The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a Riemannian manifold [ROSENBERG].)


My question is will it spread all over the whole manifold? It seems that the bump function cut it off. But this should not happen for a heat kernel that it just vanish outside a neighbourhood. Because physically, the heat should be able to conduct to everywhere.



Heat kernel on Riemannian manifold
This bump function is not supported on an arbitrary neighborhood, but one very specific to the construction of $H(t,x,y)$. Rosenberg's argument proceeds by constructing the parametrix $H\_k(t,x,y)$ in a neighborhood of the diagonal $M\_{\text{diag}}\subset M\times M$. Namely the neighborhood $U\_\epsilon=\{(x,y)\in M\times M:d(x,y)<\epsilon\}$.


The idea is that the heat kernel $H(t,x,y)$ vanishes for small $t$, provided that $x\neq y$. In fact, you should expect it vanish like $t^{-m/2}\text{exp}\left(-\frac{d(x,y)^2}{4t}\right)$, when comparing it with the Euclidean heat kernel. For this reason, you don't expect points in $(0,\infty)\_t\times (M\times M\setminus U\_\epsilon)$ to contribute much to the value of the Heat kernel, and the parametrix construction relies on this to some degree. The parametrix $H\_k$ does not agree pointwise with the true heat kernel $H$, but instead differs by some small error (see theorem 3.22)


What he proves is that $H(t,x,y)=H\_k(t,x,y)+Q\_k\ast H\_k(t,x,y)$, where the error function $Q\_k$ satisfies $|Q\_k|<Ct^{k-m/2}$, and that $Q\_k\in C^l$ as long as $k>l+\frac{n}{2}$. In particular, on a closed manifold, convolution with $Q\_k$ defines a compact operator (it is in this sense that the error is "small").


