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?

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?