Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912030786584576 |
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| author | Zhu, Jonathan J. |
| author_facet | Zhu, Jonathan J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_01633 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers Zhu, Jonathan J. Differential Geometry 53E10, 53C42 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. |
| title | Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers |
| topic | Differential Geometry 53E10, 53C42 |
| url | https://arxiv.org/abs/2011.01633 |