Flow by powers of the Gauss curvature in space forms

In this paper, we prove that convex hypersurfaces under the flow by powers α>0 of the Gauss curvature in space forms Nn+1(κ) of constant sectional curvature κ(κ=±1) contract to a point in finite time T⁎. Moreover, convex hypersurfaces under the flow by power α>1n+2 of the Gauss curvature conve...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2024-04, Vol.442, p.109579, Article 109579
Main Authors: Chen, Min, Huang, Jiuzhou
Format: Article
Language:English
Subjects:
Entropy
Gauss curvature
Monotonicity
Regularity estimates
Space forms
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Summary:In this paper, we prove that convex hypersurfaces under the flow by powers α>0 of the Gauss curvature in space forms Nn+1(κ) of constant sectional curvature κ(κ=±1) contract to a point in finite time T⁎. Moreover, convex hypersurfaces under the flow by power α>1n+2 of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in Nn+1(κ). This extends the known results in Euclidean space to space forms.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2024.109579