A note on lower bounds for the first eigenvalue of the Witten-Laplacian

TL;DR

This paper extends existing methods to derive optimal lower bounds for the first eigenvalue of the Witten-Laplacian on compact Bakry-Emery manifolds with certain curvature and symmetry conditions.

Contribution

It provides a new, optimal lower bound estimate for the first eigenvalue of the Witten-Laplacian in Bakry-Emery geometry under specific curvature and symmetry assumptions.

Findings

Derived lower bounds are optimal within the method used.

Extended Ling's arguments to Bakry-Emery geometry.

Applicable to manifolds with negative Bakry-Emery Ricci curvature and symmetric eigenfunctions.

Abstract

In this note, by extending the arguments of Ling (Illinois J. Math. 51, 853-860, 2007) to Bakry-Emery geometry, we shall give lower bounds for the first nonzero eigenvalue of the Witten-Laplacian on compact Bakry-Emery manifolds in the case that the Bakry-Emery Ricci curvature has some negative lower bounds and the manifold has the symmetry that the minimum of the first eigenfunction is the negative of the maximum. Our estimate is optimal among those obtained by a self-contained method.