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To build a little on what Agol said:

For mean curvature flow of hypersurfaces, the analogous question is at least partially known to be true (sort of). The advantage of the mean curvature flow over the Ricci flow in this case is that there are already are good notions of weak solution--that is solutions defined through singularities but that agree with classical solutions when the latter exist. The two most important are the Brakke flow and the level set flow. In principal the latter does exactly what you want, namely given a $\Sigma_0$ a closed hypersurface in $\mathbb{R}^{n+1}_x$ there is a ``hypersurface'' $\mathcal{M}$ in $\mathbb{R}^{n+1}\times \mathbb{R}^{\geq 0}$. So that $\partial \mathcal{M}=\Sigma_0 \subset \mathbb{R}^{n+1} \times 0$ and each level set $\lbrace x_{n+2}=t \rbrace \cap \mathcal{M}$ can be interpreted as the flow of $\Sigma_0$ at time $t$ (indeed if $\Sigma_0$ is smooth these level sets agree with the usual flow up to the first singular time, thereafter they continue to exist while the classical flow ceases to make sense). It should be pointed out that in principal $\mathcal{M}$ is quite singular and there are some subtleties about when the level sets have non-empty interior. If you are willing to start with a mean convex $\Sigma_0$, White has shown that the latter never occurs and that (I believe) the levels are almost everywhere smooth.

On the other hand the surgery question for mean curvature flow is a bit trickier. Huisken and Sinestrari give such a surgery when (if I recall correctly) $n\geq 3$ and the initial surface is 2-convex (i.e. the sum of the lowest two principal curvatures is positive...a stronger condition than mean convex). Recently, there were a couple of papers on the arxiv where it was shown that if you took the surgery times (of the H-S surgery procedure) closer and closer to the singular time then you you would limit to the level set flow.

This seems to be the flavor of what you are looking for, though it should be pointed out that the regularity of the higher dimensional guy $\mathcal{M}$ is somewhat unclear in general.

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To build a little on what Agol said:

For mean curvature flow of hypersurfaces, the analogous question is at least partially known to be true (sort of). The advantage of the mean curvature flow over the Ricci flow in this case is that there are already are good notions of weak solution--that is solutions defined through singularities but that agree with classical solutions when the latter exist. The two most important are the Brakke flow and the level set flow. In principal the latter does exactly what you want, namely given a $\Sigma_0$ a closed hypersurface in $\mathbb{R}^{n+1}_x$ there is a ``hypersurface'' $\mathcal{M}$ in $\mathbb{R}^{n+1}\times \mathbb{R}^{\geq 0}$. So that $\partial \mathcal{M}=\Sigma_0 \subset \mathbb{R}^{n+1} \times 0$ and each level set $\lbrace x_{n+2}=t \rbrace \cap \mathcal{M}$ can be interpreted as the flow of $\Sigma_0$ at time $t$ (indeed if $\Sigma_0$ is smooth these level sets agree with the usual flow up to the first singular time, thereafter they continue to exist while the classical flow ceases to make sense). It should be pointed out that in principal $\mathcal{M}$ is quite singular and there are some subtleties about when the level sets have non-empty interior. If you are willing to start with a mean convex $\Sigma_0$, White has shown that the latter never occurs and that (I believe) the levels are almost everywhere smooth.

On the other hand the surgery question for mean curvature flow is a bit trickier. Huisken and Sinestrari give such a surgery when (if I recall correctly) $n\geq 3$ and the initial surface is 2-convex (i.e. the sum of the lowest two principal curvatures is positive...a stronger condition than mean convex). Recently, there were a couple of papers on the arxiv where it was shown that if you took the surgery times (of the H-S surgery procedure) closer and closer to the singular time then you you would limit to the level set flow.

This seems to be what you are looking for, though it should be pointed out that the regularity of the higher dimensional guy $\mathcal{M}$ is somewhat unclear in general.