Heat kernel estimate in a conical singular space

TL;DR

This paper establishes heat kernel upper bounds on conical spaces with a specific potential, using Hadamard parametrix and wave propagation techniques, extending understanding of heat behavior in singular geometries.

Contribution

It introduces a novel approach combining Hadamard parametrix and wave speed to estimate heat kernels on conical manifolds with potential.

Findings

Heat kernel is bounded above under specified conditions.

The method applies Hadamard parametrix and finite propagation speed.

Results extend heat kernel estimates to conical singular spaces.

Abstract

Let $(X,g)$ be a product cone with the metric $g=dr^2+r^2h$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the upper boundedness of heat kernel associated with the operator $L_V=-\Delta_g+V_0 r^{-2}$, where $-\Delta_g$ is the positive Friedrichs extension Laplacian on $X$ and $V=V_0(y) r^{-2}$ and $V_0\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $-\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$.The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on $Y$.