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Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below. (English) Zbl 1239.53085
Summary: 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.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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