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

Takegoshi, Kensho

doi:10.1090/S0002-9939-98-04609-7

A maximum principle for $P$-harmonic maps with $L^q$ finite energy
HTML articles powered by AMS MathViewer

by Kensho Takegoshi PDF

Proc. Amer. Math. Soc. 126 (1998), 3749-3753 Request permission

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

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
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 385749, DOI 10.1002/cpa.3160280303
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
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
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, DOI 10.1007/BF02392380
Nakauchi,N., A Liouville type theorem for $p$-harmonic maps, preprint.
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 438388, DOI 10.1007/BF02568161
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, DOI 10.1515/crll.1984.348.23
Takegoshi,K., A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds, to appear in Nagoya Mathematical Journal.
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 417452, DOI 10.1512/iumj.1976.25.25051
Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880