Celebratio Mathematica — Uhlenbeck

[1] K. K. Uhlenbeck: The calculus of variations and global analysis. Ph.D. thesis, Brandeis University, 1968. Advised by R. S. Palais. MR 2617502 phdthesis

[2] K. Uhlenbeck: “Morse theory on Banach manifolds,” Bull. Am. Math. Soc. 76 : 1 (1970), pp. 105–106. MR 253381 Zbl 0199.43102 article

[3] K. Uhlenbeck: “Integrals with nondegenerate critical points,” Bull. Am. Math. Soc. 76 : 1 (1970), pp. 125–128. MR 254873 Zbl 0198.43403 article

[4] 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.

[5] K. Uhlenbeck: “Regularity theorems for solutions of elliptic polynomial equations,” pp. 225–231 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 16. American Mathematical Society (Providence), 1970. MR 413168 Zbl 0216.38202 incollection

Stephen Smale	Related
S. S. Chern	Related	Volume

[6] K. Uhlenbeck: “Eigenfunctions of Laplace operators,” Bull. Am. Math. Soc. 78 : 6 (November 1972), pp. 1073–1076. MR 319226 Zbl 0275.58003 article

[7] K. Uhlenbeck: “Bounded sets and Finsler structures for manifolds of maps,” J. Diff. Geom. 7 : 3–4 (1972), pp. 585–595. MR 334273 Zbl 0307.58007 article

Abstract infinite dimensional manifolds modelled on Banach spaces lack much of the topological structure of both finite dimensional manifolds and their linear Banach space models. In this paper we show that certain manifolds of maps between finite dimensional manifolds, or more generally manifolds of sections of a finite dimensional fiber bundle, have an additional natural structure of sets which we call “intrinsically bounded” which have many of the properties of bounded sets in the linear model. Theorem 1 shows that these sets can be characterized in several different ways. Our results are specifically stated for the Sobolev manifolds \( L_k^p(E) \) where \( E \) is a fiber bundle over the compact manifold \( M \) of dimension less than \( pk \). We also construct a canonical Finsler structure for \( L_k^p(E) \) from geometrical structure on \( E \), and find Finsler structures which have intrinsically bounded sets as their bounded sets. The discussion of Finsler structures will be helpful in freeing the use of condition (C) of Palais and Smale in the calculus of variations from the unnaturally arbitrary choice of Finsler structre on the manifolds of maps.

[8] K. Uhlenbeck: “Morse theory on Banach manifolds,” J. Funct. Anal. 10 : 4 (August 1972), pp. 430–445. MR 377979 Zbl 0241.58002 article

Let \( f \) be a \( C^2 \) function on a \( C^2 \) Banach manifold. A critical point \( x \) of \( f \) is said to be weakly nondegenerate if there exists a neighborhood \( U \) of \( x \) and a hyperbolic linear isomorphism \[ L_x: T_x(M)\to T_x(M) \] such that in the coordinate system of \( U \), \[ df_{x+v}(L_xv) > 0 \] if \( v\neq 0 \). \( L_x \) defines an index invariantly, and it is shown that this is an extension of the usual definition of nondegeneracy and index. It is shown that this weaker nondegeneracy can be used in place of the stronger nondegeneracy conditions in Morse theory. In addition, sufficient conditions for the critical points of variational problems to be weakly nondegenerate are given.

[9] K. Uhlenbeck: “A new proof of a regularity theorem for elliptic systems,” Proc. Am. Math. Soc. 37 : 1 (January 1973), pp. 315–316. MR 315282 Zbl 0249.35026 article

[10] K. Uhlenbeck: “The Morse index theorem in Hilbert space,” J. Diff. Geom. 8 : 4 (1973), pp. 555–564. MR 350778 Zbl 0277.58002 article

When does the critical point of a calculus of variations problem minimize the integral? The classical result is due to Jacobi, who proved that for a regular problem in one independent variable, the integral is minimized at a solution of the Euler–Lagrange equation up to the first conjugate point but not after. Morse extended the theorem to give a formula for the index of a critical curve in terms of the conjugate points along the curve. This result has since been generalized by Edwards [1964], Simons [1969] and Smale [1965] to systems of higher order, minimal surfaces, and partial differential systems respectively. In this article we present an infinite dimensional proof of a general theorem on the index of a bilinear form in Hilbert space which can be applied to all these cases.

