Flow expanding by Gauss curvature to 𝐿_{𝑝} dual Minkowski problems

Wu, Di; Wu, Chuanxi; Tu, Qiang

doi:10.1090/proc/15692

Flow expanding by Gauss curvature to $L_p$ dual Minkowski problems
HTML articles powered by AMS MathViewer

by Di Wu, Chuanxi Wu and Qiang Tu PDF

Proc. Amer. Math. Soc. 150 (2022), 305-318 Request permission

Abstract:

In this paper we consider a class of parabolic flows and study the long-time behavior of this flow, which amounts to solving a class of Monge-Ampère type equations. As an application, we provide an alternative proof for the existence of solutions to the $L_p$ dual Minkowski problem when $p > 0$.

References

A. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb. N.S., 3 (1938), 27–46.
Ben Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1–34. MR 1781612, DOI 10.2140/pjm.2000.195.1
Károly J. Böröczky and Ferenc Fodor, The $L_p$ dual Minkowski problem for $p>1$ and $q>0$, J. Differential Equations 266 (2019), no. 12, 7980–8033. MR 3944247, DOI 10.1016/j.jde.2018.12.020
Károly J. Böröczky, Martin Henk, and Hannes Pollehn, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differential Geom. 109 (2018), no. 3, 411–429. MR 3825606, DOI 10.4310/jdg/1531188189
Károly J. Böröczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang, and Yiming Zhao, The dual Minkowski problem for symmetric convex bodies, Adv. Math. 356 (2019), 106805, 30. MR 4008522, DOI 10.1016/j.aim.2019.106805
Paul Bryan, Mohammad N. Ivaki, and Julian Scheuer, A unified flow approach to smooth, even $L_p$-Minkowski problems, Anal. PDE 12 (2019), no. 2, 259–280. MR 3861892, DOI 10.2140/apde.2019.12.259
Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI 10.2307/1971510
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. MR 423267, DOI 10.1002/cpa.3160290504
Chuanqiang Chen, Yong Huang, and Yiming Zhao, Smooth solutions to the $L_p$ dual Minkowski problem, Math. Ann. 373 (2019), no. 3-4, 953–976. MR 3953117, DOI 10.1007/s00208-018-1727-3
H. Chen and Q. Li, The $L_p$ dual Minkowski problem and related parabolic flows, https://pdfs.semanticscholar.org/5b54/093165a53513326520ff5d7b20d067749c94.pdf, 2021.
Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314. MR 1064876
Claus Gerhardt, Curvature problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR 2284727
Claus Gerhardt, Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 471–489. MR 3148124, DOI 10.1007/s00526-012-0589-x
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Yong Huang, Jiakun Liu, and Lu Xu, On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in $\Bbb {R}^3$, Adv. Math. 281 (2015), 906–927. MR 3366857, DOI 10.1016/j.aim.2015.02.021
Yong Huang and QiuPing Lu, On the regularity of the $L_p$ Minkowski problem, Adv. in Appl. Math. 50 (2013), no. 2, 268–280. MR 3003347, DOI 10.1016/j.aam.2012.08.005
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388. MR 3573332, DOI 10.1007/s11511-016-0140-6
Yong Huang and Yiming Zhao, On the $L_p$ dual Minkowski problem, Adv. Math. 332 (2018), 57–84. MR 3810248, DOI 10.1016/j.aim.2018.05.002
Qi-Rui Li, Weimin Sheng, and Xu-Jia Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 893–923. MR 4055992, DOI 10.4171/jems/936
Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), no. 1, 33–83. MR 2254308, DOI 10.1016/j.aim.2005.07.004
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
Erwin Lutwak and Vladimir Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), no. 1, 227–246. MR 1316557
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ dual curvature measures, Adv. Math. 329 (2018), 85–132. MR 3783409, DOI 10.1016/j.aim.2018.02.011
John I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), no. 3, 355–372. MR 1082861, DOI 10.1007/BF02571249
John I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), no. 1, 91–125. MR 1085136
Yiming Zhao, Existence of solutions to the even dual Minkowski problem, J. Differential Geom. 110 (2018), no. 3, 543–572. MR 3880233, DOI 10.4310/jdg/1542423629
Guangxian Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom. 101 (2015), no. 1, 159–174. MR 3356071