Degenerating sequences of conformal classes and the conformal Steklov spectrum

Let be a compact surface with boundary. For a given conformal class c on the functional is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on. A precise formula for the limit of when the sequence degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as, where the infimum is taken over all conformal classes c on. We show that these quantities are equal to for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components. ©