Degenerating sequences of conformal classes and the conformal Steklov spectrum

Abstract

Let $\Sigma$ be a compact surface with boundary. For a given conformal class $c$ on $\Sigma$ the functional $\sigma_k^*(\Sigma,c)$ is defined as the supremum of the $k-$th normalized Steklov eigenvalue over all metrics on $c$. We consider the behaviour of this functional on the moduli space of conformal classes on $\Sigma$. A precise formula for the limit of $\sigma_k^*(\Sigma,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as $\inf_{c}\sigma_k^*(\Sigma,c)$, where the infimum is taken over all conformal classes $c$ on $\Sigma$. We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the $k-$th normalized Steklov eigenvalue of a non-orientable surface in terms of its genus and the number of boundary components.