Thank you in advance for your comments!
Let $(M,g)$ be a Riemannian manifold (in general non-compact, connected). Then the associated Dirichlet Laplacian $\Delta_g$ generates the heat semigroup $(e^{s\Delta_g})_{s\geq 0}$ on $L^2(M,d\mu_g)$, where $\mu_g$ is the measure induced by the metric volume form as usual. Moreover, this semigroup is realized as an integral operator with a smooth integral kernel $K_s(x,y)$ (with $x,y\in M$), the heat kernel.
My question is:
Suppose there is a sequence of Riemannian metrics $(g_n)_{n\in \mathbb{N}}$ on $M$ such that all the differences $g-g_n$ have compact support within the same chart neighborhood $(U,\varphi)$. Assume further that (in the local coordinates of the chart $(U,\varphi)$), that $g_n$ and all its (chart coordinate) derivatives converge uniformly to $g$. Denoting the heat kernels associated to $g_n$ by $K^{(n)}_s(x,y)$, is it true that the new kernels converge pointwise to the original kernel, i.e. $\lim_{n\to \infty}K^{(n)}_s(x,y)=K_s(x,y)$?
I would be grateful for a proof, or a reference to one.