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

Kyeongsu Choi, Robert Haslhofer

TL;DR

This work provides a new proof of the Brendle–Choi classification for noncompact ancient noncollapsed flows in $\mathbb{R}^3$, showing that such flows without a line-splitting are selfsimilar translators, hence the bowls up to scaling and rigid motion. The approach merges the differential neck theorem (from the authors’ joint work with Hershkovits) with the rigidity case of Hamilton’s Harnack inequality to force selfsimilarity from neck-slope control. It also yields a higher-dimensional analogue by extending the neck-analysis framework via the CHHW fine neck theorem, giving a new proof of uniform two-convex classification in $\mathbb{R}^{n+1}$. The results have broad implications for singularity analysis in mean curvature flow, including implications for neighborhood conjectures and flow uniqueness, by unifying ancient noncollapsed flow behavior under a translating soliton paradigm.

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.

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

TL;DR

This work provides a new proof of the Brendle–Choi classification for noncompact ancient noncollapsed flows in , showing that such flows without a line-splitting are selfsimilar translators, hence the bowls up to scaling and rigid motion. The approach merges the differential neck theorem (from the authors’ joint work with Hershkovits) with the rigidity case of Hamilton’s Harnack inequality to force selfsimilarity from neck-slope control. It also yields a higher-dimensional analogue by extending the neck-analysis framework via the CHHW fine neck theorem, giving a new proof of uniform two-convex classification in . The results have broad implications for singularity analysis in mean curvature flow, including implications for neighborhood conjectures and flow uniqueness, by unifying ancient noncollapsed flow behavior under a translating soliton paradigm.

Abstract

In this short paper, we give a new proof of the classification theorem for noncompact ancient noncollapsed flows in 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.
Paper Structure (3 sections, 3 theorems, 9 equations)

This paper contains 3 sections, 3 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Background and differential neck theorem
  3. Proof of the classification theorem

Key Result

Theorem 1.1

Any noncompact ancient noncollapsed flow in $\mathbb{R}^3$ is, up to scaling and rigid motion, either the static plane, the round shrinking cylinder, or the rotationally symmetric bowl.

Theorems & Definitions (8)

  • Theorem 1.1: Brendle-Choi BC
  • Theorem 1.2: differential neck theorem
  • Theorem 1.4: selfsimilarity
  • Remark 1.5: higher dimensions
  • Remark 2.8: intuitive explanation
  • proof : Proof of Theorem \ref{['thm_selfsim']}
  • Claim 3.1: cylindrical scale, c.f. CHH and CHHW
  • proof : Proof of Claim \ref{['claim_cyl']}