The first section contains the abstract formulation and proof of the main theorem. The second section deals with single integral problems and the third with multiple integral problems. In the applications we assume less differentiability than the previous results.

[11] K. Uhlenbeck: “Lorentz geometry,” pp. 235–242 in Global analysis and its applications (Trieste, Italy, 4 July–25 August 1972), vol. 3. IAEA Proceedings Series. International Atomic Energy Agency (Vienna), 1974. MR 443820 Zbl 0303.53027 incollection

[12] K. Uhlenbeck: “A Morse theory for geodesics on a Lorentz manifold,” Topology 14 : 1 (March 1975), pp. 69–90. MR 383461 Zbl 0323.58010 article

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

[14] 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

[15] K. Uhlenbeck: “Regularity for a class of non-linear elliptic systems,” Acta Math. 138 : 3–4 (1977), pp. 219–240. MR 474389 Zbl 0372.35030 article

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

[17] 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.

[18] 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

[19] K. K. Uhlenbeck: “Variational problems for gauge fields,” pp. 455–464 in Seminar on differential geometry. Edited by S.-T. Yau. Annals of Mathematics Studies 102. Princeton University Press and University of Tokyo Press, 1982. A later article with the same title was published in Proceedings of the International Congress of Mathematicians (1984). MR 645753 Zbl 0481.58016 incollection

[20] 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

[21] 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

[22] 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.

[23] 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

[24] K. K. Uhlenbeck: “Equivariant harmonic maps into spheres,” pp. 146–158 in Harmonic maps (New Orleans, 15–19 December 1980). Edited by U. R. J. Knill, M. Kalka, and H. C. J. Sealey. Lecture Notes in Mathematics 949. Springer (New York), 1982. MR 673590 Zbl 0505.58015 incollection

Morris Kalka	Related
H. C. J. Sealey	Related
U. R. J. Knill	Related

[25] 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

[26] 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

[27] K. Uhlenbeck: “Conservation laws and their application in global differential geometry,” pp. 103–115 in Emmy Noether in Bryn Mawr (Bryn Mawr, PA, 17–19 March 1982). Edited by B. Srinivasan and J. Sally. Springer (New York), 1983. MR 713794 Zbl 0524.53049 incollection

Bhama Srinivasan	Related
E. Noether	Related	Volume
Judith D. Sally	Related

[28] 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

[29] 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

[30] 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

[31] 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

[32] K. K. Uhlenbeck: “Variational problems in nonabelian gauge theories,” pp. 443–471 in Proceedings of the 1981 Shanghai symposium on differential geometry and differential equations (Shanghai and Hefei, China, 20 August–13 September 1981). Edited by C. Gu. Science Press, 1984. MR 825291 Zbl 0697.58016 incollection

[33] 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

[34] 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

[35] D. Frid and K. Ulenbek: Instantony i chetyrekhmernye mnogoobraziya [Instantons and four-manifolds]. Mir (Moscow), 1988. Translated from the English and with a preface by Yu. P. Solov’ev. Russian translation of 1984 English original. MR 955496 book

Yurii Petrovich Solov’ev	Related
Daniel Stuart Freed	Related

[36] K. Uhlenbeck: “Moment maps in stable bundles,” AWM Newsletter 18 : 3 (May–June 1988), pp. 2. remarks taken from 1988 Noether Lecture. Reprinted in Complexities: Women in mathematics (2005). article

[37] 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

[38] K. Uhlenbeck: “Harmonic maps into Lie groups (classical solutions of the chiral model),” J. Diff. Geom. 30 : 1 (1989), pp. 1–50. Dedicated to R. F. Williams. MR 1001271 Zbl 0677.58020 article

[39] 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

[40] 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

[41] K. K. Uhlenbeck: Applications of nonlinear analysis in topology, 1990. 60 minute videocassette, American Mathematical Society ICM Series. A plenary address presented at the International Congress of Mathematicians held in Kyoto, August 1990. This was published as an article in Proceedings of the International Congress of Mathematicians (1990). MR 1127164 misc

[42] D. S. Freed and K. K. Uhlenbeck: Instantons and four-manifolds, 2nd edition. Mathematical Sciences Research Institute Publications 1. Springer (New York), 1991. MR 1081321 Zbl 0559.57001 book

