TITLE: Does Ricci flow depend continuously on the initial metric?
QUESTION [25 upvotes]: 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 instantaneously complete Ricci flow on $\mathbb R^2$ studied by Giesen and Topping.

REPLY [5 votes]: A complete proof of continuous dependence of Ricci flow was published recently by Eric Bahuaud, Christine Guenther & James Isenberg. Let me record their main theorem:

Theorem A (Continuous Dependence of the Ricci Flow) Let $(M, g_0)$ be a smooth
compact Riemannian manifold. Let $g_0(t)$ be the maximal solution of the Ricci flow
(1) for $t\in [0,\tau (g_0))$, $\tau (g_0) \le \infty$. Choose $0 < \tau < \tau(g_0)$. Let $k \ge  2$. There exist positive constants $r$ and $C$ depending only on $g_0$ and $\tau$ such that if
$$\|g_1 − g_0\|_{h^{k+2,\alpha}} \le r,$$
then for the unique solution $g_1(t)$ of the Ricci flow starting at $g_1$, the maximal existence time satisfies $\tau (g_1)\ge \tau$, and
$$\|g_1(t) − g_0(t)\|_{h^{k,\alpha}} \le C\|g_1 − g_0\|_{h^{k+2,\alpha}}$$
for all $t \in [0,\tau ]$.

Their proof involves the De-Turck trick and standard semi-group theory in parabolic PDE. They also cite this MO post in their paper.