Fundamental gaps of spherical triangles

TL;DR

This paper explicitly computes eigenvalues for spherical lunes and triangles, revealing how the fundamental gap behaves as angles shrink, and shows the spherical equilateral triangle as a local minimizer of the gap.

Contribution

It extends the understanding of spectral gaps from planar triangles to spherical triangles, providing explicit eigenvalue computations and new minimizer results.

Findings

Fundamental gap tends to infinity as lune angle approaches zero.

Spherical equilateral triangle of diameter π/2 is a local minimizer of the fundamental gap.

Explicit eigenvalues for spherical lunes and triangles are derived.

Abstract

We compute Dirichlet eigenvalues and eigenfunctions explicitly for spherical lunes and the spherical triangles which are half the lunes, and show that the fundamental gap goes to infinity when the angle of the lune goes to zero. Then we show the spherical equilateral triangle of diameter $\frac{\pi}{2}$ is a strict local minimizer of the fundamental gap on the space of the spherical triangles with diameter $\frac{\pi}{2}$, which partially extends Lu-Rowlett's result from the plane to the sphere.