Generic mean curvature flow I; generic singularities
Generic mean curvature flow I; generic singularities
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DOI：
10.4007/annals.2012.175.2.7
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
T. Colding;W. Minicozzi
中科院分区：
文献类型：
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作者：
T. Colding;W. Minicozzi
长期以来，人们已经证明，从R^3中的一般光滑闭嵌入曲面开始，平均曲率流保持光滑，直到它到达奇点，在奇点的邻域中，流看起来像同心球或圆柱。也就是说，一般流的唯一奇点是球形或圆柱形的。我们将在这里和后续讨论这个猜想。更高维度的情况将在别处讨论。证明这一猜想的关键是要证明收缩球、圆柱和平面是在平均曲率流下唯一稳定的自收缩体。我们在所有维度上都证明了这一点。这样做的一个简单的结果是，除了球面和柱面之外，所有其他的奇点都可以被摄动掉。
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.