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

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 ${\lambda }_2-{\lambda }_1\ge \frac{3{\pi }^2}{D^2}$ for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than $\frac{3}{2}\frac{{\pi }^2}{D^2}$, even though the first eigenfunction of the Laplace operator is not log-concave.

13 pages, 1 figure. Comments are welcome