Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below

摘要

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M, g) such that: (a) (M, g) is non-collapsed (i.e. the volume of an arbitrary ball of radius one is bounded from below by v > 0), (b) the Ricci curvature of (M, g) is bounded from below by k, (c) the geometry at infinity of (M, g) is not too extreme (or (M, g) is compact). Given such initial data (M, g) we show that a Ricci flow exists for a short time interval [0;T), where T = T(v; k) > 0. This enables us to construct a Ricci flow of any (possibly singular) metric space (X; d) which arises as a Gromov-Hausdorff (GH) limit of a sequence of 3-manifolds which satisfy (a), (b) and (c) uniformly. As a corollary we show that such an X must be a manifold. This shows that the conjecture of M. Anderson-J. Cheeger-T. Colding-G. Tian is correct in dimension three.