$ ; and if $A_{0} + a$ is in a slice transversal to the automorphism group’s orbit through $A_{0}$ (and sufficiently close to $A_{0}$ ), then $a$ will obey a differential equation (by virtue of $A_{0} + a$ having anti-self dual curvature) that takes the schematic form $D_{0} a + Π (a \land a) = 0$ where $D_{0}$ is a first order differential operator with elliptic symbol that is independent of $A_{0}$ . Meanwhile, $Π (a \land a)$ denotes the anti-self dual projection of $a \land a$ . The key point in this regard is that $D_{0}$ is Fredholm when its range and domain are completed using suitable Banach space norms.

With regards to compactness of the orbit space: The equation $D_{0} a + Π (a \land a) = 0$ suggests an iterative scheme to prove compactness for the space of automorphism group orbits of anti-self dual connections; it goes something like this: Suppose that some minimal bound is known for $a$ , and that this bound leads to a suitable bound for the quadratic term $Π (a \land a)$ . If so, then the equation $D_{0} a + Π (a \land a)$ says that some linear combination of first derivatives of $a$ obeys the bound that $Π (a \land a)$ obeys; which suggests in turn (because $D_{0}$ is elliptic) that all of the derivatives of $a$ might also obey that same bound. If that were the case, then the equation $D_{0} a + Π (a \land a)$ could be differentiated termwise to obtain an equation for the first derivatives of $a$ which would be an equation having the schematic form $D_{0} \partial a + Π (\partial a \land a) + Π (a \land \partial a) + r \cdot a = 0 .$ (What is denoted by $r$ here comes from the metric’s curvature tensor.) This last equation suggests that some linear combination of the second derivatives of $a$ obeys a suitable bound, and if this is the case for all of the second derivatives (which is reasonable because $D_{0}$ is elliptic), then one can differentiate the equation again to obtain third derivative bounds, and so on. Continuing in this vein would get uniform bounds for all higher derivatives from an initial bound for just $a$ . Then, by virtue of the Arzelà–Ascoli theorem, any bounded sequence of $a$ ’s would have a convergent subsequence (which is the desired compactness conclusion).

The problem with implementing the preceding is this: The equation $D_{0} a + Π (a \land a)$ is an equation for the difference between two anti-self dual connections, one being the original $A_{0}$ and the other being $A_{0} + a$ ; and it can be exploited (and, in fact, derived in the first place) only in the event that their respective automorphism orbits are sufficiently close (which guarantees that $a$ is small to begin with). In particular, if the automorphism group orbits of a sequence of connections with anti-self dual curvature is not converging, then (this is essentially the nonconvergence assumption) no infinite subset of the sequence can stay uniformly close to any given automorphism group orbit in the space of connections.

More by way of geometric input is needed to proceed. Here is the input: If $A$ is a connection on a principal bundle with compact structure group, then the pointwise norm of its curvature 2-form is invariant under the action of the automorphism group. This is because the curvature 2-form at any given point can be viewed as a 2-form with values in the Lie algebra of the structure group. Viewed in this light, an automorphism of the principal bundle acts on the curvature 2-form at that point by conjugation of the Lie algebra factors with an element in the group. It is the adjoint action on the Lie algebra. Therefore, if an adjoint-invariant norm on the Lie algebra (for example, the trace norm in the case of a matrix group) is used to define the norm of the curvature 2-form, then that norm is invariant with respect to the action of the automorphism group.

It follows from the preceding observations that if some functional of the pointwise norms of the curvature diverges along a sequence of connections, then the corresponding sequence of automorphism orbits does not converge. Karen Uhlenbeck’s brilliant results (from [7] and [8]) were a converse which is (in part) this: Suppose that ${A_{n}}_{n = 1, 2, \dots}$ is a sequence of anti-self dual connections with an a priori bound only on the integral of the square of the norm of the curvature (thus, an a priori bound for the $L^{2}$ norm of the curvatures along the sequence). Then there exists the following data: First, a new principal bundle over the manifold and a connection on it with anti-self dual curvature 2-form. (The new connection is denoted by $A$ .) Second, a finite set of points in the manifold (an upper bound for the number can be given a priori in terms of the bound for the integrals of the squares of the curvature 2-forms). This finite set is denoted by $Ω$ . Third, a sequence (to be denoted by ${g_{n}}_{n = 1, 2, \dots}$ ) of isomorphisms on the complement of $Ω$ from the new bundle to the original bundle. These are such that the sequence ${g_{n}^{*} A_{n}}_{n = 1, 2, \dots}$ has a subsequence that converges to the connection $A$ on compact sets in the complement of $Ω$ . Karen Uhlenbeck’s analysis in [7] and [8] can also say precisely what happens to the sequence near the points of $Ω$ . Some words about the proof of Karen Uhlenbeck’s results appear momentarily.

