Long-time behavior of 3 dimensional Ricci flow -- A: Generalizations of Perelman's long-time estimates

TL;DR

This paper extends Perelman's long-time estimates for 3D Ricci flows with surgery to manifolds with boundary and establishes new estimates for collapsed metrics, advancing understanding of Ricci flow behavior over time.

Contribution

It generalizes Perelman's estimates to manifolds with boundary and introduces new long-time bounds for collapsed Ricci flows.

Findings

Perelman's long-time estimates are extended to manifolds with boundary.

New estimates are established for Ricci flows with collapsed metrics.

A fixed notion of Ricci flows with surgery is introduced for future analysis.

Abstract

This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman's techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed.