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The
quermassintegral-preserving mean curvature flow in the
sphere
Esther Cabezas-Rivas and Julian Scheuer |
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Vol. 17 (2024), No. 10, 3589–3621
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DOI: 10.2140/apde.2024.17.3589
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Abstract |
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We introduce a mean curvature flow with global term of convex hypersurfaces in the sphere, for which the global term can be chosen to keep any quermassintegral fixed. Then, starting from a strictly convex initial hypersurface, we prove that the flow exists for all times and converges smoothly to a geodesic sphere. This provides a workaround to an issue present in the volume-preserving mean curvature flow in the sphere introduced by Huisken (1987). We also classify solutions for some constant curvature-type equations in space forms, as well as solitons in the sphere and in the upper branch of the De Sitter space. |
Keywords
volume-preserving mean curvature flow, spherical geometry
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Mathematical Subject Classification
Primary: 53C21, 53E10
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Milestones
Received: 5 December 2022
Accepted: 31 August 2023
Published: 21 November 2024
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Authors |
| Esther Cabezas-Rivas | |
| Departament de Matemàtiques Universitat de València València Spain |
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| Julian Scheuer | |
| Institut für Mathematik Goethe-Universität Frankfurt Germany |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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