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

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 $\ge 3\frac{{\pi }^2}{D^2}$ when $n\ge 3$. 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].

19 pages