In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same thing for (the Dirichlet Laplacian of) regular domains in (complete) Riemannian manifolds?

While Davies' book does have a chapter about Riemannian manifolds, it is not about domains with boundary in a manifold - which is what I am looking for.