Mathematical Research Letters

Volume 28 (2021)

Number 5

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

Pages: 1319 – 1336

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n5.a2

Authors

Theodora Bourni (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)

Julie Clutterbuck (School of Mathematics, Monash University, Victoria, Australia)

Xuan Hien Nguyen (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Alina Stancu (Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada)

Guofang Wei (Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

Valentina-Mira Wheeler (School of Mathematics and Applied Statistics, University of Wollongong, NSW, Australia)

Abstract

Motivated by an example of Shih [10], we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb{H}^2$, showing that for some of them $\lambda_2 - \lambda_1 \lt \frac{3\pi^2}{D^2}$, 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 Rn or Sn, where $\lambda_2 - \lambda_1 \geq \frac{3\pi^2}{D^2}$ for convex domains $[1, 5, 7, 9]$. 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 logconcave.

Received 28 November 2019

Accepted 25 April 2020

Published 16 August 2022