游荡奇点

摘要：

抛物线几何流在短时间内是平滑的，然而，在长时间内，奇异性通常是不可避免的，可能是非常讨厌的，并且可能无法分类。[CM6]的思想是，通过引入流的动力学性质，我们还获得了一般初始条件的大时间平滑。当与[CM1]结合时，这表明，在一个重要的特殊情况下，奇点是最简单的可能。奇点的动力学问题有两个部分。一个是：奇点附近的动力学是什么？二是：什么是长期行为？也就是说，如果流动离开奇点的邻域，它能在很晚的时间返回吗？第一个问题在[CM6]中得到了解决，第二个问题在这里得到了解决。结合[CM1]，[CM6]，我们证明了所有其他的封闭奇点比（圆）球有一个邻域，其中“几乎每个”封闭超曲面离开下流，永远不会返回，甚至到一个膨胀，旋转或平移的副本的奇点。换句话说，它是游离的。相比之下，Huisken认为，任何靠近球体的闭超曲面在每一时刻都保持接近球体的扩张或平移副本。

英文摘要：

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initial conditions. When combined with [CM1], this shows, in an important special case, the singularities are the simplest possible. The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior? That is, if the flow leaves a neighborhood of a singularity, can it return at a much later time? The first question was addressed in [CM6] and the second here. Combined with [CM1], [CM6], we show that all other closed singularities than the (round) sphere have a neighborhood where `nearly every' closed hypersurface leaves under the flow and never returns, even to a dilated, rotated or translated copy of the singularity. In other words, it wanders off. In contrast, by Huisken, any closed hypersurface near a sphere remains close to a dilated or translated copy of the sphere at each time.

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