An Extension of Barta's Theorem and Geometric Applications

Details An Extension of Barta's Theorem and Geometric Applications An Extension of Barta's Theorem and Geometric Applications

2003-08-11

arxiv.org/abs/math/0308099v4

Ann. Global Anal. Geom. 31 (2007), no. 4, 345--362.

Mathematics

Differential Geometry

23 pages. This paper is an improved version of our paper of the same Titled posted here

Scientific paper

We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng's lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a generalization of a result of Kazdan-Kramer about existence of solutions of certain quasi-linear elliptic equations.

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