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Ricci flow of almost non-negatively curved three manifolds

By Miles Simon

Abstract

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i) goes to zero as i goes to infinity, and whose diameter is bounded by a constant independent of i, and whose volume is bounded from below by a positive constant independent of i. We show for such spaces, that a solution to Ricci flow exists for a short time, and that the solution is smooth for all positive times and that it has non-negative Ricci curvature. This allows us to classify the topological type and the differential structure of the limit manifold (in view of Hamilton's Theorem on closed three manifolds with non-negative Ricci curvature)

Topics: Mathematics - Differential Geometry, Mathematics - Metric Geometry
Year: 2006
OAI identifier: oai:arXiv.org:math/0612095

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