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 nonsingular eventually and the curvature is bounded by $C{t}^{-1}$. The second result provides a qualitative description of the geometry as $t\to \infty$.