The Steklov problem on triangle-tiling graphs in the hyperbolic plane

TL;DR

This paper investigates the asymptotic behavior of Steklov eigenvalues on graphs resembling the hyperbolic plane, showing they tend to zero as the boundary size grows, using geometric and discretization techniques.

Contribution

It introduces a new approach to analyze Steklov eigenvalues on hyperbolic-like graphs by relating them to bounded hyperbolic domains through discretization.

Findings

Steklov eigenvalues tend to zero as boundary size increases

Eigenvalues decay proportionally to inverse boundary volume

Method connects hyperbolic geometry with spectral graph theory

Abstract

We introduce a graph $\Gamma$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $\Omega$ of $\Gamma$. For $(\Omega_l)_{l\geq 1}$ a sequence of subraphs of $\Gamma$ such that $|\Omega_l| \longrightarrow \infty$, we prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B_l|$. The idea of the proof consists in finding a bounded domain $N$ of the hyperbolic plane which is roughly isometric to $\Omega$, giving an upper bound for the Steklov eigenvalues of $N$ and transferring this bound to $\Omega$ via a process called discretization.