Geometric heat comparison criteria for Riemannian manifolds
  The main results of this article are small time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate them by examples. 
Leon Karp (City University of New York), Norbert Peyerimhoff   (University of Durham)