Noncompact self-shrinkers for mean curvature flow with arbitrary genus
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917768543076352 |
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| author | Buzano, Reto Nguyen, Huy The Schulz, Mario B. |
| author_facet | Buzano, Reto Nguyen, Huy The Schulz, Mario B. |
| contents | In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_06027 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Noncompact self-shrinkers for mean curvature flow with arbitrary genus Buzano, Reto Nguyen, Huy The Schulz, Mario B. Differential Geometry In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends. |
| title | Noncompact self-shrinkers for mean curvature flow with arbitrary genus |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2110.06027 |