The first Dirichlet eigenvalue and the width

TL;DR

This paper establishes an explicit inequality linking the width of a geodesic ball with non-negative Ricci curvature to the spectral gap of its first Dirichlet eigenvalue, providing a quantitative geometric-spectral relation.

Contribution

It introduces a new explicit inequality that bounds the width of geodesic balls using the spectral gap of the first Dirichlet eigenvalue, extending previous sharp bounds.

Findings

Derived a quantitative inequality relating width and spectral gap.

Provided explicit bounds for geodesic ball widths based on spectral data.

Enhanced understanding of geometric-spectral relationships in Riemannian geometry.

Abstract

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.