Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

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