Sharp lower bound for the first eigenvalue of the Weighted $p$-Laplacian II

TL;DR

This paper establishes sharp lower bounds for the first nonzero eigenvalue of the weighted p-Laplacian on compact Bakry-Émery manifolds, extending previous results to broader geometric conditions.

Contribution

It provides the first sharp lower bound estimates for the eigenvalues of the weighted p-Laplacian under general Bakry-Émery Ricci curvature conditions, including boundary cases.

Findings

Derived sharp lower bounds for eigenvalues.

Extended previous results to manifolds with boundary.

Applicable to a wide class of weighted manifolds.

Abstract

Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-\'Emery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq \kappa \, g$ for some $\kappa \in \mathbb{R}$.