Heat kernel

Summary

The heat kernel is the solution of the heat equation that starts from a unit amount of heat concentrated at one point. On $\R^n$ it is the Gaussian $\Phi(x, t) = (4 \pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$. Every bounded continuous initial temperature evolves by convolution with it. On domains and on Riemannian manifolds the heat kernel encodes the spectrum of the Laplacian. It is a basic tool in geometric analysis and index theory.