Long-time behavior of 3 dimensional Ricci flow -- D: Proof of the main results

Abstract

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 $t \to \infty$.