Communications in Analysis and Geometry

Volume 29 (2021)

Number 5

Fundamental gap estimate for convex domains on sphere — the case $n=2$

Pages: 1095 – 1125

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n5.a3

Authors

Xianzhe Dai (Department of Mathematics, ECNU, Shanghai, China; and Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

Shoo Seto (Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

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

Abstract

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb{S}^n$ is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$. We prove the same result when $n = 2$. In fact our proof works for all dimension. We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the model in [SWW16].

Received 8 March 2018

Accepted 15 January 2019

Published 1 December 2021