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arXiv · 2608.30787

Ancient cylindrical flows with fast or slow convergence

Kyeongsu Choi·Robert Haslhofer
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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).

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Kyeongsu Choi, Robert Haslhofer. 2026-08-31. Ancient cylindrical flows with fast or slow convergence. https://arxiv.org/abs/2608.30787

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