论文标题
在标准高斯空间中的子手机的平均曲率流$^†$上
On the mean curvature flow of submanifolds in the standard Gaussian space $^†$
论文作者
论文摘要
在本文中，我们研究了标准高斯度量空间$（{\ Mathbb r}^{m+p}，e^{ - | x | x | x |^2/m} \ ol g）$（{ $ x \ in {\ mathbb r}^{m+p} $表示位置向量。请注意，作为特殊的Riemannian歧管，$（{\ Mathbb r}^{m+p}，e^{ - | x |^2/m} \ ol G）$具有无限的曲率。在$ m^m $上的一个差异性家庭中，我们在这里考虑的平均曲率流量相当于我们先前介绍的``{\ em em sonformal平均曲率流} \''的特殊变化。本文的主要定理表明，在标准高斯空间中，任何沉浸在标准高斯空间中的紧凑型亚体（其平均值不等于$ m $），在平均曲率流下有限的时间，其位置或曲率膨胀到无限;此外，根据这个主定理，间隔$ [0，t）$的时间，其中流动的子手机保持规则具有一定的最佳上限，并且只有当初始子序列缩小到原点或均匀扩展到该流程下的无穷大时，它才能达到界限。除了主要定理外，我们还获得了其他一些有趣的结论，这些结论不仅在证明主要定理方面发挥了关键作用，而且还部分特征了流动的几何行为，具有独立的意义。
In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard Euclidean space and $x\in{\mathbb R}^{m+p}$ denotes the position vector. Note that, as a special Riemannian manifold, $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ has an unbounded curvature. Up to a family of diffeomorphisms on $M^m$, the mean curvature flow we considered here turns out to be equivalent to a special variation of the ``{\em conformal mean curvature flow}\,'' which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to $m$, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval $[0,T)$ of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.
