论文标题
平均曲率流下有界直径
Bounded Diameter Under Mean Curvature Flow
论文作者
论文摘要
我们证明，对于闭合嵌入式超曲面的平均曲率流动，固有直径保持均匀界限，只要流动在第一个奇异时间，只要所有奇异性都是颈部或圆锥形类型。特别是，假设伊尔曼纳（Ilmanen）的多重性一个猜想，而没有圆柱体的猜想，我们得出结论，在二维情况下，直径始终保持界限。我们还获得了针对曲率的尖锐$ l^{n-1} $。我们证明的关键要素是洛贾西维奇（Lojasiewicz）的不平等现象，微米和乔多什·舒尔兹（Chodosh-Schulze）以及cho，haslhofer，hershkovits和white的均值凸纽群落的解决方案。我们的结果改善了Gianniotis-Haslhofer的先前结果，在该流量是全球两种凸口的更严格的假设下，直径和曲率控制已获得。
We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen's multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case, the diameter always stays bounded. We also obtain sharp $L^{n-1}$ bound for the curvature. The key ingredients for our proof are the Lojasiewicz inequalities by Colding-Minicozzi and Chodosh-Schulze, and the solution of the mean-convex neighbourhood conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the prior results by Gianniotis-Haslhofer, where diameter and curvature control has been obtained under the more restrictive assumption that the flow is globally two-convex.
