Estimates for low Steklov eigenvalues of surfaces with several boundary components

2022/11/02 by Perrin, Hélène
#35P15 #58C40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.01043

Abstract

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue σ₁ of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

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