Eigenvalue estimates for Beltrami-Laplacian under Bakry-\'Emery Ricci curvature condition

TL;DR

This paper establishes lower bounds for positive eigenvalues of the Beltrami-Laplacian on closed Riemannian manifolds with Bakry-Émery Ricci curvature bounds, depending on geometric and curvature parameters.

Contribution

It provides new eigenvalue estimates for the Beltrami-Laplacian under Bakry-Émery curvature conditions, extending previous results to include the manifold's dimension and diameter.

Findings

Lower bounds depend on curvature, gradient bounds, dimension, and diameter.

Results apply to manifolds with Ricci curvature bounded from below.

Volume of the manifold does not influence the bounds.

Abstract

On closed Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded from below and bounded gradient of the potential function, we obtain lower bounds for all positive eigenvalues of the Beltrami-Laplacian instead of the drifted Laplacian. The lower bound of the $k$th eigenvalue depends on $k$, Bakry-\'Emery Ricci curvature lower bound, the gradient bound of the potential function, and the dimension and diameter upper bound of the manifold, but the volume of the manifold is not involved. Especially, these results apply to closed manifolds with Ricci curvature bounded from below.