Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
  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 $k$. 
Mikhail Karpukhin, Gerasim Kokarev, and Iosif Polterovich
math.DG math.SP