Geometric Bounds for Low Steklov Eigenvalues of Finite Volume Hyperbolic Surfaces.

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the corresponding Steklov eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.