[43] K. Uhlenbeck: “Applications of nonlinear analysis in topology,” pp. 261–279 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Mathematical Society of Japan (Tokyo), 1991. A video recording of this plenary address was published in 1990. MR 1159217 Zbl 0753.53001 incollection

[44] K. Uhlenbeck: “On the connection between harmonic maps and the self-dual Yang–Mills and the sine-Gordon equations,” J. Geom. Phys. 8 : 1–4 (March 1992), pp. 283–316. MR 1165884 Zbl 0747.58025 article

[45] 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

[46] Global analysis in modern mathematics: Proceedings of the symposium in honor of Richard Palais’ sixtieth birthday (Orono, ME, 8–10 August 1991 and Waltham, MA, 12 August 1992). Edited by K. Uhlenbeck. Publish or Perish (Houston, TX), 1993. MR 1278744 Zbl 0920.00058 book

[47] K. Uhlenbeck: “Preface. Global analysis: A subject before its time,” pp. vii–xvii in Global analysis in modern mathematics: A symposium in honor of Richard Palais’ sixtieth birthday (Orono, ME, 8–10 August 1991 and Waltham, MA, 12 August 1992). Edited by K. Uhlenbeck. Publish or Perish (Houston, TX), 1993. MR 1278745 incollection

[48] M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. MacLane, B. B. Mandelbrot, D. Ruelle, A. Schwarz, K. Uhlenbeck, R. Thom, E. Witten, and C. Zeeman: “Responses to ‘Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., New Ser. 30 : 2 (April 1994), pp. 178–207. Zbl 0803.01014 ArXiv math/9404229 article

M. F. Atiyah	Related	Volume
Benoit B. Mandelbrot	Related
Edward Witten	Related
David Ruelle	Related
René Thom	Related
Daniel Harry Friedan	Related
E. Christopher Zeeman	Related
Arthur Jaffe	Related
Armand Borel	Related
James G. Glimm	Related
Jeremy J. Gray	Related
Albert S. Schwarz	Related	Volume
Frank Stringfellow Quinn, III	Related
S. Mac Lane	Related	Volume
Gregory John Chaitin	Related
Morris W. Hirsch	Related

@article {key0803.01014z,
	AUTHOR = {Atiyah, Michael and Borel, Armand and
		 Chaitin, G. J. and Friedan, Daniel and
		 Glimm, James and Gray, Jeremy J. and
		 Hirsch, Morris W. and MacLane, Saunders
		 and Mandelbrot, Benoit B. and Ruelle,
		 David and Schwarz, Albert and Uhlenbeck,
		 Karen and Thom, Ren\'e and Witten, Edward
		 and Zeeman, Christopher},
	TITLE = {Responses to ``Theoretical mathematics:
		 {T}oward a cultural synthesis of mathematics
		 and theoretical physics'', by {A}.~{J}affe
		 and {F}.~{Q}uinn},
	JOURNAL = {Bull. Am. Math. Soc., New Ser.},
	FJOURNAL = {Bulletin of the American Mathematical
		 Society. New Series},
	VOLUME = {30},
	NUMBER = {2},
	MONTH = {April},
	YEAR = {1994},
	PAGES = {178--207},
	DOI = {10.1090/S0273-0979-1994-00503-8},
	NOTE = {ArXiv:math/9404229. Zbl:0803.01014.},
	ISSN = {0273-0979},
}

[49] G. D. Daskalopoulos and K. K. Uhlenbeck: “An application of transversality to the topology of the moduli space of stable bundles,” Topology 34 : 1 (January 1995), pp. 203–215. MR 1308496 Zbl 0835.58005 article

[50] Geometry and quantum field theory (Park City, UT, 22 June–20 July 1991). Edited by D. S. Freed and K. K. Uhlenbeck. IAS/Park City Mathematics Series 1. American Mathematical Society (Providence, RI), 1995. MR 1338390 Zbl 0819.00004 book

[51] G. Daskalopoulos, K. Uhlenbeck, and R. Wentworth: “Moduli of extensions of holomorphic bundles on Kähler manifolds,” Comm. Anal. Geom. 3 : 3–4 (1995), pp. 479–522. MR 1371207 Zbl 0852.58014 article

Georgios G. Daskalopoulos	Related
Richard Alan Wentworth	Related

[52] R. Mazzeo, D. Pollack, and K. Uhlenbeck: “Connected sum constructions for constant scalar curvature metrics,” Topol. Methods Nonlinear Anal. 6 : 2 (1995), pp. 207–233. Dedicated to Louis Nirenberg on the occasion of his 70th birthday. MR 1399537 Zbl 0866.58069 article

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making “analytic” connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen’s [1988] well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.

