Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

Mikhail Karpukhin

doi:10.3934/era.2017.24.011

Article Contents

2017, Volume 24: 100-109. Doi: 10.3934/era.2017.24.011

This volume Previous Article Real orientations, real Gromov-Witten theory, and real enumerative geometry Next Article Central limit theorems in the geometry of numbers

Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

Mikhail Karpukhin

Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9

The author is grateful to Iosif Polterovich for fruitful discussions and comments on the initial versions of the manuscript. The author thanks Spiro Karigiannis for providing the reference [13]

Received: May 03, 2017

Revised: August 15, 2017

Published: August 2017

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Abstract

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.

Keywords:

Steklov eigenvalues,
geometric optimisation.

Mathematics Subject Classification: 35P15(Primary), 58J50(Secondary).

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References

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