The heat kernel on a two-dimensional manifold $M$ has the well-known expression
$$H(p,q,t) = \sum_{i=1}^\infty e^{\lambda_i t}\phi_i(p)\phi_i(q)$$ where $\phi_i, \lambda_i$ are the eigenfunctions and eigenvalues of the Laplace-Beltrami operator on $M$.
This expression reveals the fact that for large times, the behavior of the low-magnitude eigenvalue eigenfunctions determine the value of the heat kernel.
Unfortunately the above formula is impractical for computing $H$ for small values of $t$: the formula becomes increasingly dependent on the high-frequency eigenfunctions, which are hard to compute stably.
Is there an alternative formula that can be used for $t$ near zero?