Deforming a map into a harmonic map

Yang, Deane

doi:10.1090/S0002-9947-99-02402-2

Deforming a map into a harmonic map
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Trans. Amer. Math. Soc. 352 (2000), 1021-1038 Request permission

Abstract:

This paper presents some existence and uniqueness theorems for harmonic maps between complete noncompact Riemannian manifolds. In particular, we obtain as a corollary a recent result of Hardt–Wolf on the existence of harmonic quasi-isometries of the hyperbolic plane.

References

J. P. Anderson, Ph. D. dissertation, 1991.
Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
R. R. Coifman and S. Semmes, Interpolation of Banach spaces, Perron processes, and Yang-Mills, Amer. J. Math. 115 (1993), no. 2, 243–278. MR 1216432, DOI 10.2307/2374859
Wei Yue Ding and You De Wang, Harmonic maps of complete noncompact Riemannian manifolds, Internat. J. Math. 2 (1991), no. 6, 617–633. MR 1137089, DOI 10.1142/S0129167X9100034X
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Robert Hardt and Michael Wolf, Harmonic extensions of quasiconformal maps to hyperbolic space, Indiana Univ. Math. J. 46 (1997), no. 1, 155–163. MR 1462800, DOI 10.1512/iumj.1997.46.1351
Willi Jäger and Helmut Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, Manuscripta Math. 28 (1979), no. 1-3, 269–291. MR 535705, DOI 10.1007/BF01647975
Willi Jäger and Helmut Kaul, Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds, Math. Ann. 240 (1979), no. 3, 231–250. MR 526846, DOI 10.1007/BF01362313
Jürgen Jost and Hermann Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), no. 1, 27–77 (German, with English summary). MR 679120, DOI 10.1007/BF01168235
—, Almost linear functions and a priori estimates for harmonic maps, Global Riemannian Geometry (T.J. Willmore and N.J. Hitchin, eds.), Ellis Horwood Ser. Math. Appl., E. Horwood, Chichester, 1984, pp. 148–155.
Serge Lang, Differential and Riemannian manifolds, 3rd ed., Graduate Texts in Mathematics, vol. 160, Springer-Verlag, New York, 1995. MR 1335233, DOI 10.1007/978-1-4612-4182-9
Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549
Richard Schoen and Shing Tung Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), no. 3, 333–341. MR 438388, DOI 10.1007/BF02568161
Luen-Fai Tam and Tom Y.-H. Wan, On quasiconformal harmonic maps, Pacific J. Math. 182 (1998), 359–383.
Deane Yang, Convergence of Riemannian manifolds with integral bounds on curvature. II, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 2, 179–199. MR 1169351
Ju. M. Movsisjan, Generalized distributivity identities of nonzero rank, Erevan. Gos. Univ. Učen. Zap. Estestv. Nauki 3 (124) (1973), 11–16 (Russian, with Armenian summary). MR 0432514