Monotonicity results for the first Steklov eigenvalue on compact surfaces

TL;DR

This paper establishes monotonicity properties of the first Steklov eigenvalue on compact surfaces, demonstrating how it varies with topology, and applies these results to minimal immersions and boundary component analysis.

Contribution

It provides new monotonicity results for the first Steklov eigenvalue related to surface topology changes, including genus and boundary components.

Findings

Strict monotonicity in the genus of surfaces.

Monotonicity in the number of boundary components.

Implications for free boundary minimal immersions.

Abstract

We show several results comparing sharp eigenvalue bounds for the first Steklov eigenvalue on surfaces under change of the topology. Among others, we obtain strict monotonicity in the genus. Combined with results of the second named author \cite{petrides_2} this implies the existence of free boundary minimal immersions from higher genus surfaces into Euclidean balls. Moreover, we can also give a new proof of a result by Fraser and Schoen that shows monotonicity in the number of boundary components.