Mathematische Annalen

, Volume 365, Issue 3–4, pp 1603–1632 | Cite as

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

  • A. Hurtado
  • S. Markvorsen
  • V. Palmer


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.

Mathematics Subject Classification

Primary 58C40 Secondary 53C20 



We thank the referee for useful remarks and references.


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Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.DTU Compute, MathematicsKgs. LyngbyDenmark
  3. 3.Departament de Matemàtiques-INITUniversitat Jaume ICastellóSpain

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