Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity
Samuel Audet-Beaumont
TL;DR
The paper proves that the first Steklov eigenvalue on compact surfaces with boundary can have arbitrarily large multiplicity by adapting the Burger–Colbois construction to the Steklov problem. It builds a family of surfaces S_p using a Cayley graph Γ_p of the group G_p=Z_p ⋊ Z_p^*, gluing building blocks along Γ_p to form S_p with genus g_p=1+p(p−1) and 2p boundary components. An isometric G_p‑action on S_p induces a representation on the first Steklov eigenspace E1(S_p), which is shown to contain a high-degree irreducible component, forcing dim E1(S_p) ≥ p−1 and hence m1(S_p) ≥ p−1. The argument combines variational estimates, a Steklov–Neumann comparison, and detailed representation theory of G_p to separate degree-1 from degree-(p−1) representations, establishing unbounded multiplicity as p grows.
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 $G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$ 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 $σ_1(S_p)$, leading to the desired result.