As it turns out, the $L^{2}$ bound is more important than the anti-self dual condition: Here is what Karen Uhlenbeck’s theorems from [7] and [8] imply when there is nothing but an $L^{2}$ bound: Suppose that ${A_{n}}_{n = 1, 2, \dots}$ is a sequence of connections with only an a priori bound on the integral of the square of the norm of the curvature. Then there exists a new principal bundle and now a Sobolev class $L_{1}^{2}$ connection on that bundle (to be denoted by $A$ ). There is again a finite set of points (denoted here also by $Ω$ ) and a sequence of isomorphisms from the new bundle to the old bundle on the complement of $Ω$ . This is ${g_{n}}_{n = 1, 2, \dots}$ . This data is such that ${g_{n}^{*} A_{n}}_{n = 1, 2, \dots}$ converges in the weak $L_{1}^{2}$ -topology to $A$ on open sets with compact closure in the complement of $Ω$ . (A connection on a principal bundle in the Sobolev class $L_{1}^{2}$ differs from a smooth connection by a measurable Lie algebra valued 1-form with measurable first derivatives whose norm is square integrable and whose first derivatives have square integrable norm.)

By way of a parenthetical remark: The $L^{2}$ norm bound is the borderline case for convergence in two respects. First, Karen Uhlenbeck’s theorems in [8] also prove this: If ${A_{n}}_{n = 1, 2, \dots}$ is a sequence of anti-self dual connections such that the integrals of any power greater than 2 of the norms of the curvature 2-forms along the sequence is bounded, then the corresponding sequence of automorphism group orbits has a convergent subsequence. Second, there are known sequences of anti-self dual connections with a priori $L^{2}$ norm bound that do not converge without removing some finite set of points. The next paragraph describes an example.

The Levi-Civita connection on the round 4-dimensional sphere induces a corresponding connection on the principal $S O (3)$ bundle of oriented orthonormal frames in the bundle of self dual 2-forms for this round metric. And, this $S O (3)$ connection has nonzero, anti-self dual curvature 2-form. Since the curvature is nonzero, the integral of the square of its norm is likewise not zero. Keeping the preceding in mind, note that the anti-self duality condition is conformally invariant, as is the integral of the square of the norm of the curvature. (A conformal change of metric changes the metric by multiplying it by a positive function.) Therefore, if the 4-sphere admits a conformal diffeomorphism (the pull-back of the metric is conformal to the original), then the pull-back of the connection by this diffeomorphism has anti-self dual curvature whose $L^{2}$ norm is the same as the original. As it turns out, the four dimensional sphere admits a 1-parameter group of conformal diffeomorphisms that act by pushing points towards the south pole along the great circles. This family of conformal diffeomorphism is the image via the inverse of stereographic projection (from the south pole) of the Euclidean space rescaling map $x \to λ x$ with $λ$ being a number greater than 1. Since this family pushes points to the south pole, the corresponding sequence of anti-self dual connections (all with the same curvature $L^{2}$ norm) cannot have a convergent subsequence on the whole of $S^{4}$ . (But, it does converge on the complement of the north pole; and the limit is a flat connection.)

It is fortuitous (and crucial for Donaldson’s applications) that anti-self dual connections on any given principal bundle over any given compact 4-dimensional manifold have an a priori $L^{2}$ bound on their curvature. In fact, their curvatures have identical $L^{2}$ norms, and here is why: For any connection, the integral of the square of the norm of the self-dual part of the curvature minus that of the anti-self dual part is a characteristic number that classifies (in part) the principal bundle (it is an integer up to a canonical factor). Therefore, all connections on the same principal bundle with anti-self dual curvature 2-form have the following property: The integral of the square of the norm of their curvature 2-form is independent of the connection; it is the aforementioned characteristic number which is predetermined by the principal bundle.

