The Steklov Problem on Triangle-Tiling Graphs in the Hyperbolic Plane

Abstract

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