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An extension of Barta’s Theorem and geometric applications

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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.

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Authors and Affiliations

  1. Departamento de Matematica, Universidade Federal do Ceará-UFC, Campus do Pici, Bloco. 914, 60455-760, Fortaleza-Ceará, Brazil

    G. Pacelli Bessa & J. Fábio Montenegro

Authors
  1. G. Pacelli Bessa
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  2. J. Fábio Montenegro
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Correspondence to G. Pacelli Bessa.

Additional information

Bessa and Montenegro were partially supported by CNPq Grant.

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Pacelli Bessa, G., Fábio Montenegro, J. An extension of Barta’s Theorem and geometric applications. Ann Glob Anal Geom 31, 345–362 (2007). https://doi.org/10.1007/s10455-007-9058-8

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  • DOI: https://doi.org/10.1007/s10455-007-9058-8

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