It is also fortuitous (and important for Donaldson’s applications) that Karen Uhlenbeck’s $L^{2}$ compactness theorems can be turned around (so to speak) and used to construct connections with anti-self dual curvature 2-forms. The point here is that Karen Uhlenbeck’s theorems also describe the behavior of the connections in neighborhoods of the points of the set $Ω$ that appears in the statement of her theorem. What happens is this (to a first approximation): A sequence of coordinate rescalings in small balls about any given point in $Ω$ (use Gaussian normal coordinates) produces a connection on 4-dimensional Euclidean space with anti-self dual curvature which comes from the 4-dimensional round sphere via the inverse of the stereographic projection map. (Just as in the example two paragraphs back). Knowing this, one can take a connection on the round sphere with anti-self dual curvature, pull it back to Euclidean space, rescale it sufficiently (using the $x \to λ x$ rescaling) and then graft the result into any oriented 4-manifold using Gaussian normal coordinates. This gives a connection with curvature concentrated around the given point whose curvature is almost anti-self dual. If this is done cleverly at sufficiently many points (taking into account the freedom to choose the points and the Gaussian normal coordinates and the scaling parameter), then a small perturbation can be made so that the result is a connection on a principal bundle with identically anti-self dual curvature. The full details of how this is done can be found in [e5].

What follows directly are the promised words about the proof of Karen Uhlenbeck’s results. (What is said does not necessarily reflect Karen Uhlenbeck’s thinking.) There are two essential observations, one geometric and straight forward. The other is brilliant via analysis insight from Karen Uhlenbeck. The geometric insight is this: Any given principal Lie group bundle over a ball is isomorphic to the product bundle (the product of the ball with the group); and any given connection on the product bundle over a ball with zero curvature 2-form is equivalent via an automorphism to the product connection. Thus, modulo automorphisms and isomorphisms, there is just one principal bundle over a ball, and there is just one connection with zero curvature. This connection will be the connection $A_{0}$ to use when writing other connections over the ball as $A_{0} + a$ . Now suppose that $A$ is another connection on the given principal bundle over the given ball. If its curvature 2-form is nearly zero, then it stands to reason that there is an automorphism/isomorphism that writes it as $A_{0} + a$ with $a$ being small. Karen Uhlenbeck proved that nearly zero means the following when the dimension is four (she proved analogous results for other dimensions): There exist $ε_{0} > 0$ and $κ > 0$ (that are independent of the radius of the ball supposing it is small) such that if $A$ is a connection on the ball with the integral of the square of the norm its curvature 2-form being less than $ε_{0}^{2}$ , then there is an isomorphism between the product principal bundle over the ball and the given bundle (to be denoted by $g$ ) such that $g^{*} A = A_{0} + a$ with $a$ being small and orthogonal to the orbit through $A_{0}$ of the automorphism group of the product principal bundle. (The notion of “small” here means that the integral over the ball of the sum of the square of $a$ ’s norm and the squares of the norms of $a$ ’s first derivatives is at most $κ$ times the integral over the ball of the square of the norm of $A$ ’s curvature.) If it is also the case that $A$ has anti-self dual curvature, then the corresponding Lie algebra valued 1-form $a$ obeys the equation $D_{0} a + Π (a \land a) = 0$ on the given ball; and, as a consequence, there are a priori bounds for its pointwise norm and those of its derivatives to any given order on a slightly smaller radius, concentric ball. (The latter are derived in an iterative manner by taking derivatives of the equation $D_{0} a + Π (a \land a) = 0$ in the manner outlined previously.)

