Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$

TL;DR

This paper calculates the fundamental gap for convex domains in hyperbolic space, revealing cases where it is smaller than Euclidean and spherical bounds, and explores eigenfunction properties.

Contribution

It provides explicit fundamental gap estimates for convex domains in hyperbolic space, contrasting with known bounds in Euclidean and spherical geometries.

Findings

Some convex hyperbolic domains have a fundamental gap less than 3π²/D².

The fundamental gap of Shih's example exceeds 1.5 times π²/D².

Eigenfunctions in the example are not log-concave, despite the gap size.

Abstract

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $\lambda_2 - \lambda_1 < \frac{3\pi^2}{D^2}$, where $D$ is the diameter of the domain and $\lambda_1$, $\lambda_2$ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in $\mathbb R^n $ or $\mathbb S^n$, where $\lambda_2 - \lambda_1 \geq \frac{3 \pi^2}{D^2}$ for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than $\tfrac 32 \frac{\pi^2}{D^2}$, even though the first eigenfunction of the Laplace operator is not log-concave.