Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T02:59:04.915Z Has data issue: false hasContentIssue false
  • English
  • Français

A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

Published online by Cambridge University Press:  22 January 2016

Kensho Takegoshi
Kensho Takegoshi*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka, 560-0043Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 151 , September 1998 , pp. 25 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[B-K] Baikoussis, C. H. and Koufogiorgos, T. H., Harmonic maps into a cone, Arch. Math., 40 (1983), 372–376.Google Scholar
[C-L] Cheng, K.-S. and Lin, J.-T., On the elliptic equations Δu = K(x)uσ and Δu = K(x)e2u , Trans. Amer. Math. Soc., 304 (1987), 639–668.Google Scholar
[C-Y] Cheng, S. Y. and Yau, S. T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975), 333–354.Google Scholar
[C-X] Chen, Q. and Xin, Y. L., A generalized maximum principle and its applications in geometry, Amer. J. Math., 114 (1992), 355–366.Google Scholar
[G-W] Greene, R. E. and Wu, Y., Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer-Verlag.Google Scholar
[H] Hiriart-Urruty, J. B., A short proof of the variational principle for approximate solutions of a minimaization problem, Amer. Math. Monthly, 90 (1983), 206–207.Google Scholar
[H-J-W] Hildebrandt, J., Jost, J. and Widman, K. O, Harmonic mappings and minimal submanifolds, Invent. Math., 62 (1980), 269–298.Google Scholar
[Kr] Karp, L., Differential inequalities on complete Riemannian manifolds and applications, Math. Ann., 272 (1985), 449–459.Google Scholar
[Ks] Kasue, A., Estimates for solutions of Poisson equations and their applications to submanifolds, Lecture Notes in Math. Differential Geometry of submanifolds, 1090, Springer-Verlag (1984), pp. 1–14.Google Scholar
[L] Li, P., Uniqueness of L1 solutions for the Laplace equation and the heat equation on Riemannian manifolds, J. Differential Geom., 20 (1984), 447–457.CrossRefGoogle Scholar
[L-S] Li, P. and Schoen, R., Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math., 153 (1984), 279–301.CrossRefGoogle Scholar
[M-Y] Mok, N. and Yau, S. T., Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, Proc. Sympos. Pure Math., 39, Amer. Math. Soc, Providence, R. I. (1983), pp. 41–59.Google Scholar
[N] Ni, W.-M., On the elliptic equation Δu + K(x)e2u = 0 and conformal metrics with prescribed Gaussian curvature, Invent. Math., 66 (1982), 343–352.Google Scholar
[Om] Omori, H., Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19 (1967), 205–214.Google Scholar
[Os] Osserman, R., On the inequality Δu ≥ f(u), Pacific J. Math., 7 (1957), 1641–1647.CrossRefGoogle Scholar
[R-R] Ratto, A. and Rigoli, M., Elliptic differential inequalities with applications to harmonic maps, J. Math. Soc. Japan, 45 (1993), 321–337.Google Scholar
[R-R-S] Ratto, A., Rigoli, M. and Setti, G., On the Omori-Yau maximum principle and its applications to differential equations and geometry, J. Funct. Anal., 134 (1995), 486–510.CrossRefGoogle Scholar
[Ry] Royden, H. L., The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv., 55 (1980), 547–558.Google Scholar
[Y-1] Yau, S. T., Harmonic function on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201–228.Google Scholar
[Y-2] Yau, S. T., Some function theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 656–670.Google Scholar
[Y-3] Yau, S. T., A general Schwarz lemma for Kähler manifolds, Amer. J. Math., 100 (1978), 197–203.Google Scholar