Volume growth and on-diagonal heat kernel bounds on Riemannian manifolds with an end

TL;DR

This paper studies lower bounds on the heat kernel for large times on Riemannian manifolds with at least one end, linking geometric structure to heat diffusion behavior.

Contribution

It provides new heat kernel lower bounds on manifolds with ends, extending understanding of heat diffusion in non-compact geometric settings.

Findings

Established lower bounds for the heat kernel on manifolds with ends.

Connected geometric properties of ends to heat kernel estimates.

Extended previous results to a broader class of non-compact manifolds.

Abstract

We investigate heat kernel estimates of the form $p_{t}(x, x)\geq c_{x}t^{-\alpha},$ for large enough $t,$ where $\alpha$ and $c_{x}$ are positive reals and $c_{x}$ may depend on $x,$ on manifolds having at least one end.