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Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

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Abstract

Given a manifold \(M\), we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.

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Acknowledgments

We would like to thank Pedro Antunes for his help with the numerical computations in Sect. 6.

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Author notes

  1. Jing Mao

    Present address: Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai , 264209, China

Authors and Affiliations

  1. Department of Mathematics, Faculty of Human Kinetics, Universidade de Lisboa, Lisbon, Portugal

    Pedro Freitas

  2. Group of Mathematical Physics, Universidade de Lisboa, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, 1649-003 , Lisbon, Portugal

    Pedro Freitas

  3. Centro de Física das Interacções Fundamentais, Instituto Superior Técnico, Universidade de Lisboa, Edifício Ciência, Piso 3, Av. Rovisco Pais, 1049-001 , Lisbon, Portugal

    Jing Mao & Isabel Salavessa

Authors
  1. Pedro Freitas
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  2. Jing Mao
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Corresponding author

Correspondence to Pedro Freitas.

Additional information

Communicated by L. Ambrosio.

The first author was partially supported by Fundação para a Ciência e Tecnologia, Portugal (FCT) through projects PEst-OE/MAT/UI0208/2011 and PTDC/MAT/101007/2008; the second author was supported by FCT through a doctoral fellowship SFRH/BD/60313/2009; the third author was partially supported by FCT through projects PTDC/MAT/101007/2008 and PTDC/MAT/118682/2010.

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Freitas, P., Mao, J. & Salavessa, I. Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds. Calc. Var. 51, 701–724 (2014). https://doi.org/10.1007/s00526-013-0692-7

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  • DOI: https://doi.org/10.1007/s00526-013-0692-7

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