The Fundamental Gap of Simplices

Zhiqin Lu,Julie Rowlett

doi:10.1007/s00220-013-1670-9

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.

Highlights

Our first theorem is a compactness result for the gap function on the moduli space of n-simplices
If we consider heating the domain at some initial time and keeping the boundary of the domain fixed at zero temperature, the fundamental gap determines the rate at which the overall heat in the domain vanishes as time tends to infinity
If the gap function is restricted to a certain moduli space of convex domains, what are its properties?

Summary

Introduction

For any convex domain in Rn, the gap function is bounded below by 3π 2. If the gap function is restricted to a certain moduli space of convex domains, what are its properties?. Since any triangle with unit diameter is a graph over the unit interval, this theorem implies that there exists at least one triangle which minimizes the gap function on the moduli space of triangles. The following result shows that the gap function on triangular domains is more than twice as large as the gap function on a generic convex domain; the theorem was conjectured in [2]. Our Theorem 3 implies that the equilateral triangle can be heard with a realistic ear, because Theorem 3 demonstrates that the gap function alone uniquely distinguishes the equilateral triangle within the moduli space of all triangles.

A Compactness Result for the Gap of Simplices

Theorem 3 is True for Short Triangles

The Equilateral Triangle is a Strict Local Gap Minimizer

Theorem 3 is True for Almost Equilateral Triangles

Findings

Proof of Theorem 3