Given the result above on balls, then uniform analytic control on connections with a priori $L^{2}$ bounds on curvature on the whole 4-dimensional manifold can be had by covering the given manifold by balls with radii chosen so that the $L^{2}$ norm of the curvature on each is less than the number $ε_{0}$ from the preceding paragraph. If ${A_{n}}_{n = 1, 2, \dots}$ is a sequence of connections with a priori bound on the integral of the square of the corresponding curvature 2-forms, then convergence of a subsequence will follow automatically unless ever smaller radius balls are needed to obtain the desired $ε_{0}^{2}$ bound on all balls along the sequence. But note in this regard that if the $L^{2}$ norm of the curvature is bounded along the sequence by a given number $E$ , then at most $O (E^{2} / ε_{0}^{2})$ very small radius balls are needed for any given connection in the sequence. This bound on the number of small balls implies that there is curvature concentration along a suitable subsequence at just a finite set of points (this is the set $Ω$ , it has at most $O (E^{2} / ε_{0}^{2})$ elements); and consequential there is convergence for the subsequence on the complement of this same finite set. Karen Uhlenbeck’s paper Removable singularities in Yang–Mills fields [7] explains why the limit connection that is obtained on the complement of the finite set $Ω$ comes from a connection on a principal bundle over the whole manifold.

By way of a parenthetical remark: The paper Removable singularities in Yang–Mills fields [7] just noted had some influence on the development of high energy particle physics in the early 1980s. By way of background: Electrons and subatomic particles called quarks are the fundamental constituents of the matter around us. Almost all of the mass of the matter is from the quarks and the force that holds them together. Connections on the product $S O (3)$ principal bundle over $R^{4}$ supply the force. Now for the mathematics: A theorem in that paper said in effect that any compact Lie group connection over 4-dimensional Euclidean space $(R^{4})$ with the integral of the square of the norm of the curvature being finite comes via stereographic projection from a connection on the 4-dimensional sphere. Karen showed in particular that this has the following implication: Suppose that $A$ is a connection that solves the Yang–Mills equation on $R^{4}$ or on just the complement of a ball in $R^{4}$ . Assume also that $A$ has $L^{2}$ curvature 2-form. Then, the norm of $A$ ’s curvature 2-form falls off at large distance on $R^{4}$ as fast or faster than $| x |^{- 4}$ . (The Yang–Mills equations ask that the covariant exterior derivative defined by $A$ of the Hodge dual of the $A$ ’s curvature 2-form is zero. Connections with anti-self dual or self-dual curvature 2-form always solve the Yang–Mills equations.) For a physicist, this $| x |^{- 4}$ fall off of the curvature implies that the force generated by the connection on subatomic particles is very small when the particles are far apart. And, the latter fact killed a proposed explanation for the fact (experimentally verified) that the force between the fundamental quarks grows as quarks try to separate.

Here is another parenthetical remark with regards to solutions to the Yang–Mills equations on $R^{4}$ : It was conjectured for some time by physicists that the only solutions with the $L^{2}$ norm of the curvature being finite were connections with curvature either self-dual or anti-self dual. Karen Uhlenbeck with Lesley Sibner and Bob Sibner constructed a counterexample to this conjecture (see [19]).

Acknowledgement

Karen Uhlenbeck has been an inspiration to me and a crucial role model (mathematically and otherwise) since my days as a graduate student.

Selected papers by Karen Uhlenbeck on geometric analysis topics

Commentaries

[2] K. Uhlenbeck: “Generic properties of eigenfunctions,” Am. J. Math. 98 : 4 (1976), pp. 1059–1078. MR 464332 Zbl 0355.58017

[20] K. Uhlenbeck: “Commentary on ‘analysis in the large’,” pp. 357–359 in A century of mathematics in America, part 2. Edited by P. L. Duren, R. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003144

[22] K. Uhlenbeck: “Instantons and their relatives,” pp. 467–477 in Mathematics into the twenty-first century (Providence, RI, 8–12 August 1988). Edited by F. E. Browder. American Mathematical Society Centennial Publications 2. American Mathematical Society (Providence, RI), 1992. MR 1184623 Zbl 1073.53505

[23] C.-L. Terng and K. Uhlenbeck: “Introduction,” pp. 5–19 in Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Cambridge, MA), 1998. Zbl 0938.35182

Selected papers on minimal surfaces

[3] J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of two-spheres,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 1033–1036. MR 448408 Zbl 0375.49016

[6] J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of 2-spheres,” Ann. Math. (2) 113 : 1 (January 1981), pp. 1–24. MR 604040 Zbl 0462.58014

