Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that \begin{eqnarray} 0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots\to\infty\\ \Delta\phi_i+\lambda_i\phi_i=0\\ \end{eqnarray} and $(\phi_i)_i$ forms a complete orthonormal system of $L^2(M)$.
Then, my textbook says that \begin{equation} H(x,y,t):=\sum_{i=0}^\infty e^{-\lambda_it}\phi_i(x)\phi_i(y) \end{equation} is the fundamental solution of the heat equation $\Delta u-\dfrac{\partial u}{\partial t}=0$.
I cannot understand what the sum means. Does the sum converge in $L^2(M\times M)$ for all $t$? Or does it converge pointwise on $M\times M\times (0,\infty)$? I want to know the mean of the $H$ as a function.
It converges uniformly in $M\times M$ for each $t>0$. If I remember correctly, a proof is given in Rosenberg's Laplacian on Riemannian manifolds. Chavel's book must also have something on this. In any case the proof is straightforward if you use Weyl's law (Please try).