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

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 $({\Omega }_l{)}_{l\ge 1}$ a sequence of subraphs of $\Gamma$ such that $|{\Omega }_l|⟶\infty$, 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.

27 pages