Weighted heat kernel comparison theorems and its applications in spectral geometry

TL;DR

This paper develops weighted heat kernel comparison theorems for complete manifolds with bounded radial curvatures and applies these results to derive eigenvalue comparison theorems for the Witten-Laplacian in spectral geometry.

Contribution

It introduces new weighted heat kernel comparison theorems and applies them to obtain eigenvalue comparison results for the Witten-Laplacian.

Findings

Weighted heat kernel comparison theorems established.

Eigenvalue comparison theorems for the Witten-Laplacian derived.

Applications in spectral geometry demonstrated.

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.