Estimates for the first eigenvalues of Bi-drifted Laplacian on smooth metric measure space

TL;DR

This paper derives lower bounds for the first eigenvalues of the Bi-drifted Laplacian on compact smooth metric measure spaces with boundary, considering various curvature conditions and boundary types.

Contribution

It provides new eigenvalue estimates for the Bi-drifted Laplacian under curvature bounds and boundary conditions, extending previous results to more general settings.

Findings

Lower bounds for eigenvalues under Bakry-Emery Ricci curvature conditions

Eigenvalue estimates for Wentzell-type boundary problems

Extension of eigenvalue bounds to boundary-including smooth metric measure spaces

Abstract

In this paper, we obtain lower bounds for the first eigenvalue to some kinds of the eigenvalue problems for Bi-drifted Laplacian operator on compact manifolds (also called a smooth metric measure space) with boundary and $m$-Bakry-Emery Ricci curvature or Bakry-Emery Ricci curvature bounded below. We also address the eigenvalue problem with Wentzell-type boundary condition for drifted Laplacian on smooth metric measure space.