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)



Bounded harmonic maps on a class of manifolds

Authors: Chiung-Jue Sung, Luen-fai Tam and Jiaping Wang
Journal: Proc. Amer. Math. Soc. 124 (1996), 2241-2248
MSC (1991): Primary 58E20
MathSciNet review: 1307567
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Without imposing any curvature assumptions, we show that bounded harmonic maps with image contained in a regular geodesic ball share similar behaviour at infinity with the bounded harmonic functions on the domain manifold.

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

  • [A-C-M] P. Avilés, H.I. Choi, and M. Micallef, Boundary behavior of harmonic maps on non-smooth domains and complete negatively curved manifolds, J. Functional Anal. 99 (1991), 293--331. MR 92j:58025
  • [Cai] M. Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. AMS 24 (1991), 371--377. MR 92f:53045
  • [C-G 2] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 92 (1972), 413--443. MR 46:8121
  • [Cg] S. Y. Cheng, Liouville Theorem for Harmonic Maps, Proc. of Symposia in Pure Math. 36 (1980), 147--151. MR 81i:58021
  • [C-Y] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333--354. MR 52:6608
  • [Ch] H. I. Choi,, On the Liouville theorem for harmonic maps, Proc. AMS 85 (1982), 91--94. MR 83j:53073
  • [Go] W. B. Gordon, Convex functions and harmonic maps, Proc. AMS 33 (1972), 433--437. MR 45:1075
  • [G-W] R. E. Green and H. Wu, Function theory on Manifolds which possess a pole, Lecture Notes in Math. 699 (1979).
  • [H-K-W] S. Hildebrandt, H. Kaul and K. -O. Widman, An existence theory for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), 1--16. MR 55:6478
  • [J-K] Jäger, W. and Kaul, H., Uniqueness and stability of harmonic maps and their Jocobi fields, Manu. Math. 28 (1979), 269--291. MR 80j:58030
  • [K 1] A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Scient. Ec. Norm. Sup. 21 (1988), 593--622. MR 90d:53049
  • [K 2] A. Kasue, Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I, Geometry and Analysis on Manifolds, Springer-Verlag Lecture Notes in Mathematics 1339, 1988, pp. 158--181. MR 89i:53030
  • [Ke] W. S. Kendall,, Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence, Proc. London Math. Soc. 61 (3) (1990), 371--406. MR 91g:58062
  • [L-T 1] P. Li and L. F. Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set., Annals Math. 125 (1987), 171--207. MR 88m:58039
  • [L-T 2] 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 3] P. Li and L. F. Tam, Complete surfaces with finite total curvature, J. Diff. Geom. 33 (1991), 139--168. MR 92e:53051
  • [L-T 4] P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Diff. Geom. 35 (1992), 359--383. MR 93b:53033
  • [L-T 5] P. Li and L. F. Tam, Green's Functions, Harmonic Functions and Volume Comparison, J. Diff. Geom., vol. 41, 1995, pp. 277--318.
  • [Liu] Z-D. Liu, Ball covering on manifolds with nonnegative Ricci curvature near infinity, preprint.
  • [S] C. J. Sung, Harmonic functions under quasi-isometry, to appear in J. Geom. Anal..
  • [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): 58E20

Retrieve articles in all journals with MSC (1991): 58E20

Additional Information

Chiung-Jue Sung
Affiliation: Department of Mathematics, National Chung Cheng University, Chia-Yi, Taiwan 62117

Luen-fai Tam
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Jiaping Wang
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): December 16, 1994
Additional Notes: The first author was partially supported by NSC grant# 830208M194030.
The second author was partially supported by NSF grant #DMS9300422 .
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society