Question

Consider a version of Ricci flow for which short time existence and uniqueness are known, e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the initial metric?

I thought the answer is "yes" for Ricci flow on a closed manifold but I cannot see why. My immediate interest is the same question for the instanteneously complete Ricci flow on $\mathbb R^2$ studied by Giesen and Topping.

Answer 1

If you use the right topology on the space of metrics, the answer is yes. Basically, this is always true and a consequence of the proof for any theorem on the existence, uniqueness, and regularity of solutions to the initial value problem of a time-dependent PDE. The "right" topology is the one used in the proof.

ADDED: If you can deal with learning the statement of the Nash-Moser implicit function theorem, as say presented in Hamilton's expository article in the Bulletin of the AMS, then his original paper on the 3-d Ricci flow provides a proof for closed manifolds (without boundary). A much simpler proof, relying on standard estimates for the heat equation, was given shortly afterward by DeTurck, and I believe this appears in the same issue as Hamilton's paper.

There is a paper by Shi in JDG that extends this to a complete Riemannian manifold, and I give a different proof of this theorem in papers of mine on $L_p$ convergence of Riemannian manifolds.

I don't know if there is a proof of short-time existence and uniqueness of the Ricci flow in one of Ben Chow's books, but if there is, I'm sure it's a really good and careful presentation.

MORE: I probably overstated the claim that continuous dependence on parameters is proved in these papers. It is more accurate to say that this is a consequence of the arguments in the papers cited. And it is indeed a general principle for PDE's. Almost every proof of existence of solutions to a PDE involves identifying an initial or boundary value problem for which the PDE has a unique solution, and the same techniques used in the proof can be used to show that the solution depends continuously (and, if everything is smooth, smoothly) on the initial or boundary data.

With the Ricci flow, there is a result like the following: Fix a closed manifold and a smooth Riemannian metric $g_0$. Suppose that the Ricci flow $g(t)$ with $g(0) = g_0$ exists for a time interval $[0,T)$ and fix $\tau \in [0,T)$. Let $\|\cdot\|_k$ denote the L_2 Sobolev norm with $k$ derivatives. Given any $\epsilon > 0$, there exists $\delta > 0$ (which depends not only on $\epsilon$ but everything else mentioned so far) such that if $\hat{g}_0$ is a smooth Riemannian metric such that $\|\hat{g}_0-g_0\|_k < \delta$, then the Ricci flow $\hat{g}(t)$ with $\hat{g}(0) = \hat{g}_0$ exists on the interval $[0,\tau]$ and $\|\hat{g}(\tau) - g(\tau)\|_k < \epsilon$.

To prove this, you can't just study the PDE satisfied by the difference of the two metrics, because like the Ricci flow itself, this PDE is highly degenerate due to the invariance under the action of diffeomorphisms. You have to use the DeTurck trick or some variant of it to make the PDE an honest nonlinear heat equation. Once you do that, the above follows by applying $L_2$ energy estimates satisfied by the "normalized" difference.

Now that I've written this, I guess I can see why this is a reasonable question. Somebody probably should write up the details (not me, I'm way oversubscribed already).

COMMENT: There are many people who are much more expert in the Ricci flow than me, and I had always taken it for granted that these people understand the existence and uniqueness proof using PDE theory at least as well as me. I'm beginning to realize that all the experts know how to study the long time behavior of the Ricci flow (much better than me) but are not so familiar with the technical details of the short-time argument.

FINAL COMMENT: It appear to me that Terry Tao's remarks below answer the question rather succintly and better than me. I went a bit astray.

YET ONE MORE: Terry Tao is obviously a counterexample to my statement above about experts on the Ricci flow.