Long-time behavior of 3 dimensional Ricci flow -- Introduction

TL;DR

This paper proves that 3D Ricci flows with properly performed surgeries have finitely many surgeries and exhibit curvature decay over time, confirming Perelman's conjecture and describing the asymptotic geometry.

Contribution

It establishes the finiteness of surgeries and curvature bounds in 3D Ricci flows, providing a detailed long-time geometric description, thus confirming a key conjecture of Perelman.

Findings

Finitely many surgeries occur in the flow.

Curvature is bounded by Ct^{-1} after some time.

Provides a qualitative description of the geometry as t approaches infinity.

Abstract

In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$.