Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-18T01:52:27.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic maps from noncompact Riemannian manifolds with non-negative Ricci curvature outside a compact set

Published online by Cambridge University Press:  14 November 2011

Youde Wang
Youde Wang
Affiliation:
Institute of Mathematics, Academia Sinica, Beijing, China
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete, noncompact Riemannian manifold (M, g) with Sobolev constant S2(M) > 0 and Ricci curvature Ric (M) ≧ 0 outside some compact subset, into a complete manifold of nonpositive curvature or a regular ball. In particular, we prove the uniqueness and existence of bounded harmonic functions on (M, g).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1259 - 1275
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Berger, M., Gauduchon, P. and Mazet, E.. Le spectre d'une variété Riemannienne. Lecture Notes in Mathematics 194 (1971), 134–138.Google Scholar
2Burstall, F.. Harmonic maps of finite energy from non-compact manifolds. J. London Math. Soc. 30 (1984), 361–370.CrossRefGoogle Scholar
3Cheng, S. Y.. Liouville theorem for harmonic maps. Proc. Sympos. Pure Math. 36 (1980), 147–151.CrossRefGoogle Scholar
4Choi, H. I.. On the Liouville theorem for harmonic maps. Proc. Amer. Math. Soc. 85 (1982), 91–94.CrossRefGoogle Scholar
5Ding, W. Y. and Wang, Y.. Harmonic maps of complete noncompact Riemannian manifolds (Preprint, 1991).CrossRefGoogle Scholar
6Donnelly, H.. Bounded harmonic functions and positive Ricci curvature. Math. Z. 191 (1986), 559–565.CrossRefGoogle Scholar
7Eells, J. and Lemaire, L.. Another report on Harmonic maps. Bull. London Math. Soc. 20 (1988), 385–524.CrossRefGoogle Scholar
8Eells, J. and Sampson, J. H.. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160.CrossRefGoogle Scholar
9Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).Google Scholar
10Hamilton, R.. Harmonic Maps of Manifolds with Boundary, Lecture Notes in Mathematics 471 (Berlin: Springer, 1975).CrossRefGoogle Scholar
11Hildebrandt, S.. Harmonic Mappings and Minimal Immersions, Lecture Notes in Mathematics 1161 (New York: Springer, 1984).Google Scholar
12Jager, W. and Kaul, H.. Uniqueness and stability of harmonic maps and their Jacobi fields. Manuscripta Math. 28 (1979), 269–291.CrossRefGoogle Scholar
13Jost, J.. Harmonic Mappings between Riemannian Manifolds, Proc. Centre Math. Anal. 4 (Canberra: Australian National University Press, 1983).Google Scholar
14Li, P.. On the structure of complete Kahler manifolds with nonnegative curvature near infinity. Inventionaes Math. 99 (1990), 579–600.CrossRefGoogle Scholar
15Li, P. and Tam, L. F.. Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set. Ann. of Math. 125 (1987), 171–207.CrossRefGoogle Scholar
16Li, P. and Tam, L. F.. The heat equation and harmonic maps of complete manifolds (Preprint, 1989).Google Scholar
17Schoen, R. and Uhlenbeck, K.. A regularity theory for harmonic maps. J. Differential Geometry 17 (1982), 307–335.CrossRefGoogle Scholar
18Schoen, R. and Uhlenbeck, K.. Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geometry 18 (1983), 253–268.CrossRefGoogle Scholar
19Schoen, R. and Yau, S. T.. Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature. Comm. Math. Helv. 39 (1976), 333–341.CrossRefGoogle Scholar
20Schoen, R. and Yau, S. T.. Compact group actions and the topology of manifolds with nonpositive curvature. Topology 18 (1979), 361–380.CrossRefGoogle Scholar
21Yau, S. T.. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar
22Yau, S. T.. Some function-theoretic properties of complete Riemannian manifolds and their application to geometry. Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar
23Yau, S. T.. Survey on partial differential equations in differential geometry. Ann. of Math. Stud. 102 (1982), 3–71.Google Scholar