Heat kernel on a Riemannian manifold

Remarks

If $\phi_0, \phi_1, \ldots$ is an orthonormal basis of $L^2(M)$ of eigenfunctions with $\Delta \phi_j = \lambda_j \phi_j$, then $$K(t, x, y) = \sum_{j \ge 0} e^{-t \lambda_j} \phi_j(x) \phi_j(y),$$ the series converging uniformly for $t \ge t_0 > 0$. So $K(t, \cdot, \cdot)$ is the integral kernel of the operator $e^{-t\Delta}$, and $K$ is smooth, symmetric in $x$ and $y$, and positive. The same definition, with $\Delta$ replaced by a Laplace-type operator on a vector bundle, gives heat kernels of operators such as the Hodge Laplacian. On a complete noncompact manifold one takes the minimal positive fundamental solution; for $\R^n$ it is the Gaussian heat kernel.

Example: on $S^1 = \R / 2\pi\Z$ with $\Delta = -d^2/d\theta^2$, the eigenvalues are $k^2$ with eigenfunctions $e^{ik\theta}$, and $$K(t, \theta, \theta') = \frac{1}{2\pi} \sum_{k \in \Z} e^{-k^2 t} e^{ik(\theta - \theta')}.$$ Each term solves the heat equation. For fixed $t$, $K(t, \theta, \cdot)$ is the sum over $m \in \Z$ of the Gaussian kernel of $\R$ translated by $2\pi m$. It is a positive kernel of total mass $1$ that concentrates at $\theta$ as $t \to 0^+$, which gives clause 3.

Non-example: on $S^1$ the constant $K \equiv 1/(2\pi)$ satisfies clauses 1 and 2, but $\int K f \to \frac{1}{2\pi}\int f$, not $f(x)$, so clause 3 fails.