Mathematical Research Letters

Volume 28 (2021)

Number 5

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

Pages: 1319 – 1336



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)


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.

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The research of Theodora Bourni was supported by grant 707699 of the Simons Foundation and by NSF Grant DMS 2105026. The research of Julie Clutterbuck was supported by grant FT1301013 of the Australian Research Council. The research of Xuan Hien Nguyen was supported by grant 579756 of the Simons Foundation. The research of Alina Stancu was supported by NSERC Discovery Grant RGPIN 327635. The research of Guofang Wei was supported by NSF Grant DMS 1811558. The research of Valentina-Mira Wheeler was supported by grant DP180100431 and DE190100379 of the Australian Research Council.

Received 28 November 2019

Accepted 25 April 2020

Published 16 August 2022