Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity

2024/12/10 by Audet-Beaumont, Samuel
#35P15 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2412.07692

Abstract

We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces S_p with a specific subgroup of isometry G_p:= ℤ_p \rtimes ℤ_p^* for each prime p. We do so by gluing surfaces with boundary following the structure of the Cayley graph of G_p. We then exploit the properties of G_p and S_p in order to show that an irreducible representation of high degree (depending on p) acts on the eigenspace of functions associated with σ₁(S_p), leading to the desired result.

Related

• Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
• Higher Steklov eigenvalues of graphs on surfaces
• Large Steklov eigenvalues on hyperbolic surfaces
• Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups
• Monotonicity results for the first Steklov eigenvalue on compact surfaces