This work will study the perturbation of heat equations under the change of metric/distance function. The investigator will use the analytical approach, especially the knowledge and techniques developed by Saloff-Coste and Grigor'yan[1]. The goal is to gain a sufficient condition on the conformal distance, similar to that of the weighted functions in Tasena[5], for which the Gaussian behavior of the original heat kernel would imply that of the perturbed heat kernel. Using Sturm's results[2,3,4], this is equivalent to proving doubling property and Poincare inequality of the conformal Dirichlet spaces.
