Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Harmonic maps with finite total energy


Authors: Shiu-Yuen Cheng, Luen-Fai Tam and Tom Y.-H. Wan
Journal: Proc. Amer. Math. Soc. 124 (1996), 275-284
MSC (1991): Primary 53C99; Secondary 31C05, 58E20
DOI: https://doi.org/10.1090/S0002-9939-96-03170-X
MathSciNet review: 1307503
Full-text PDF

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 $M$, every harmonic function with finite energy is bounded, then every harmonic map with finite total energy from $M$ into a Cartan-Hadamard manifold must also have bounded image. No assumption on the curvature of $M$ is required. As a consequence, we will generalize some of the uniqueness results on homotopic harmonic maps by Schoen and Yau.


References [Enhancements On Off] (What's this?)

  • 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

Luen-Fai Tam
Email: ltam@math.uci.edu

Tom Y.-H. Wan
Email: tomwan@cuhk.hk

DOI: https://doi.org/10.1090/S0002-9939-96-03170-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
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society