Hot spots on cones and warped product manifolds
Lawford Hatcher
TL;DR
This work analyzes extrema of heat-flow solutions on warped-product manifolds and their noncompact cone limits. By exploiting separation of variables along the fiber and a detailed heat-kernel analysis on the cone via modified Bessel functions, the authors establish Rauch's hot spots conjecture for a broad class of warped products under monotonic, non-constant warping with simple second eigenvalue, and they classify the long-time hot-spot behavior on the infinite cone in terms of the fibre spectral gap. The results reveal four distinct asymptotic regimes dictated by the fibre’s second eigenvalue relative to $2n$ and $n-1$, and they provide explicit leading-order asymptotics and precise spatial localization in each regime. This advances understanding of how geometry (warping and cone structure) and spectral data control the ultimate location of hot spots for heat flow on manifolds with and without boundary.
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 $(M,h)$.