  We obtain precise asymptotics for the Steklov eigenvalues on a compact
Riemannian surface with boundary. It is shown that the number of connected
components of the boundary, as well as their lengths, are invariants of the
Steklov spectrum. The proofs are based on pseudodifferential techniques for the
Dirichlet-to-Neumann operator and on a number-theoretic argument.

Spectral Theory;Differential Geometry