Fundamental Gap of Convex Domains in the Spheres (with Appendix B by Qi S. Zhang)

TL;DR

This paper extends a known lower bound on the eigenvalue gap for convex domains in the sphere from diameter at most π/2 to any diameter less than π, broadening the applicability of the result.

Contribution

It generalizes the eigenvalue gap estimate for convex domains in the sphere to include those with diameter less than π, beyond the previous restriction of π/2.

Findings

Eigenvalue gap bound holds for convex domains with diameter less than π

Extension of previous results to larger convex domains in the sphere

Provides a broader understanding of spectral gaps in spherical geometry

Abstract

In [SWW], S. Seto, L. Wang and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter, provided that the domain has the diameter at most $\pi/2$. In this paper, we extend Seto-Wang-Wei's result to convex domains in the unit sphere with diameter less than $\pi$.