Pointwise monotonicity of heat kernels

TL;DR

This paper proves a pointwise radial monotonicity property of heat kernels across Euclidean spaces, spheres, and hyperbolic spaces, using an elementary parabolic maximum principle argument.

Contribution

It introduces a new proof of heat kernel monotonicity that applies to various geometric spaces, expanding understanding of heat kernel behavior.

Findings

Heat kernels are radially monotonic in Euclidean, spherical, and hyperbolic spaces.

The proof employs a simple application of the parabolic maximum principle.

Monotonicity holds from specific points on revolution hypersurfaces.

Abstract

In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.