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

Abstract

We introduce a graph $Γ$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $Ω$ of $Γ$. For $(Ω_l)_{l\geq 1}$ a sequence of subraphs of $Γ$ such that $|Ω_l| \longrightarrow \infty$, 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 $Ω$, giving an upper bound for the Steklov eigenvalues of $N$ and transferring this bound to $Ω$ via a process called discretization.