Louis Nirenberg	Related
Daniel Pollack	Related
Rafe Roys Mazzeo	Related

[53] K. Uhlenbeck: “Adiabatic limits and moduli spaces,” Notices Am. Math. Soc. 42 (1995), pp. 41–42. This is one section of larger article, “A celebration of women in mathematics”. article

[54] R. Mazzeo, D. Pollack, and K. Uhlenbeck: “Moduli spaces of singular Yamabe metrics,” J. Am. Math. Soc. 9 : 2 (1996), pp. 303–344. MR 1356375 Zbl 0849.58012 article

Complete, conformally flat metrics of constant positive scalar curvature on the complement of \( k \) points in the \( n \)-sphere, \( k \ge 2 \), \( n \ge 3 \), were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension \( k \). For a generic set of nearby conformal classes the moduli space is shown to be a \( k \)-dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

Daniel Pollack	Related
Rafe Roys Mazzeo	Related

[55] K. Uhlenbeck: “Coming to grips with success: A profile of Karen Uhlenbeck,” Math Horizons 3 : 4 (1996). Also pubished in Journeys of women in science and engineering (1997). article

[56] C. Henrion: “Karen Uhlenbeck (1942–),” pp. 25–46 in Women in mathematics: The addition of difference. Indiana University Press (Bloomington and Indianapolis, IN), 1997. incollection

[57] K. Uhlenbeck: “Karen Uhlenbeck,” pp. 395–398 in Journeys of women in science and engineering: No universal constants. Edited by S. Ambrose, K. L. Dunkle, B. B. Lazarus, I. Nair, and D. A. Harkus. Labor & Social Change 71. Temple University Press (Philadelphia), 1997. Also published in Math Horizons 3:4 (1996). incollection

[58] Geometry, topology and physics: Proceedings of the first Brazil–USA workshop (Campinas, Brazil, 30 June–7 July 1996). Edited by B. N. Apanasov, S. B. Bradlow, W. A. Rodrigues, Jr., and K. K. Uhlenbeck. de Gruyter (Berlin), 1997. MR 1605264 Zbl 0883.00022 book

Boris Nikolaevich Apanasov	Related
Waldyr Alves Rodrigues, Jr.	Related
Steven Benjamin Bradlow	Related

[59] Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Cambridge, MA), 1998. MR 1726558 Zbl 0918.00013 book

[60] C.-L. Terng and K. Uhlenbeck: “Poisson actions and scattering theory for integrable systems,” pp. 315–402 in Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Boston), 1998. MR 1726931 Zbl 0935.35163 ArXiv dg-ga/9707004 incollection

Conservation laws, hierarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS \( n{\times}n \) hierarchy (which includes the non-linear Schrödinger equation, modified KdV, and the \( n \)-wave equation). We first find a simple model Poisson group action that contains flows for systems with a Lax pair whose terms all decay on \( R \). Bäcklund transformations and flows arise from subgroups of this single Poisson group. For the ZS-AKNS \( n{\times}n \) hierarchy defined by a constant \( a\in\mathfrak{u}(n) \), the simple model is no longer correct. The \( a \) determines two separate Poisson structures. The flows come from the Poisson action of the centralizer \( H_a \) of \( a \) in the dual Poisson group (due to the behaviour of \( e^{a\lambda x} \) at infinity). When \( a \) has distinct eigenvalues, \( H_a \) is abelian and it acts symplectically. The phase space of these flows is the space \( S_a \) of left cosets of the centralizer of \( a \) in \( D_- \), where \( D_- \) is a certain loop group. The group \( D_- \) contains both a Poisson subgroup corresponding to the continuous scattering data, and a rational loop group corresponding to the discrete scattering data. The \( H_a \)-action is the right dressing action on \( S_a \). Bäcklund transformations arise from the action of the simple rational loops on \( S_a \) by right multiplication. Various geometric equations arise from appropriate choice of \( a \) and restrictions of the phase space and flows. In particular, we discuss applications to the sine-Gordon equation, harmonic maps, Schrödinger flows on symmetric spaces, Darboux orthogonal coordinates, and isometric immersions of one space-form in another.

