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arXiv 2011.016332020-11math.DG

柱形自收缩子的Łojasiewicz不等式、唯一性与刚性

Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

Jonathan J. Zhu

备注：: 39 pages; simplified some proofs, added details of iteration scheme

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AI总结：

该研究针对平均曲率流的一类柱形自收缩子建立Łojasiewicz不等式，证明对应奇点爆破的唯一性与这类柱体的孤立性，解答了Colding-Minicozzi的相关猜想，其直接微扰分析方法对圆柱情形也具创新性。

摘要：

我们针对平均曲率流的一类柱形自收缩子建立了Łojasiewicz不等式，这类自收缩子包括任意余维数下的圆柱以及以Abresch-Langer曲线为底的柱体。我们推导了以这类柱体为模型的奇点处爆破的唯一性，且证明了所有此类柱体在自收缩子空间中都是孤立的。其中Abresch-Langer情形解答了Colding-Minicozzi的一个猜想。我们的证明采用了对收缩子平均曲率的直接微扰分析，因此即便对于圆柱而言也是全新的方法。

英文摘要：

We establish Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.

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