The vanishing of the fundamental gap of convex domains in $\mathbb H^n$

TL;DR

This paper proves that in hyperbolic space, convex domains can have arbitrarily small fundamental gap times diameter squared, challenging assumptions about spectral gaps in such geometries.

Contribution

It demonstrates that the fundamental gap times the square of the diameter can be made arbitrarily small in convex domains within hyperbolic space, revealing new spectral properties.

Findings

Fundamental gap times diameter squared can be arbitrarily small.

This phenomenon occurs for convex domains of any diameter in hyperbolic space.

Results challenge existing spectral gap bounds in non-Euclidean geometries.

Abstract

For the Laplace operator with Dirichlet boundary conditions on convex domains in $\mathbb H^n$, $n\geq 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter.