On monotonicity of heat kernels: a new example and counterexamples

Abstract: We discover a new, non-radial example of a manifold whose heat kernel decreases monotonically along all minimal geodesics. We also classify the flat tori with this monotonicity property. Furthermore, we show that for a generic metric on any smooth manifold the monotonicity property fails at large times. This answers a recent question of Alonso-Or\'an, Chamizo, Mart\'inez, and Mas.