[61] L. Taylor: “Karen Uhlenbeck,” pp. 261–266 in Notable women in mathematics: A biographical dictionary. Edited by C. Morrow and T. Perl. Greenwood Press (Westport, CT), 1998. incollection

Charlene Morrow	Related
Teri Hoch Perl	Related

[62] 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

[63] C.-L. Terng and K. Uhlenbeck: “Bäcklund transformations and loop group actions,” Comm. Pure Appl. Math. 53 : 1 (2000), pp. 1–75. MR 1715533 Zbl 1031.37064 article

We construct a local action of the group of rational maps from \( \mathbb{S}^2 \) to \( \mathrm{GL}(n,\mathbb{C}) \), on local solutions of flows of the ZS-AKNS \( \mathfrak{sl}(n,\mathbb{C}) \)-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different factorizations of a quadratic element. We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities. However, the action of the subgroup of the rational maps that satisfy the \( U(n) \)-reality condition \[ g(\overline{\lambda})*g(\lambda) = I \] on the space of global rapidly decaying solutions of the flows in the \( \mathfrak{u}(n) \)-hierarchy is global, and the action of a simple element gives a global Bäcklund transformation. The actions of certain elements in the rational loop group on the vacuum give rise to explicit time-periodic multisolitons (multibreathers). We show that this theory generalizes the classical Bäcklund theory of the sine-Gordon equation. The group structures of Bäcklund transformations for various hierarchies are determined by their reality conditions. We identify the reality conditions (the group structures) for the \( \mathfrak{sl}(n,\mathbb{R}) \), \( \mathfrak{u}(k,n-k) \), KdV, Kupershmidt–Wilson, and Gel’fand–Dikii hierarchies. The actions of linear fractional transformations that satisfy a reality condition, modulo the center of the group of rational maps, give Bäcklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under this reality condition, the action is again local, and Bäcklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed.

[64] C.-L. Terng and K. Uhlenbeck: “Geometry of solitons,” Notices Am. Math. Soc. 47 : 1 (2000), pp. 17–25. cover article. MR 1733063 Zbl 0987.37072 article

[65] N.-H. Chang, J. Shatah, and K. Uhlenbeck: “Schrödinger maps,” Comm. Pure Appl. Math. 53 : 5 (2000), pp. 590–602. MR 1737504 Zbl 1028.35134 article

Nai-Heng Chang	Related
Jalal M. Shatah	Related

[66] K. K. Uhlenbeck and J. A. Viaclovsky: “Regularity of weak solutions to critical exponent variational equations,” Math. Res. Lett. 7 : 5–6 (2000), pp. 651–656. MR 1809291 Zbl 0977.58020 article

[67] “Two mathematicians awarded National Medals of Science,” MAA FOCUS 21 : 1 (January 2001), pp. 3. article

[68] G. Warfield: “Uhlenbeck receives National Medal of Science,” AWM Newsletter 31 : 1 (January–February 2001), pp. 9–10. Reprinted in Complexities: Women in mathematics (2005). article

[69] A. Nahmod, A. Stefanov, and K. Uhlenbeck: “On Schrödinger maps,” Comm. Pure Appl. Math. 56 : 1 (2003), pp. 114–151. An erratum was published in Comm. Pure Appl. Math. 57:6 (2004). MR 1929444 Zbl 1028.58018 article

Andrea Rica Nahmod	Related
Atanas Stefanov	Related

[70] A. Nahmod, A. Stefanov, and K. Uhlenbeck: “On the well-posedness of the wave map problem in high dimensions,” Comm. Anal. Geom. 11 : 1 (2003), pp. 49–83. MR 2016196 Zbl 1085.58022 article

Andrea Rica Nahmod	Related
Atanas Stefanov	Related

[71] Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck (Cambridge, MA, 3–5 May 2002). Edited by S.-T. Yau. Surveys in Differential Geometry 8. International Press (Somerville, MA), 2003. Zbl 1034.53003 book

H. Blaine Lawson, Jr.	Related
Yum-Tong Siu	Related
Shing-Tung Yau	Related
Eugenio Calabi	Related

[72] A. Nahmod, A. Stefanov, and K. Uhlenbeck: “Erratum: ‘On Schrödinger maps’,” Comm. Pure Appl. Math. 57 : 6 (2004), pp. 833–839. Erratum to article published in Comm. Pure Appl. Math. 56:1 (2003). MR 2038118 article

