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Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

Part of: Global differential geometry Elliptic equations and systems

Published online by Cambridge University Press:  01 November 2021

Li Chen ,
Qiang Tu ,
Di Wu  and
Ni Xiang
Li Chen
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Qiang Tu
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Di Wu
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Ni Xiang
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
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Abstract

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

Keywords

Gauss curvature flowconvex hypersurfaceMonge-Ampère type equation

MSC classification

Primary: 35J96: Elliptic Monge-Ampère equations
Secondary: 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 148 - 162
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Andrews, B.. Contraction of convex hypersurfaces by their affine normal. Jour. Diff. Geom. 43 (1996), 207–230.Google Scholar
Andrews, B.. Evolving convex curves. Calc. Vara. Part. Diff. Equa. 7 (1998), 315–371.10.1007/s005260050111CrossRefGoogle Scholar
Andrews, B.. Gauss curvature flow the fate of the rolling stones. Invent. Math. 138 (1999), 151–161.10.1007/s002220050344CrossRefGoogle Scholar
Andrews, B.. Motion of hypersurfaces by Gauss curvature. Paci. Jour. Math. 195 (2000), 1–34.10.2140/pjm.2000.195.1CrossRefGoogle Scholar
Andrews, B.. Classification of limiting shapes for isotropic curve flows. Jour. Amer. Math. Soc. 16 (2003), 443–459.10.1090/S0894-0347-02-00415-0CrossRefGoogle Scholar
Andrews, B. and Chen, X.. Surface moving by powers of Gauss curvature. Pure. Appl. Math. Q. 8 (2012), 825–834.10.4310/PAMQ.2012.v8.n4.a1CrossRefGoogle Scholar
Andrews, B., Guan, P. F. and Ni, L.. Flow by powers of the Gauss curvature. Adv. Math. 299 (2016), 174–201.10.1016/j.aim.2016.05.008CrossRefGoogle Scholar
Böröczky, K., Lutwak, E., Yang, D. and Zhang, G.. The logarithmic Minkowski problem. Jour. Amer. Math. Soc. 26 (2013), 831–852.10.1090/S0894-0347-2012-00741-3CrossRefGoogle Scholar
Brendle, S., Choi, K. and Daskalopoulos, P.. Asymptotic behavior of flows by powers of the Gaussian curvature. Acta. Math. 219 (2017), 1–16.10.4310/ACTA.2017.v219.n1.a1CrossRefGoogle Scholar
Bryan, P., Ivaki, M. and Scheuer, J.. A unified flow approach to smooth,even $L_p$-Minkowski problems. Anal. PDE. 12 (2018), 259–280.10.2140/apde.2019.12.259CrossRefGoogle Scholar
Chen, C., Huang, Y. and Zhao, Y.. Smooth solutions to the $L_p$ dual Minkowski problem. Math. Ann. 373 (2019), 953–976.10.1007/s00208-018-1727-3CrossRefGoogle Scholar
Chen, H. and Li, Q.. The $L_p$ dual Minkowski problems and related parabolic flows, preprint.Google Scholar
Chou, K. and Wang, X.. A logarithmic Gauss curvature flow and the Minkowski problem. Ann. Inst. H. Poincariäe Anal. Non Liniäeaire 17 (2000), 733–751.CrossRefGoogle Scholar
Chow, B.. Deforming convex hypersurfaces by the n-th root of the Gaussian curvature. Jour. Diff. Geom. 22 (1985), 117–138.Google Scholar
Firey, W.. Shapes of worn stones. Mathematika 21 (1974), 1–11.10.1112/S0025579300005714CrossRefGoogle Scholar
Guan, P. F. and Ni, L.. Entropy and a convergence theorem for Gauss curvature flow in high dimension. Jour. Euro. Math. Soc. 19 (2017), 3735–3761.CrossRefGoogle Scholar
Huang, Y., Lutwak, E., Yang, D. and Zhang, G.. Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math. 216 (2016),325–388.CrossRefGoogle Scholar
Li, Q., Sheng, W. and Wang, X.. Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J. Eur. Math. Soc. 22 (2020), 893–923.10.4171/JEMS/936CrossRefGoogle Scholar
Liu, Y. and Li, J.. A flow method for the dual Orlicz-Minkowski problem. Trans. Amer. Math. Soc. 373 (2020), 5833–5853.CrossRefGoogle Scholar
Tso, K.. Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math. 38 (1985), 867–882.10.1002/cpa.3160380615CrossRefGoogle Scholar
Urbas, J.. An expansion of convex hypersurfaces. Jour. Diff. Geom. 33 (1991), 91–125.Google Scholar
Wang, X.. Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature. Trans. Amer. Math. Soc. 348 (1996), 4501–4524.CrossRefGoogle Scholar
Zhao, Y.. The Dual Minkowski Problem for Negative Indices. Calc. Vara. Part. Diff. Equa. 56 (2017) Art. 18, 16 pp.CrossRefGoogle Scholar
Zhu, B., Xing, S. and Ye, D.. The dual Orlicz-Minkowski problem. Jour. Geom. Anal. 430 (2015), 810–829.Google Scholar