This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by $C t^{ -1}$. The second result provides a qualitative description of the geometry as $t \to \infty$.
Categories: math.DG math.AP math.GT