Andrea Rica Nahmod	Related
Atanas Stefanov	Related

[73] C.-L. Terng and K. Uhlenbeck: “\( 1+1 \) wave maps into symmetric spaces,” Comm. Anal. Geom. 12 : 1–2 (2004), pp. 345–388. MR 2074882 Zbl 1082.37068 article

We explain how to apply techniques from integrable systems to construct \( 2k \)-soliton homoclinic wave maps from the periodic Minkowski space \( \mathbb{S}^1\times\mathbb{R}^1 \) to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the \( -1 \) flow equation associated to a compact Lie group \( G \) and wave maps into \( G \). We use Bäcklund transformations to construct explicit \( 2k \)-soliton breather solutions for the \( -1 \) flow equation and show that the corresponding wave maps are periodic and homoclinic. The compact symmetric space \( G/K \) can be embedded as a totally geodesic submanifold of \( G \) via the Cartan embedding. We prescribe the constraint condition for the \( -1 \) flow equation associated to \( G \) which insures that the corresponding wave map into \( G \) actually lies in \( G/K \). For example, when \[ G/K = \mathrm{SU}(2)/\mathrm{SO}(2) = \mathbb{S}^2 ,\] the constrained \( -1 \)-flow equation associated to \( \mathrm{SU}(2) \) has the sine-Gordon equation (SGE) as a subequation and classical breather solutions of the SGE are 2-soliton breathers. Thus our result generalizes the result of Shatah and Strauss that a classical breather solution of the SGE gives rise to a periodic homoclinic wave map to \( \mathbb{S}^2 \). When the group \( G \) is non-compact, the bi-invariant metric on \( G \) is pseudo-Riemannian and Bäcklund transformations of a smooth solution often are singular. We use Bäcklund transformations to show that there exist smooth initial data with constant boundary conditions and finite energy such that the Cauchy problem for wave maps from \( \mathbb{R}^{1,1} \) to the pseudo-Riemannian manifold \( \mathrm{SL}(2,\mathbb{R}) \) develops singularities in finite time.

[74] C.-L. Terng and K. Uhlenbeck: “Schrödinger flows on Grassmannians,” pp. 235–256 in Integrable systems, geometry, and topology. Edited by C.-L. Terng. AMS/IP Studies in Advanced Mathematics 36. American Mathematical Society (Providence, RI), 2006. MR 2222517 Zbl 1110.37056 ArXiv math/9901086 incollection

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold \( M \) of \( k \)-planes in \( C^n \) is the evolution equation on the space \( C(\mathbb{R},M) \) of paths on \( M \): \[ J_{\lambda}(\gamma_{\lambda}) = \nabla_{\gamma_x}\gamma_x, \] where \( \nabla \) is the Levi–Civita connection of the Kähler metric and \( J \) is the complex structure. GNLS is the Hamiltonian equation for the energy functional on \( C(\mathbb{R},M) \) with respect to the symplectic form induced from the Kähler form on \( M \). It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrödinger equation (MNLS) for \( q \) from \( \mathbb{R}^2 \) to the space of complex \( k{\times}(n-k) \) matrices: \[ q_t = \tfrac{i}{2}(q_{xx} + 2qq^*q). \] We construct via gauge transformations an isomorphism from \( C(\mathbb{R},M) \) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order \( k \) symplectic structures correspond to the order \( k-2 \) symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice.

[75] B. Dai, C.-L. Terng, and K. Uhlenbeck: “On the space-time monopole equation,” pp. 1–30 in Essays in geometry in memory of S. S. Chern. Edited by S.-T. Yau. Surveys in Differential Geometry 10. International Press (Somerville, MA), 2006. MR 2408220 Zbl 1157.53016 incollection

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang–Mills equation on \( \mathbb{R}^{2,2} \). A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas–Ioannidou, Villarroel and Zakhorov–Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

Chuu-Lian Terng	Related
S. S. Chern	Related	Volume
Shing-Tung Yau	Related
Bo Dai	Related

[76] K. Uhlenbeck: “Moment maps in stable bundles,” pp. 193–195 in Complexities: Women in mathematics. Edited by B. A. Case and A. Leggett. Princeton University Press, 2006. Based on a piece published in AWM Newsletter 18:3 (1988). incollection

