Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Abstract

We prove two explicit bounds for the multiplicities of Steklov eigenvalues ${\sigma }_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues ${\sigma }_k$ are uniformly bounded in $k$.

final version, 17 pages, to appear in Annales de l'Institut Fourier