Celebratio Mathematica

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.

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

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

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 \[ \mathbb{R}^4\cup\{\infty\} = \mathbb{S}^4 .\]

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

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.

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

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

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

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.

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

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

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 incollection

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 incollection

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

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

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

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. A note on this article was published in Commun. Pure Appl. Math. 42:5 (1989). MR 861491 Zbl 0615.58045 incollection

Cathleen Morawetz	Related	Volume
Shing-Tung Yau	Related

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 article

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

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 incollection

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

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. A note on an article published in Commun. Pure Appl. Math. 39:S1 (1986). MR 997570 Zbl 0678.58041 article

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 incollection

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 incollection

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 article

Karen Uhlenbeck

Karen Uhlenbeck's contributions to gauge theoretic analysis

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