Anne M. Leggett	Related
Bettye Anne Case	Related

[77] G. Warfield: “Uhlenbeck receives National Medal of Science,” pp. 195–196 in Complexities: Women in mathematics. Edited by B. A. Case and A. Leggett. Princeton University Press, 2006. Reprinted from AWM Newsletter 31:1 (2001). incollection

Anne M. Leggett	Related
Bettye Anne Case	Related

[78] 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

[79] “2007 Steele Prizes,” Notices Am. Math. Soc. 54 : 4 (April 2007), pp. 514–518. Zbl 1142.01310 article

Henry P. McKean, Jr.	Related
David Bryant Mumford	Related

[80] M. Vajiac and K. Uhlenbeck: “Virasoro actions and harmonic maps (after Schwarz),” J. Diff. Geom. 80 : 2 (2008), pp. 327–341. MR 2454896 Zbl 1153.53046 article

John Henry Schwarz	Related
Mihaela Brînduşa Vajiac	Related

[81] C.-L. Terng and K. Uhlenbeck: “The \( n{\times}n \) KdV hierarchy,” J. Fix. Point Theory A. 10 : 1 (2011), pp. 37–61. MR 2825739 Zbl 1251.37070 article

[82] M. Gagliardo and K. Uhlenbeck: “Geometric aspects of the Kapustin–Witten equations,” J. Fix. Point Theory Ap. 11 : 2 (2012), pp. 185–198. MR 3000667 Zbl 1260.53002 article

[83] Regularity and evolution of nonlinear equations: Essays dedicated to Richard Hamilton, Leon Simon, and Karen Uhlenbeck. Edited by H.-D. Cao, R. Schoen, and S.-T. Yau. Surveys in Differential Geometry 19. International Press (Somerville, MA), 2015. MR 3380570 Zbl 1317.53001 book

Richard Melvin Schoen	Related
Richard Streit Hamilton	Related
Shing-Tung Yau	Related
Leon Melvyn Simon	Related
Huai-Dong Cao	Related

[84] C.-L. Terng and K. Uhlenbeck: “Tau function and Virasoro action for the \( n{\times}n \) KdV hierarchy,” Comm. Math. Phys. 342 : 1 (2016), pp. 81–116. MR 3455146 Zbl 1354.37068 article

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group \( L \) as positive and negative subgroups \( L_{\pm} \) and a commuting sequence in the Lie algebra \( \mathcal{L}_+ \) of \( L_+ \). Given \( f\in L_- \), there is a formal inverse scattering solution \( u_f \) of the hierarchy. When there is a 2 co-cycle on \( \mathcal{L} \) that vanishes on both \( \mathcal{L}_+ \) and \( \mathcal{L}_- \), Wilson constructed for each \( f\in L_- \) a tau function \( \tau_f \) for the hierarchy. In this third paper, we prove the following results for the \( n{\times}n \) KdV hierarchy:

The second partials of \( \ln\tau_f \) are differential polynomials of the formal inverse scattering solution \( u_f \). Moreover, \( u_f \) can be recovered from the second partials of \( \ln\tau_f \).
The natural Virasoro action on \( \ln\tau_f \) constructed in the second paper is given by partial differential operators in \( \ln\tau_f \).
There is a bijection between phase spaces of \( n{\times}n \) KdV hierarchy and Gelfand–Dickey (\( \mathrm{GD}_n \)) hierarchy on the space of order \( n \) linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection.
Our Virasoro action on the \( n{\times}n \) KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the \( \mathrm{GD}_n \) hierarchy under the bijection.

[85] C.-L. Terng and K. Uhlenbeck: “Tau functions and Virasoro actions for soliton hierarchies,” Comm. Math. Phys. 342 : 1 (2016), pp. 117–150. MR 3455147 Zbl 1346.37058 article

There is a general method for constructing a soliton hierarchy from a splitting \( L_{\pm} \) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \( \mathcal{L}_+ \). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each \( f \) in the negative subgroup \( L_- \) a solution \( u_f \) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \( \tau_f \) for each element \( f\in L_- \). In this paper, we give integral formulas for variations of \( \ln \tau_f \) and second partials of \( \ln \tau_f \), discuss whether we can recover solutions \( u_f \) from \( \tau_f \), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \( \mathrm{GL}(n,\mathbb{C}) \)-hierarchy.

Complete Bibliography