Ricci flow of almost non-negatively curved three manifolds

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), (i)g), i is an element of N, whose Ricci curvature is bigger than -1/i, and whose diameter is less than d(0) (independent of i) and whose volume is bigger than v(0) > 0 (independent of i). We show for such spaces, that a solution to Ricci flow exists for a short time t is an element of (0, T), that the solution is smooth for t > 0, and has Ricci (g(t)) >= 0 and Riem (g(t)) <= c/t for t is an element of (0, T) (for some constant c = c(v(0), d(0), n)). This allows us to classify the topological type and the differential structure of the limit manifold (in view of the theorem of Hamilton [10] on closed three manifolds with non-negative Ricci curvature).