# Ancient asymptotically cylindrical flows and applications

@article{Choi2022AncientAC,
title={Ancient asymptotically cylindrical flows and applications},
author={Kyeongsu Choi and Robert Haslhofer and Or Hershkovits and Brian White},
journal={Inventiones mathematicae},
year={2022}
}
• Published 1 October 2019
• Mathematics
• Inventiones mathematicae
In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the…
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