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Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

Kröner, Heiko and Scheuer, Julian 2019. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature. Mathematical News / Mathematische Nachrichten 292 (7) , pp. 1514-1529. 10.1002/mana.201700370

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We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form urn:x-wiley:0025584X:media:mana201700370:mana201700370-math-0001, where urn:x-wiley:0025584X:media:mana201700370:mana201700370-math-0002 and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature urn:x-wiley:0025584X:media:mana201700370:mana201700370-math-0003. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power urn:x-wiley:0025584X:media:mana201700370:mana201700370-math-0004 loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Wiley-VCH Verlag
ISSN: 0025-584X
Funders: Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
Date of First Compliant Deposit: 8 October 2020
Date of Acceptance: 5 October 2018
Last Modified: 27 Feb 2021 04:51

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