Conformal Perturbation of Heat Kernels with applications

Let $(M,g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 2$. Consider the conformal perturbation $\tilde{g}=hg$ where $h$ is a smooth bounded positive function on $M$. Denote by ${\tilde{p}}_t(x,y)$ the heat kernel of manifolds $(M,\tilde{g})$. In this paper, we derive the upper bounds and gradient estimates of ${\tilde{p}}_t(x,y)$.