[10] J. Sacks and K. Uhlenbeck: “Minimal immersions of closed Riemann surfaces,” Trans. Am. Math. Soc. 271 : 2 (1982), pp. 639–652. MR 654854 Zbl 0527.58008

[13] K. K. Uhlenbeck: “Closed minimal surfaces in hyperbolic 3-manifolds,” pp. 147–168 in Seminar on minimal submanifolds. Edited by E. Bombieri. Annals of Mathematics Studies 103. Princeton University Press, 1983. MR 795233 Zbl 0529.53007

[14] K. K. Uhlenbeck: “Minimal spheres and other conformal variational problems,” pp. 169–176 in Seminar on minimal submanifolds. Edited by E. Bombieri. Annals of Mathematics Studies 103. Princeton University Press, 1983. MR 795234 Zbl 0535.53050

[24] A. Gonçalves and K. Uhlenbeck: “Moduli space theory for constant mean curvature surfaces immersed in space-forms,” Comm. Anal. Geom. 15 : 2 (2007), pp. 299–305. MR 2344325 Zbl 1136.53048 ArXiv math/0611295

Selected papers on harmonic maps

[1] K. Uhlenbeck: “Harmonic maps: A direct method in the calculus of variations,” Bull. Am. Math. Soc. 76 : 5 (1970), pp. 1082–1087. MR 264714 Zbl 0208.12802

[5] K. Uhlenbeck: “Morse theory by perturbation methods with applications to harmonic maps,” Trans. Am. Math. Soc. 267 : 2 (1981), pp. 569–583. MR 626490 Zbl 0509.58012

[9] R. Schoen and K. Uhlenbeck: “A regularity theory for harmonic maps,” J. Diff. Geom. 17 : 2 (1982), pp. 307–335. MR 664498 Zbl 0521.58021

[11] R. Schoen and K. Uhlenbeck: “Boundary regularity and the Dirichlet problem for harmonic maps,” J. Diff. Geom. 18 : 2 (1983), pp. 253–268. MR 710054 Zbl 0547.58020

[12] R. Schoen and K. Uhlenbeck: “Correction to: ‘A regularity theory for harmonic maps’,” J. Diff. Geom. 18 : 2 (1983), pp. 329. MR 710058

[16] R. Schoen and K. Uhlenbeck: “Regularity of minimizing harmonic maps into the sphere,” Invent. Math. 78 : 1 (February 1984), pp. 89–100. MR 762354 Zbl 0555.58011

Selected papers on Hermitian Yang–Mills connections

[18] K. Uhlenbeck and S.-T. Yau: “On the existence of Hermitian-Yang–Mills connections in stable vector bundles,” pp. S257–S293 in Proceedings of the Symposium on Frontiers of the Mathematical Sciences: 1985 (New York, October 1985), published as Comm. Pure Appl. Math. 39 : Supplement S1. Issue edited by C. Morawetz. J. Wiley and Sons (New York), 1986. MR 861491 Zbl 0615.58045

[21] K. Uhlenbeck and S. T. Yau: “A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles,” Commun. Pure Appl. Math. 42 : 5 (1989), pp. 703–707. MR 997570 Zbl 0678.58041

Selected papers on gauge theory

[4] K. K. Uhlenbeck: “Removable singularities in Yang–Mills fields,” Bull. Am. Math. Soc. (N.S.) 1 : 3 (May 1979), pp. 579–581. MR 526970 Zbl 0416.35026

[7] K. K. Uhlenbeck: “Removable singularities in Yang–Mills fields,” Comm. Math. Phys. 83 : 1 (February 1982), pp. 11–29. MR 648355 Zbl 0491.58032

[8] K. K. Uhlenbeck: “Connections with $L^{p}$ bounds on curvature,” Comm. Math. Phys. 83 : 1 (February 1982), pp. 31–42. MR 648356 Zbl 0499.58019

[17] K. K. Uhlenbeck: “The Chern classes of Sobolev connections,” Comm. Math. Phys. 101 : 4 (December 1985), pp. 449–457. MR 815194 Zbl 0586.53018

