Kröner, Heiko and Scheuer, Julian ORCID: https://orcid.org/0000-0003-2664-1896 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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Abstract
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 |
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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: | 12 Nov 2023 21:40 |
URI: | https://orca.cardiff.ac.uk/id/eprint/135468 |
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