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

TL;DR

This paper proves that properly performed surgeries in 3D Ricci flows lead to non-singular evolution with curvature bounds and describes the geometric structure as time progresses to infinity.

Contribution

It establishes the long-time behavior and geometric limits of 3D Ricci flows with surgery, completing the series of results on this topic.

Findings

Flow becomes non-singular eventually with curvature bounded by Ct^{-1}

Provides a qualitative description of the geometry as t approaches infinity

Validates the correctness of surgery procedures for long-time flow behavior

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 $t \to \infty$.