Hot spots on cones and warped product manifolds

TL;DR

This paper investigates the behavior of hot spots, or extrema of heat solutions, on warped product manifolds and infinite cones, proving conjectures under certain conditions and analyzing long-term dynamics based on spectral properties.

Contribution

It establishes conditions under which Rauch's hot spots conjecture holds for warped product manifolds and characterizes long-term hot spot behavior on infinite cones.

Findings

Rauch's hot spots conjecture verified for certain warped product manifolds.

Long-time hot spot behavior depends on the spectral gap of the fiber manifold.

Different asymptotic behaviors identified for infinite cones based on initial conditions.

Abstract

We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on the warping function $f$, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber $(M,h)$.