AI 中文总结
本文针对平均凸邻域猜想的两种情形给出替代论证，采用基础方法证明了不同收敛速度下古老渐近柱形平均曲率流的类型，并简化了颈奇点猜想的证明。
AI 中文摘要
在近期的一项突破性研究中，Bamler-Lai证明了所有维度下的平均凸邻域猜想，其核心结论为：任何非平凡的古老渐近柱形平均曲率流，在拆分欧几里得因子后，要么是平移碗，要么是古老卵形，要么是平移卵形碗。本文针对上述三种情形中的两种提供了一种简短的替代论证。具体而言，本文采用更基础/传统的方法证明：若解向圆圆柱的收敛速度快，则该解为一个碗乘以一个欧几里得因子；若收敛速度慢，则该解为一个古老卵形。此外，本文还给出了颈奇点平均凸邻域猜想的新证明，该证明大幅简化并梳理了作者此前与Hershkovits合作（发表于《Acta》2022年）及与Hershkovits-White合作（发表于《Inventiones》2022年）的研究方法。
英文摘要
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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