Degenerating sequences of conformal classes and the conformal Steklov spectrum

Medvedev, Vladimir

Canadian journal of mathematics, Tome 74 (2022) no. 4, pp. 1093-1136

Résumé

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 kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior 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 kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

Détail

DOI : 10.4153/S0008414X21000171

Mots-clés : Spectral geometry, Steklov spectrum, upper bounds, conformal Steklov spectrum, cobordisms