[19] L. M. Sibner, R. J. Sibner, and K. Uhlenbeck: “Solutions to Yang–Mills equations that are not self-dual,” Proc. Natl. Acad. Sci. U.S.A. 86 : 22 (November 1989), pp. 8610–8613. MR 1023811 Zbl 0731.53031

References

[e1]S. K. Donaldson: “An application of gauge theory to four-dimensional topology,” J. Differential Geom. 18 : 2 (1983), pp. 279–315. MR 710056 Zbl 0507.57010

[e2]S. K. Donaldson: “Connections, cohomology and the intersection forms of 4-manifolds,” J. Differential Geom. 24 : 3 (1986), pp. 275–341. MR 868974 Zbl 0635.57007

[e3] A. Floer: “An instanton invariant for 3-manifolds,” Commun. Math. Phys. 118 (1988), pp. 215–240. MR 956166 Zbl 0684.53027 article

[e4]S. K. Donaldson: “Polynomial invariants for smooth four-manifolds,” Topology 29 : 3 (1990), pp. 257–315. MR 1066174 Zbl 0715.57007

[e5]S. K. Donaldson and P. B. Kronheimer: The geometry of four-manifolds. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press (New York), 1990. MR 1079726 Zbl 0820.57002

[e6]S. K. Donaldson: Floer homology groups in Yang–Mills theory. Cambridge Tracts in Mathematics 147. Cambridge University Press, 2002. With the assistance of M. Furuta and D. Kotschick. MR 1883043 Zbl 0998.53057

Works

[1] K. Uhlenbeck: “Harmonic maps: A direct method in the calculus of variations,” Bull. Am. Math. Soc. 76 : 5 (1970), pp. 1082–1087. MR 264714 Zbl 0208.12802 article

Sampson and Eells [1964] have shown the existence of harmonic maps in any homotopy class of maps from a compact Riemannian manifold with nonpositive sectional curvature. (An imbedding condition is necessary if the image manifold is not compact.) These results were extended by Hartman [1967] to include a uniqueness result if the sectional curvature is negative. The original proofs of these existence and uniqueness theorems for harmonic maps, which are the solutions of nonlinear elliptic systems, rely on the properties of the related nonlinear parabolic equations. We present here a direct method, which uses a perturbation of the energy integral to an integral which can be shown to satisfy condition (C) of Palais and Smale. We then automatically get existence theorems for the new integrals and we show that the maps which minimize these new integrals converge to a minimizing function of the original integral. Regularity theorems for critical points seem to be essential for this method to work. The uniqueness theorem can be derived from Morse theory or directly from Ljusternik–Schnirelman theory.

[2] K. Uhlenbeck: “Generic properties of eigenfunctions,” Am. J. Math. 98 : 4 (1976), pp. 1059–1078. MR 464332 Zbl 0355.58017 article

[3] J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of two-spheres,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 1033–1036. A related article with almost the same title was published in Ann. Math. 113:1 (1981). MR 448408 Zbl 0375.49016 article

[4] K. K. Uhlenbeck: “Removable singularities in Yang–Mills fields,” Bull. Am. Math. Soc. (N.S.) 1 : 3 (May 1979), pp. 579–581. A related article with the same title was published in Comm. Math. Phys. 83:1 (1982). MR 526970 Zbl 0416.35026 article

In the last several years, the study of gauge theories in quantum field theory has led to some interesting problems in nonlinear elliptic differential equations. One such problem is the local behavior of Yang–Mills fields over Euclidean 4-space. Our main result is a local regularity theorem: A Yang–Mills field with finite energy over a 4-manifold cannot have isolated singularities. Apparent point singularities (including singularities in the bundle) can be removed by a gauge transformation. In particular, a Yang–Mills field for a bundle over $ \mathbb{R}^4 $ which has finite energy may be extended to a smooth field over a smooth bundle over \[ \mathbb{R}^4\cup\{\infty\} = \mathbb{S}^4 .\]

[5] K. Uhlenbeck: “Morse theory by perturbation methods with applications to harmonic maps,” Trans. Am. Math. Soc. 267 : 2 (1981), pp. 569–583. MR 626490 Zbl 0509.58012 article

[6] J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of 2-spheres,” Ann. Math. (2) 113 : 1 (January 1981), pp. 1–24. A related article with almost the same title was published in Bull. Am. Math. Soc. 83:5 (1977). MR 604040 Zbl 0462.58014 article

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifolds via a complete existence theory for a perturbed variational problem. Convergence of the critical maps of the pertured problem is sufficient to produce at least one harmonic map of the sphere into the Riemannian manifold. A harmonic map from a sphere is in fact a conformal branched minimal immersion.

