|
Harmonic maps with finite total energy
Author(s):
Shiu-Yuen
Cheng;
Luen-Fai
Tam;
Tom
Y.-H.
Wan
Journal:
Proc. Amer. Math. Soc.
124
(1996),
275-284.
MSC (1991):
Primary 53C99;
Secondary 31C05, 58E20
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We will give a criteria for a nonnegative subharmonic function with finite energy on a complete manifold to be bounded. Using this we will prove that if on a complete noncompact Riemannian manifold  , every harmonic function with finite energy is bounded, then every harmonic map with finite total energy from  into a Cartan-Hadamard manifold must also have bounded image. No assumption on the curvature of  is required. As a consequence, we will generalize some of the uniqueness results on homotopic harmonic maps by Schoen and Yau.
References:
- Cg
- S. Y. Cheng, Liouville theorem for harmonic maps, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 147--151, MR 81i:58021.
- Ci
- H. I. Choi, On the Liouville theorem for harmonic maps, Proc. Amer. Math. Soc. 85 (1982), 91--94, MR 83j:53073.
- Gr
- A. A. Grigor
 yan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1) (1991), 55--87; English transl., Math. USSR-Sb. 72 (1992), no. 1, 47--77, MR 92h:58189. - K
- A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4) 21 (1988), 593--622, MR 90d:53049.
- Ke
- W. S. Kendall, Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence, Proc. London Math. Soc. (3) 61 (1990), 371--406, MR 91g:58062.
- L-T 1
- P. Li and L. F. Tam, Symmetric Green's functions on complete manifolds, Amer. J. Math. 109 (1987), 1129--1154, MR 89f:58129.
- L-T 2
- ------, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1--46, MR 93e:58039.
- L-T 3
- ------, Green's functions, harmonic functions and volume comparison, J. Differential Geom. 41 (1995), 277--318.
- Ly
- T. J. Lyons, private communication.
- SC
- L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417--450, MR 93m:58122.
- S-S-G
- L. Sario, M. Schiffer, and M. Glasner, The span and principal functions in Riemannian spaces, J. Analyse Math. 15 (1965), 115--134, MR 32:1655.
- S-Y 1
- R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature, Comment. Math. Helv. 39 (1976), 333--341, MR 55:11302.
- S-Y 2
- ------, Compact group actions and the topology of manifolds with non-positive curvature, Topology 18 (1979), 361--380, MR 81a:53044.
- S-T-W
- J.-T. Sung, L.-F. Tam, and J.-P. Wang, Bounded harmonic maps on a class of manifolds, Proc. Amer. Math. Soc. (to appear).
- V
- N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. (821--837), MR 85a:58103.
- W
- J.-P. Wang, private communication.
- Y
- S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201--228, MR 55:4042.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(1991):
53C99,
31C05, 58E20
Retrieve articles in all Journals with MSC
(1991):
53C99,
31C05, 58E20
Additional Information:
Shiu-Yuen
Cheng
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90024 - Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Luen-Fai
Tam
Affiliation:
Department of Mathematics, University of California, Irvine, California 92717-3875
Address at time of publication:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email:
ltam@math.uci.edu
Tom
Y.-H.
Wan
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email:
tomwan@cuhk.hk
DOI:
10.1090/S0002-9939-96-03170-X
PII:
S 0002-9939(96)03170-X
Received by editor(s):
July 28, 1994
Additional Notes:
The first and the third authors are partially supported by Earmarked Grant, Hong Kong, and the second author is partially supported by NSF grant \#DMS9300422.
Communicated by:
Peter Li
Copyright of article:
Copyright
1996,
American Mathematical Society
|
Comments: webmaster@ams.org