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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.09199 |
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| _version_ | 1866908821154168832 |
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| author | Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas |
| author_facet | Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas |
| contents | In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology $S^{2n-1} \times R$. The example wraps around a Clifford torus $S^{2n-1} \times S^1$ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On mean curvature flow solitons in the sphere Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas Differential Geometry In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology $S^{2n-1} \times R$. The example wraps around a Clifford torus $S^{2n-1} \times S^1$ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form. |
| title | On mean curvature flow solitons in the sphere |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.09199 |