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Fundamental gap estimate for convex domains on sphere — the case $n=2$
Communications in Analysis and Geometry  (IF0.736),  Pub Date : 2021-12-01, DOI: 10.4310/cag.2021.v29.n5.a3
Xianzhe Dai, Shoo Seto, Guofang Wei

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].