Abstract

Given a solution to the heat equation on a Euclidean domain, Riemannian manifold, or discrete graph, it is natural to ask where the hottest and coldest points are located over large time scales. Rauch’s hot spots conjecture states that, in the Euclidean setting with Neumann boundary conditions, these points should all tend toward the boundary of the domain as time tends to infinity. Motivated by this conjecture, we study the long-time distance between the sets of hottest and coldest points, i.e. the hot-cold distance, in several geometric settings. In several cases, these distances agree with the geodesic diameter of the object in question, leading to an affirmative resolution of Rauch’s conjecture. Counterintuitively, however, we also construct some situations in which the hot-cold distance is strikingly small with respect to the diameter.