Existence and nonexistence results for eigenfunctions of the Laplacian in unbounded domains of H^n

TL;DR

This paper studies when eigenfunctions of the Laplacian exist or not in unbounded hyperbolic domains, revealing conditions related to the domain's boundary at infinity and providing specific results for hyperballs.

Contribution

It establishes new criteria for the existence and nonexistence of Laplacian eigenfunctions in unbounded hyperbolic domains, especially concerning boundary behavior at infinity.

Findings

No positive bounded eigenfunctions in domains contained in a horoball.

Existence of solutions that vanish at infinity when the domain's asymptotic boundary contains an open set.

Results specifically apply to hyperballs.

Abstract

We investigate, for the Laplacian operator, the existence and nonexistence of eigenfunctions of eigenvalue between zero and the first eigenvalue of the hyperbolic space H^n, for unbounded domains of H^n. If a domain is contained in a horoball, we prove that there is no positive bounded eigenfunction that vanishes on the boundary. However, if the asymptotic boundary of a domain contains an open set of the asymptotic boundary of H^n, there is a solution that converges to 0 at infinity and can be extended continuously to the asymptotic boundary. In particular, this result holds for hyperballs.