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Weighted heat kernel comparison theorems and its applications in spectral geometry

Jing Mao

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.

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.
Paper Structure (6 sections, 15 theorems, 145 equations)

This paper contains 6 sections, 15 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and some useful facts
  3. Weighted heat kernel comparison theorems
  4. Applications in spectral geometry
  5. Appendix A
  6. Appendix B

Key Result

Theorem 1.6

Given a complete Riemannian $n$-manifold $M^n$, $n\geq2$, and assuming that the potential function $\phi$ satisfies Property 1, we can obtain: (1) if $M^n$ has a radial Ricci curvature lower bound $(n-1)\kappa(s)$ w.r.t. some point $q\in M^n$, then, for $r_{0}<\min\{\ell(q),l\}$, the inequality holds for all $(y,t)\in B(q,r_{0})\times(0,\infty)$ with $d_{M^n}(q,y)=d_{M^{-}}(q^{-},z)$ for any $z\i

Theorems & Definitions (58)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3: $\phi$-heat kernel for steady Gaussian soliton WW1
  • Example 1.4: Mehler heat kernel for shrinking Gaussian soliton GA
  • Example 1.5: Mehler heat kernel for expanding Gaussian soliton GA
  • Theorem 1.6
  • Example 1.7: $1$-dimensional model
  • Remark 1.8
  • Example 1.9
  • Remark 1.10
  • ...and 48 more