Ancient cylindrical flows with fast or slow convergence
Abstract
In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is -- up to splitting Euclidean factors -- either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).
Related
- Mean curvature flows with cylindrical singularities II: stability and genericity
- Morse resolution of mean curvature flows with cylindrical singularities
- Parametrization of ancient Ricci flows with smooth compact tangent shrinkers
- Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold
- Symmetry Breaking and Moduli for Ancient Yamabe Flows on the Sphere
BibTeX
@misc{choi2026ancient,
title = {{Ancient cylindrical flows with fast or slow convergence}},
author = {Kyeongsu Choi and Robert Haslhofer},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30787},
eprint = {2608.30787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30787},
}
RIS
TY - UNPB
TI - Ancient cylindrical flows with fast or slow convergence
AU - Kyeongsu Choi
AU - Robert Haslhofer
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30787
UR - https://arxiv.org/abs/2608.30787
AB - In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is -- up to splitting Euclidean factors -- either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).
ER -