On monotonicity of heat kernels: a new example and counterexamples

TL;DR

This paper presents a new example of a manifold with a heat kernel that decreases monotonically along all minimal geodesics, classifies flat tori with this property, and shows that generic metrics usually lack this monotonicity at large times.

Contribution

It introduces a non-radial example of a manifold with monotonic heat kernel behavior and classifies flat tori with this property, addressing a recent open question.

Findings

New non-radial manifold example with monotonic heat kernel

Classification of flat tori with monotonic heat kernels

Monotonicity generally fails for generic metrics at large times

Abstract

We discover a new, non-radial example of a manifold whose heat kernel decreases monotonically along all minimal geodesics. We also classify the flat tori with this monotonicity property. Furthermore, we show that for a generic metric on any smooth manifold the monotonicity property fails at large times. This answers a recent question of Alonso-Or\'an, Chamizo, Mart\'inez, and Mas.