I read that Ricci flow is "a nonlinear heat flow for the Riemannian metric". Can someone explain what this means?
Nonlinear heat flow has a wikipedia page but I don't understand how this differs from the traditional heat flow on Euclidean space. My guess is to replace the Laplacian with its version on a Riemannian manifold.
This might also be related, but I recently learned that heat flow is locally a mean curvature flow. I'm not sure what the relationship between MCF and Ricci flow is, but this seems like a piece of the puzzle.
Best Answer
$\newcommand{\R}{\mathbb{R}}$ The basic heat equation on a domain in $\R^n$ is $$ \partial_tu = \sum_{i=1}^n (\partial_i)^2u $$ The inhomogeneous version is $$ \partial_tu = \sum_{i=1}^n (\partial_i)^2u + f, $$ where $f$ is a function of $x \in \R^n$ only. This all can be generalized to the linear PDE $$ \partial_tu = a^{ij}(x,t) \partial^2_{ij}u + b^k(x,t)\partial_ku + c u+ f(x), $$ where the matrix $A = [ a_{ij}]$ is positive definite. This is the most general version of what is often called a linear heat equation on a domain in $\R^n$.
A quasilinear heat equation is simply one where the functions $a^{ij}$, $b^k$, $c$, and $f$ are functions of not only $x$, $t$, but also $u$.
On a Riemannian manifold, one possible linear heat equation is of the form $$ \partial_tu = \Delta_g u + b^k\partial_k u + cu + f, $$ where $\Delta_g u = g^{ij}\nabla^2_{ij}u$.
All of the above assumes $u$ is a scalar function on its domain.
Where things get more complicated is when $u$ is either a map $u: M \rightarrow N$, as it is for the harmonic map heat flow or the Riemannian metric itself, as it is for the Ricci flow. If you write out the formulas for these PDEs in local coordinates, you will see that the coefficients of the PDE depend on the unknown map or metric and therefore the PDE is nonlinear. The fact that the harmonic map heat flow can be written simply as $$ \partial_t u = \Delta u $$ is misleading, because here $\Delta u$ is a nonlinear function of its first and second partial derivatives.
As for why the Ricci flow is a nonlinear heat equation is longer story, but after you use the so-called DeTurck trick, the Ricci flow looks like $$ \partial_t g_{ij} = g^{pq}\partial^2_{pq}g_{ij} + \text{ lower order terms} $$ which is a nonlinear heat equation for the metric $g$.
There are similar stories for other geometric heat flows, such as the mean curvature flow.