On the rotational symmetry of 3-dimensional $\kappa$-solutions

TL;DR

This paper offers an alternative proof to Brendle's uniqueness of the Bryant soliton among 3D $$-solutions and demonstrates that compact $$-solutions are rotationally symmetric, advancing understanding of Ricci flow singularities.

Contribution

It provides a new proof of the Bryant soliton's uniqueness and establishes rotational symmetry of compact $$-solutions, connecting to stability in Ricci flows.

Findings

Alternative proof of Bryant soliton uniqueness

Compact $$-solutions are rotationally symmetric

Connections to Strong Stability Theorem for singular Ricci flows

Abstract

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $\kappa$-solutions. In this paper, we present an alternative proof for this fact and show that compact $\kappa$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.