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# 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

#### 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**].

The first-named author was partially supported by NSF DMS and NSFC.

The second-named author was partially supported by a Simons Travel Grant.

The third-named author was partially supported by NSF DMS 1506393.

Received 8 March 2018

Accepted 15 January 2019

Published 1 December 2021