The sharp lower bound of the first Dirichlet eigenvalue for geodesic balls

TL;DR

This paper establishes the exact lower bound for the first Dirichlet eigenvalue of geodesic balls in noncompact Riemannian manifolds with non-negative Ricci curvature, enhancing understanding of spectral geometry.

Contribution

It provides the sharp lower bound for the first Dirichlet eigenvalue in this setting, improving previous inequalities and implications for Poincare inequalities.

Findings

Sharp lower bound for the first Dirichlet eigenvalue

Implication of the bound for sharp Poincare inequality

Extension of Li-Schoen's uniform Poincare inequality

Abstract

On complete noncompact Riemannian manifolds with non-negative Ricci curvature, Li-Schoen proved the uniform Poincare inequality for any ge odesic ball. In this note, we obtain the sharp lower bound of the first Dirichlet eigenvalue of such geodesic balls, which implies the sharp Poincare inequality for geodesic balls.