Original Paper

Annals of Global Analysis and Geometry

, Volume 31, Issue 4, pp 345-362

First online:

An extension of Barta’s Theorem and geometric applications

  • G. Pacelli BessaAffiliated withDepartamento de Matematica, Universidade Federal do Ceará-UFC Email author 
  • , J. Fábio MontenegroAffiliated withDepartamento de Matematica, Universidade Federal do Ceará-UFC

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Abstract

We prove an extension of a theorem of Barta and we give some 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 show that the spectrum of the Nadirashvili bounded minimal surfaces in \(\mathbb{R}^{3}\) have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations.

Keywords

Bartas’s Theorem Cheng’s Eigenvalue Comparison Theorem Spectrum of Nadirashvili minimal surfaces Stability of minimal hypersurfaces

Mathematics Subject Classification (2000)

58C40 53C42