Profile expansion for the first nontrivial Steklov eigenvalue in Riemannian manifolds

TL;DR

This paper investigates the asymptotic behavior of the first nontrivial Steklov eigenvalue in Riemannian manifolds, providing local and global comparison principles related to scalar curvature and addressing challenges in domain existence and eigenvalue degeneracy.

Contribution

It offers the first detailed expansion of the Steklov eigenvalue profile as volume approaches zero, including local and global comparison principles on Riemannian manifolds.

Findings

Derived local comparison principles based on scalar curvature.

Established global expansion and comparison principles for closed surfaces.

Addressed challenges due to non-existence and degeneracy of maximizing domains.

Abstract

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.