Multiplicit\'e du spectre de Steklov sur les surfaces et nombre chromatique

TL;DR

This paper investigates the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary, improving bounds, establishing sharpness on small genus surfaces, and exploring a new chromatic invariant related to eigenvalue bounds.

Contribution

It advances understanding of eigenvalue multiplicities on surfaces and confirms a conjecture linking a new chromatic invariant to eigenvalue bounds when the bounds are sharp.

Findings

Bounds on the multiplicity of the first Steklov eigenvalue are improved.

Sharpness of bounds is demonstrated on small genus surfaces.

A conjecture relating a chromatic invariant to eigenvalue bounds is proven when the bounds are sharp.

Abstract

We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some surfaces of small genus. In a previous article, we defined a new chromatic invariant of surfaces with boundary and conjectured that this invariant is related to the bound on the first eigenvalue. In the present article, we study this invariant, and prove that the conjecture is true when the known bound is sharp.