Let $M$ be a compact Riemannian manifold without a boundary. Let $p\_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p\_t(x,y)| \leq C$$
for all $x,y \in M$ and $t > 1$.


Thanks for any help.



Heat kernel upper bound on compact Riemannian manifold
Let $C$ be the maximum of $p\_1(x,y)$ over both $x$ and $y$ (this exists because $p$ is smooth). Then from the semigroup property one obtains
$$p\_{1+t}(x,y) = \int p\_1(x,z)p\_t(z,y)\mathrm{d}z \le C\int p\_t(z,y)\mathrm{d}z = C,$$
so that you get the upper bound you where looking for.


This argument shows that the maximum is non-increasing.


More detailed upper and lower bounds are available. If you're interested you should definitely take a look at [Grigoryan's book](http://books.google.com.uy/books/about/Heat_Kernel_and_Analysis_on_Manifolds.html?id=X7QQcVa2EWsC&redir_esc=y) (which is massive but pretty self-contained) and also the [book by Chavel](https://books.google.com.uy/books?id=0v1VfTWuKGgC&printsec=frontcover&dq=eigenvalues%20in%20riemannian%20geometry&hl=en&sa=X&ei=zg61VJ-bCo7dsATx2YFo&ved=0CB0Q6AEwAA#v=onepage&q=eigenvalues%20in%20riemannian%20geometry&f=false) (much cleaner but you don't get the nitty gritty details from it).


