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We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author.

Mean curvature flow with surgery of mean convex surfaces in $$\mathbb {R}^3$$ R 3