论文标题
旋转对称性超曲面的逆平均曲率流
Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces
论文作者
论文摘要
我们证明，在$ \ mathbb {r}^{n+1} $中，非星形的，均值键入的球体的平均值曲率流与轴上有对称性，并且有史以来较长，较厚的脖子在所有时间上存在，并且在圆形的范围内，$ t \ frightrow commentery的脖子有足够长，厚实的脖子。我们的方法基于抛物线最大原理的局部版本。 我们还提供了此结果的两个应用。第一个是Minkowski不等式的扩展，以$ \ Mathbb {r}^{n+1} $中相应的非星形均值键域的扩展。第二个是IMCF和最小表面理论之间的联系。根据Meeks和Yau的先前工作以及IMCF的叶子的使用，我们建立了解决高原问题的解决方案的嵌入性，并在$ \ Mathbb {r}^{3} $中为某些Jordan曲线的稳定沉浸式最小磁盘的有限属性属性。
We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in $\mathbb{R}^{n+1}$ with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as $t \rightarrow \infty$. Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in $\mathbb{R}^{n+1}$. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in $\mathbb{R}^{3}$.
