Convergence of fundamental solutions of linear parabolic equations under Cheeger-Gromov convergence

TL;DR

This paper proves that fundamental solutions of linear parabolic equations converge under Cheeger-Gromov convergence of manifolds, given an $L^1$-bound, and establishes a local integral estimate for these solutions.

Contribution

It demonstrates the convergence of fundamental solutions under geometric convergence and provides a new local integral estimate for these solutions.

Findings

Fundamental solutions converge under Cheeger-Gromov convergence.

Established a local integral estimate for fundamental solutions.

Provided conditions involving $L^1$-bounds for convergence.

Abstract

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.