consider any smooth Riemannian manifold $(N,g)$, an open subset $U\subset N$ and the Dirichlet heat kernel $p(t;x,y)$ for $U$. I am wondering, if it is true that $\int_U p(t;x,x)dx <\infty$ for any $t>0$? For domains in euclidean space this is definitely correct, but I do not know if it is true in general?
Best wishes. I would appreciate any help
If the manifold is compact, its true because a continuous function has bounded integral over a compact manifold. Otherwise, it's wrong in general. For instance, if the manifold is a symmetric space (like ${\mathbb R}^n$), then the function $p(t,x,x)$ is independent of $x$, i.e., a constant $>0$ and if you integrate a constant and expect the integral to be finite, the space must have finite measure, which is wrong for ${\mathbb R}^n$ and other non-compact symmetric spaces. But there are also instances of non-compact manifolds with finite measure, where the integral will not be finite, like Hadamard manifolds.
