论文标题
平均曲率流的自扩展器
Self-expanders of the mean curvature flow
论文作者
论文摘要
我们研究自扩展解决方案$ M^m \ subset \ mathbb {r}^{n} $的平均曲率流量。我们的主要结果之一是，当且仅当功能$ | | a |^2/| h |^2 $时，完全平均凸自扩展的超膨胀性是自动扩展曲线和平面子空间的产物，其中$ a $ a $表示第二个基本形式，而$ h $ $ h $ $ h $ $ m $ $ m $。如果Pricipal正常$ξ= H/| H | $在正常捆绑包中是平行的，则类似的结果在函数$ | a^ξ|^2/| h |^2 $中具有更高的编成编成，其中$ a^ξ$是相对于$ξ$的第二个基本形式。作为推论，如果它们平稳地渐近地渐近呈渐近率，则可以获得完全平均凸的自扩散器，即可获得严格的正标曲率。特别是，在尺寸$ 2 $中，任何平均凸的自我远加者都必须严格地凸出。
We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function $|A|^2/|H|^2$ attains a local maximum, where $A$ denotes the second fundamental form and $H$ the mean curvature vector of $M$. If the pricipal normal $ξ=H/|H|$ is parallel in the normal bundle, then a similar result holds in higher codimension for the function $|A^ξ|^2/|H|^2$, where $A^ξ$ is the second fundamental form with respect to $ξ$. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension $2$ any mean convex self-expander that is asymptotic to a cone must be strictly convex.
