Flow expanding by Gauss curvature to \(L_p\) dual Minkowski problems. (English) Zbl 1487.53118
The \(L_p\) dual Minkowski problem is a fundamental problem in modern convex geometry and geometric analysis, associated to a class of elliptic Monge-Ampère-type equations on the sphere.
In this paper, the authors consider a class of normalised anisotropic inverse Gauss curvature flows, that is, parabolic Monge-Ampère-type equations for the support function of the hypersurfaces. Based on a priori estimates, they obtain the long-time existence and convergence of the flow. Different from the known elliptic methods, the existence of solutions to the \(L_p\) dual Minkowski problem is proved by using geometric flows for the case \(p>0\). Also see [H. Chen and Q.-R. Li, J. Funct. Anal. 281, No. 8, Article ID 109139, 65 p. (2021; Zbl 1469.35115)].
In this paper, the authors consider a class of normalised anisotropic inverse Gauss curvature flows, that is, parabolic Monge-Ampère-type equations for the support function of the hypersurfaces. Based on a priori estimates, they obtain the long-time existence and convergence of the flow. Different from the known elliptic methods, the existence of solutions to the \(L_p\) dual Minkowski problem is proved by using geometric flows for the case \(p>0\). Also see [H. Chen and Q.-R. Li, J. Funct. Anal. 281, No. 8, Article ID 109139, 65 p. (2021; Zbl 1469.35115)].
Reviewer: Jiguang Bao (Beijing)
Citations:
Zbl 1469.35115References:
| [1] | A1 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. |
| [2] | Andrews, Ben, Motion of hypersurfaces by Gauss curvature, Pacific J. Math., 195, 1, 1-34 (2000) · Zbl 1028.53072 · doi:10.2140/pjm.2000.195.1 |
| [3] | B\"{o}r\"{o}czky, K\'{a}roly J.; Fodor, Ferenc, The \(L_p\) dual Minkowski problem for \(p>1\) and \(q>0\), J. Differential Equations, 266, 12, 7980-8033 (2019) · Zbl 1437.52002 · doi:10.1016/j.jde.2018.12.020 |
| [4] | B\"{o}r\"{o}czky, K\'{a}roly J.; Henk, Martin; Pollehn, Hannes, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differential Geom., 109, 3, 411-429 (2018) · Zbl 1397.52005 · doi:10.4310/jdg/1531188189 |
| [5] | B\"{o}r\"{o}czky, K\'{a}roly J.; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming, The dual Minkowski problem for symmetric convex bodies, Adv. Math., 356, 106805, 30 pp. (2019) · Zbl 1427.52006 · doi:10.1016/j.aim.2019.106805 |
| [6] | Bryan, Paul; Ivaki, Mohammad N.; Scheuer, Julian, A unified flow approach to smooth, even \(L_p\)-Minkowski problems, Anal. PDE, 12, 2, 259-280 (2019) · Zbl 1401.53048 · doi:10.2140/apde.2019.12.259 |
| [7] | Caffarelli, Luis A., Interior \(W^{2,p}\) estimates for solutions of the Monge-Amp\`ere equation, Ann. of Math. (2), 131, 1, 135-150 (1990) · Zbl 0704.35044 · doi:10.2307/1971510 |
| [8] | Cheng, Shiu Yuen; Yau, Shing Tung, On the regularity of the solution of the \(n\)-dimensional Minkowski problem, Comm. Pure Appl. Math., 29, 5, 495-516 (1976) · Zbl 0363.53030 · doi:10.1002/cpa.3160290504 |
| [9] | Chen, Chuanqiang; Huang, Yong; Zhao, Yiming, Smooth solutions to the \(L_p\) dual Minkowski problem, Math. Ann., 373, 3-4, 953-976 (2019) · Zbl 1417.52008 · doi:10.1007/s00208-018-1727-3 |
| [10] | CL H. Chen and Q. Li, The \(L_p\) dual Minkowski problem and related parabolic flows, https://pdfs.semanticscholar.org/5b54/093165a53513326520ff5d7b20d067749c94.pdf, 2021. · Zbl 1469.35115 |
| [11] | Gerhardt, Claus, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32, 1, 299-314 (1990) · Zbl 0708.53045 |
| [12] | Gerhardt, Claus, Curvature problems, Series in Geometry and Topology 39, x+323 pp. (2006), International Press, Somerville, MA · Zbl 1131.53001 |
| [13] | Gerhardt, Claus, Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differential Equations, 49, 1-2, 471-489 (2014) · Zbl 1284.53062 · doi:10.1007/s00526-012-0589-x |
| [14] | Huisken, Gerhard; Ilmanen, Tom, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59, 3, 353-437 (2001) · Zbl 1055.53052 |
| [15] | Huang, Yong; Liu, Jiakun; Xu, Lu, On the uniqueness of \(L_p\)-Minkowski problems: the constant \(p\)-curvature case in \(\mathbb{R}^3\), Adv. Math., 281, 906-927 (2015) · Zbl 1329.52003 · doi:10.1016/j.aim.2015.02.021 |
| [16] | Huang, Yong; Lu, QiuPing, On the regularity of the \(L_p\) Minkowski problem, Adv. in Appl. Math., 50, 2, 268-280 (2013) · Zbl 1278.35119 · doi:10.1016/j.aam.2012.08.005 |
| [17] | Huang, Yong; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216, 2, 325-388 (2016) · Zbl 1372.52007 · doi:10.1007/s11511-016-0140-6 |
| [18] | Huang, Yong; Zhao, Yiming, On the \(L_p\) dual Minkowski problem, Adv. Math., 332, 57-84 (2018) · Zbl 1393.52007 · doi:10.1016/j.aim.2018.05.002 |
| [19] | Li, Qi-Rui; Sheng, Weimin; Wang, Xu-Jia, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc. (JEMS), 22, 3, 893-923 (2020) · Zbl 1445.53066 · doi:10.4171/jems/936 |
| [20] | Chou, Kai-Seng; Wang, Xu-Jia, The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205, 1, 33-83 (2006) · Zbl 1245.52001 · doi:10.1016/j.aim.2005.07.004 |
| [21] | Lutwak, Erwin, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38, 1, 131-150 (1993) · Zbl 0788.52007 |
| [22] | Lutwak, Erwin; Oliker, Vladimir, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41, 1, 227-246 (1995) · Zbl 0867.52003 |
| [23] | Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong, \(L_p\) dual curvature measures, Adv. Math., 329, 85-132 (2018) · Zbl 1388.52003 · doi:10.1016/j.aim.2018.02.011 |
| [24] | Urbas, John I. E., On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205, 3, 355-372 (1990) · Zbl 0691.35048 · doi:10.1007/BF02571249 |
| [25] | Urbas, John I. E., An expansion of convex hypersurfaces, J. Differential Geom., 33, 1, 91-125 (1991) · Zbl 0746.53006 |
| [26] | Zhao, Yiming, Existence of solutions to the even dual Minkowski problem, J. Differential Geom., 110, 3, 543-572 (2018) · Zbl 1406.52017 · doi:10.4310/jdg/1542423629 |
| [27] | Zhu, Guangxian, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101, 1, 159-174 (2015) · Zbl 1331.53016 |
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