Revisiting ancient noncollapsed flows in $\mathbb{R}^3$

TL;DR

This paper provides a new proof for classifying ancient noncollapsed flows in three-dimensional space, simplifying previous methods by directly establishing selfsimilarity through advanced geometric analysis techniques.

Contribution

It introduces a novel proof that combines the fine neck theorem and Hamilton's Harnack inequality to establish selfsimilarity in noncompact ancient flows.

Findings

New proof of classification theorem for ancient flows

Direct establishment of selfsimilarity

Integration of fine neck theorem and Harnack inequality

Abstract

In this short paper, we give a new proof of the classification theorem for noncompact ancient noncollapsed flows in $\mathbb{R}^3$ originally due to Brendle-Choi (Inventiones 2019). Our new proof directly establishes selfsimilarity by combining the fine neck theorem from our joint work with Hershkovits and the rigidity case of Hamilton's Harnack inequality.