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)

 

 

A maximum principle for $P$-harmonic maps
with $L^{q}$ finite energy


Author: Kensho Takegoshi
Journal: Proc. Amer. Math. Soc. 126 (1998), 3749-3753
MSC (1991): Primary 58D15, 58E20
MathSciNet review: 1469437
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show a maximum principle for $P$-harmonic maps with $L^q$-finite energy. As an application we can generalize a non-existence theorem for harmonic maps with finite Dirichlet integral by Schoen and Yau to those maps.


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

  • [C] Shiu Yuen Cheng, Liouville theorem for harmonic maps, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 147–151. MR 573431
  • [C-Y] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 0385749
  • [E-L] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, 10.1112/blms/10.1.1
  • [G-H] Samuel I. Goldberg and Zvi Har’El, A general Schwarz lemma for Riemannian-manifolds, Bull. Soc. Math. Grèce (N.S.) 18 (1977), no. 1, 141–148. MR 528427
  • [L-S] Peter Li and Richard Schoen, 𝐿^{𝑝} and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), no. 3-4, 279–301. MR 766266, 10.1007/BF02392380
  • [N] Nakauchi,N., A Liouville type theorem for $p$-harmonic maps, preprint.
  • [Sc-Y] 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 0438388
  • [Sh] Chun Li Shen, A generalization of the Schwarz-Ahlfors lemma to the theory of harmonic maps, J. Reine Angew. Math. 348 (1984), 23–33. MR 733920, 10.1515/crll.1984.348.23
  • [T] Takegoshi,K., A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds, to appear in Nagoya Mathematical Journal.
  • [Y-1] Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 0417452
  • [Y-2] Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 0486659

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58D15, 58E20

Retrieve articles in all journals with MSC (1991): 58D15, 58E20


Additional Information

Kensho Takegoshi
Affiliation: Department of Mathematics, Graduate School of Science, Machikaneyama-cho 1-16, Toyonaka-shi Osaka, 560 Japan
Email: kensho@math.wani.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04609-7
Received by editor(s): April 21, 1997
Communicated by: Peter Li
Article copyright: © Copyright 1998 American Mathematical Society