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Estimates of the first Dirichlet eigenvalue from exit time moment spectra

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Abstract

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung–Leung concerning the fundamental tones of Cartan–Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.

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Acknowledgments

We thank the referee for useful remarks and references.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    A. Hurtado

  2. DTU Compute, Mathematics, 2800, Kgs. Lyngby, Denmark

    S. Markvorsen

  3. Departament de Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain

    V. Palmer

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  1. A. Hurtado
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  2. S. Markvorsen
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  3. V. Palmer
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Corresponding author

Correspondence to A. Hurtado.

Additional information

A. Hurtado is supported by the Spanish Mineco-FEDER Grants  MTM2010-21206-C02-01 and MTM2013-48371-C2-1-P and Junta de Andalucia  Grants FQM-325 and P09-FQM-5088.  S. Markvorsen and V. Palmer are  supported by the Spanish Mineco-FEDER Grants MTM2010-21206-C02-02 and MTM2013-48371-C2-2-P. V. Palmer is suppoted by the Pla de Promoció de la Investigació de la Universitat Jaume I, Generalitat Valenciana Project PROMETEO II/2014/064 and Bancaja UJI grant Universitat Jaume I Project P1.1B2012-18.

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Hurtado, A., Markvorsen, S. & Palmer, V. Estimates of the first Dirichlet eigenvalue from exit time moment spectra. Math. Ann. 365, 1603–1632 (2016). https://doi.org/10.1007/s00208-015-1316-7

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  • DOI: https://doi.org/10.1007/s00208-015-1316-7

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