Small time heat expansion of the laplacian on an analytic hypersurface with an isolated singularity

TL;DR

This paper establishes small time heat expansion for the Laplacian on an analytic hypersurface with an isolated singularity, using local parametrizations and singular asymptotics techniques.

Contribution

It introduces a method to analyze the heat kernel near singularities on hypersurfaces by parametrizing the hypersurface and deriving local models for the Laplacian.

Findings

Existence of small time heat expansion proven.

Local parametrizations near singularities constructed.

Estimates for irregular singularities derived.

Abstract

I prove the existence of small time heat expansion for the Laplace operator on an analytic hypersurface with an isolated singularity. First we obtain a local parametrization of the hypersurface near the singularity. We introduce the notion of quasihomogeneous tangent cone. Then perturb the parametrization of the cone employing a Newton scheme and obtain a parametrization with functions of specific form. These allow us to obtain local models for the Laplace operator near the singularity. These are operators with irregular singularities. We derive the estimates required by singular asymptotics.