Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

TL;DR

This paper classifies ancient compact non-collapsed solutions to the 3D Ricci flow with symmetry, showing they are either spheres or have a unique asymptotic form, including Perelman's example.

Contribution

It establishes the unique asymptotic behavior of such solutions and provides a precise description, extending understanding of ancient Ricci flow solutions.

Findings

Solutions are either spheres or have a unique asymptotic profile.

The asymptotic behavior is explicitly characterized as $t o - abla$.

Includes analysis of Perelman's constructed solution.

Abstract

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies in particular to the solution constructed by G.Perelman