An important task in the theory of geometric flows is defining a continuation of the flow after the development of singularities in such a way that the topology of the evolving manifold can be controlled. This has been done by Hamilton (1997), who has introduced a surgery procedure for the Ricci flow of a suitable class of four-manifolds. Surgeries also play a central role in the recent work by Perelman (2003) on three-manifolds.
In this talk (joint work with G. Huisken) we introduce a similar procedure for mean curvature flow, which allows us to continue after the singular time the flow of three-manifolds with positive scalar curvature. In each surgery we remove a cylindrical region with high curvature and replace it by two spherical caps. We can prove that after a finite number of surgeries the remaining pieces are diffeomorphic to spheres or to tori.
As a corollary of our construction, we obtain the following result: Let M be a smooth closed immersed three-dimensional surface with positive scalar curvature. Then M is diffeomorphic either to a sphere or to a connected sum of tori.
The proof relies on the analysis of singularities obtained in the previous work of the authors, and on some a priori estimates, including a new gradient estimate for the second fundamental form of the evolving surface.
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