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I am sorry if this is a silly question, but I am new to Ricci flows.

Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose that $g_t$ is a normalized Ricci flow on $\Sigma$ with initial condition given by $g$, for $t \in [0, T)$. Is it true that there exists a smooth $1$-parameter family of isometric embeddings $\varphi_t : (\Sigma, g_t) \to \mathbb{R}^3$ for $t \in [0, \varepsilon)$ and some $0 < \varepsilon \leq T$?

If this is not the case, is it true if $\Sigma$ is convex?

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    $\begingroup$ For convex surfaces it is true --- indeed Ricci flow preserves nonnegativeness of Gauss curvature and therefore there is an isometric embedding at any time. (I would be surprised if one could write an evolution equation for such surfaces --- likely it is not local.) $\endgroup$
    – Anton Petrunin
    Commented Jul 5, 2022 at 16:11
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    $\begingroup$ To add a little "converse" to Anton's comment, if you could find a 1-parameter family of embedded spheres realizing the Ricci flow, that would be fantastic. $\endgroup$
    – Ryan Budney
    Commented Jul 6, 2022 at 7:44

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