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

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 >= 1) a sequence of subgraphs of gamma such that |omega(l)| --> infinity, we prove that for each k is an element of N, the k(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.

Automated summary

We introduce a graph gamma that is approximately isometric to the hyperbolic plane and investigate the Steklov eigenvalues of a subgraph with boundary omega of gamma. For a sequence of subgraphs (omega(l))(l >= 1) of gamma such that the size of omega(l) approaches infinity, we prove that the k(th) eigenvalue tends to 0 in proportion to 1/|B-l| for each k is an element of N. The proof idea involves finding a bounded domain N in the hyperbolic plane that is approximately isometric to omega, establishing an upper bound for the Steklov eigenvalues of N, and transferring this bound to omega through a process called discretization.