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arXiv 2007.03187math.DG
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关于标准高斯空间$^†$中子流形的中曲率流

On the mean curvature flow of submanifolds in the standard Gaussian space $^†$

An-Min Li, Xingxiao Li, Di Zhang

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30 pages; comments are welcome
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摘要：

本文研究了标准高斯度量空间$({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$中子流形的中曲率流的正则几何行为，其中$({\mathbb R}^{m+p},\ol g)$是标准欧氏空间，$x\in{\mathbb R}^{m+p}$表示位置向量.注意，作为一个特殊的黎曼流形，$({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$具有无界曲率。直到$M^m$上的一个同构族，我们在这里考虑的中曲率流被证明等价于我们以前介绍过的“{ \em共形中曲率流}”的一个特殊变体。本文的主要定理从几何上指出：标准高斯空间中任何位置向量的平方模不等于$m$的浸入紧致子流形，在中曲率流作用下，在有限时刻爆破，即位置或曲率爆破到无穷大;而且，利用这个主要定理，使流动子流形保持正则的时间区间$[0,T)$具有一定的最优上界，当且仅当初始子流形收缩到原点或在流动下均匀膨胀到无穷大时，它才能达到边界。除主要定理外，我们还得到了一些有趣的结论，这些结论不仅对证明主要定理起着关键作用，而且部分地刻画了流的几何行为，具有独立的意义。

英文摘要：

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.

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