The fundamental gap of simplices

TL;DR

This paper investigates the fundamental gap of simplices, proving a compactness result in moduli spaces and confirming that equilateral triangles uniquely minimize the gap among triangles.

Contribution

It establishes a compactness property for the gap function on all simplex moduli spaces and verifies a conjecture about the minimal gap in Euclidean triangles.

Findings

The gap function is compact on the moduli space of simplices.

The equilateral triangle uniquely minimizes the fundamental gap among triangles.

The results extend understanding of spectral properties of convex domains.

Abstract

The fundamental gap conjecture was recently proven by Andrews and Clutterbuck: for any convex domain in $\R^n$ normalized to have unit diameter, the difference between the first two Dirichlet eigenvalues of the Laplacian is bounded below by that of the interval. In this work, we focus on the moduli spaces of simplices in all dimensions, and later specialize to the moduli space of Euclidean triangles. Our first theorem is a compactness result for the gap function on the moduli space of simplices in any dimension. Our second main result verifies a recent conjecture of Antunes-Freitas: for any Euclidean triangle normalized to have unit diameter, the fundamental gap is uniquely minimized by the equilateral triangle.