[7] K. K. Uhlenbeck: “Removable singularities in Yang–Mills fields,” Comm. Math. Phys. 83 : 1 (February 1982), pp. 11–29. A related article with the same title was published in Bull. Am. Math. Soc. 1:3 (1979). MR 648355 Zbl 0491.58032 article

[8] K. K. Uhlenbeck: “Connections with $L^{p}$ bounds on curvature,” Comm. Math. Phys. 83 : 1 (February 1982), pp. 31–42. MR 648356 Zbl 0499.58019 article

[9] R. Schoen and K. Uhlenbeck: “A regularity theory for harmonic maps,” J. Diff. Geom. 17 : 2 (1982), pp. 307–335. A correction to this article was published in J. Diff. Geom. 18:2 (1983). MR 664498 Zbl 0521.58021 article

[10] J. Sacks and K. Uhlenbeck: “Minimal immersions of closed Riemann surfaces,” Trans. Am. Math. Soc. 271 : 2 (1982), pp. 639–652. MR 654854 Zbl 0527.58008 article

Let $ M $ be a closed orientable surface of genus larger than zero and $ N $ a compact Riemannian manifold. If $ u:M \to N $ is a continuous map, such that the map induced by it between the fundamental groups of $ M $ and $ N $ contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion $ s:M \to N $ having least area among all branched immersions with the same action on $ \pi_1(M) $ as $ u $. Uniqueness within the homotopy class of $ u $ fails in general: It is shown that for certain 3-manifolds which fiber over the circle there are at least two geometrically distinct conformal branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those 3-manifolds for which uniqueness fails.

[11] R. Schoen and K. Uhlenbeck: “Boundary regularity and the Dirichlet problem for harmonic maps,” J. Diff. Geom. 18 : 2 (1983), pp. 253–268. MR 710054 Zbl 0547.58020 article

[12] R. Schoen and K. Uhlenbeck: “Correction to: ‘A regularity theory for harmonic maps’,” J. Diff. Geom. 18 : 2 (1983), pp. 329. Correction to an article published in J. Diff. Geom. 17:2 (1982). MR 710058 article

[15] K. K. Uhlenbeck: “Variational problems for gauge fields,” pp. 585–591 in Proceedings of the International Congress of Mathematicians, vol. 2. Edited by Z. Ciesielski and C. Olech. PWN (Warsaw), 1984. An earlier article with the same title was published in Seminar on differential geometry (1982). MR 804715 Zbl 0562.53059 incollection

Czesław Olech	Related
Zbigniew Ciesielski	Related

[16] R. Schoen and K. Uhlenbeck: “Regularity of minimizing harmonic maps into the sphere,” Invent. Math. 78 : 1 (February 1984), pp. 89–100. MR 762354 Zbl 0555.58011 article

[17] K. K. Uhlenbeck: “The Chern classes of Sobolev connections,” Comm. Math. Phys. 101 : 4 (December 1985), pp. 449–457. MR 815194 Zbl 0586.53018 article

Cathleen Morawetz	Related	Volume
Shing-Tung Yau	Related

The Yang–Mills functional for connections on principle $ SU(2) $ bundles over $ S^4 $ is studied. Critical points of the functional satisfy a system of second-order partial differential equations, the Yang–Mills equations. If, in particular, the critical point is a minimum, it satisfies a first-order system, the self-dual or anti-self-dual equations. Here, we exhibit an infinite number of finite-action nonminimal unstable critical points. They are obtained by constructing a topologically nontrivial loop of connections to which min-max theory is applied. The construction exploits the fundamental relationship between certain invariant instantons on $ S^4 $ and magnetic monopoles on $ H^3 $. This result settles a question in gauge field theory that has been open for many years.

Lesley Millman Sibner	Related
Robert Jules Sibner	Related

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume