---
tag: 327NP6S7H5
kind: concept
name: Heat kernel on a Riemannian manifold
msc: [58J35, 35K08]
specializes: [KD404MF042]
sources:
  - url: https://en.wikipedia.org/wiki/Heat_kernel
---

## Summary

The heat kernel of a Riemannian manifold is the fundamental solution of the heat equation $\partial_t u + \Delta u = 0$, where $\Delta$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]].
On a closed manifold it is the integral kernel of $e^{-t\Delta}$ and is the sum of $e^{-t\lambda_j} \phi_j(x) \phi_j(y)$ over an orthonormal basis of eigenfunctions.
Its behaviour for small time encodes local geometry, such as volume and curvature, and its trace encodes the spectrum.
It is the main tool of the heat equation approach to [[index theory|675MAXBQN6]] and [[spectral geometry|QKAFRDPGE4]].

## Explanations

### L1 human

Put a unit of heat at one point $y$ of a curved surface and let it spread.
The heat kernel $K(t, x, y)$ is the temperature at $x$ after time $t$.
For short times heat has not yet felt the shape of the surface, and $K$ looks like the Gaussian of flat space; the curvature appears in correction terms.
For long times on a closed surface, the heat spreads out evenly and $K$ tends to $1/\mathrm{area}$.