Weighted heat kernel comparison theorems and its applications in spectral geometry

Abstract

In this paper, we firstly establish weighted heat kernel comparison theorems for the weighted heat equation on complete manifolds with radial curvatures bounded, and then by mainly using this conclusion, we can obtain two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian as applications in spectral geometry. Besides, as byproducts, eigenvalue comparisons for the first Dirichlet eigenvalue of the weighted $p$-Laplacian ($1<p<\infty$) have been established as well.

arXiv comment49 pages. Several minor revisions have been made to v2. Comments are welcome