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Willmore inequality on hypersurfaces in hyperbolic space


Author: Yingxiang Hu
Journal: Proc. Amer. Math. Soc. 146 (2018), 2679-2688
MSC (2010): Primary 53C42, 53C44
DOI: https://doi.org/10.1090/proc/13968
Published electronically: February 1, 2018
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Abstract: In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for a closed surface in hyperbolic $ 3$-space.


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  • [1] A. D. Alexandrov, Zur Theorie der gemischten Volumnia von konvexen Körpern, II: Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. 44 (1937), 1205-1238.
  • [2] A. D. Alexandrov, Zur Theorie der gemischten Volumnia von konvexen Körpern, III: Die Erweiterung zweeier Lehrsatze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flachen, Mat. Sb. 45 (1938), 27-46.
  • [3] William Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213-242. MR 1230930
  • [4] Alexandr A. Borisenko and Vicente Miquel, Total curvatures of convex hypersurfaces in hyperbolic space, Illinois J. Math. 43 (1999), no. 1, 61-78. MR 1665641
  • [5] Simon Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247-269. MR 3090261
  • [6] Simon Brendle, Pei-Ken Hung, and Mu-Tao Wang, A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124-144. MR 3433631
  • [7] Sun-Yung Alice Chang and Yi Wang, On Aleksandrov-Fenchel inequalities for $ k$-convex domains, Milan J. Math. 79 (2011), no. 1, 13-38. MR 2831436
  • [8] Sun-Yung Alice Chang and Yi Wang, Inequalities for quermassintegrals on $ k$-convex domains, Adv. Math. 248 (2013), 335-377. MR 3107515
  • [9] Sun-Yung A. Chang and Yi Wang, Some higher order isoperimetric inequalities via the method of optimal transport, Int. Math. Res. Not. IMRN 24 (2014), 6619-6644. MR 3291634
  • [10] Bang-yen Chen, On the total curvature of immersed manifolds. I. An inequality of Fenchel-Borsuk-Willmore, Amer. J. Math. 93 (1971), 148-162. MR 0278240
  • [11] Bang-yen Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. (4) 10 (1974), 380-385 (English, with Italian summary). MR 0370436
  • [12] Xu Cheng and Detang Zhou, Rigidity for closed totally umbilical hypersurfaces in space forms, J. Geom. Anal. 24 (2014), no. 3, 1337-1345. MR 3223556
  • [13] Levi Lopes de Lima and Frederico Girão, An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality, Ann. Henri Poincaré 17 (2016), no. 4, 979-1002. MR 3472630
  • [14] A. El Soufi and S. Ilias, Une inégalité du type ``Reilly'' pour les sous-variétés de l'espace hyperbolique, Comment. Math. Helv. 67 (1992), no. 2, 167-181 (French). MR 1161279
  • [15] W. Fenchel, Inégalités quadratiques entre les olumes mixtes des corps convexes, C. R. Acad. Sci., Paris 203 (1936), 647-650.
  • [16] Eduardo Gallego and Gil Solanes, Integral geometry and geometric inequalities in hyperbolic space, Differential Geom. Appl. 22 (2005), no. 3, 315-325. MR 2166125
  • [17] Y.-X. Ge, G.-F. Wang, J. Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I, preprint, 2013, arXiv: 1303.1714.
  • [18] Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237-260. MR 3263518
  • [19] Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487-527. MR 2879249
  • [20] G. W. Gibbons, Collapsing shells and the isoperimetric inequality for black holes, Classical Quantum Gravity 14 (1997), no. 10, 2905-2915. MR 1476553
  • [21] Pengfei Guan and Junfang Li, The quermassintegral inequalities for $ k$-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725-1732. MR 2522433
  • [22] G. Huisken and T. Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices 20 (1997), 1045-1058. MR 1486695
  • [23] 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
  • [24] Pei-Ken Hung and Mu-Tao Wang, Inverse mean curvature flows in the hyperbolic 3-space revisited, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 119-126. MR 3385155
  • [25] Haizhong Li, Yong Wei, and Changwei Xiong, A geometric inequality on hypersurface in hyperbolic space, Adv. Math. 253 (2014), 152-162. MR 3148549
  • [26] Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269-291. MR 674407
  • [27] Masao Maeda, The integral of the mean curvature, Sci. Rep. Yokohama Nat. Univ. Sect. I 25 (1978), 17-21. MR 523505
  • [28] Guohuan Qiu, A family of higher-order isoperimetric inequalities, Commun. Contemp. Math. 17 (2015), no. 3, 1450015, 20. MR 3325038
  • [29] Manuel Ritoré, Optimal isoperimetric inequalities for three-dimensional Cartan-Hadamard manifolds, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 395-404. MR 2167269
  • [30] Peter Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math. 503 (1998), 47-61. MR 1650335
  • [31] Neil S. Trudinger, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 4, 411-425 (English, with English and French summaries). MR 1287239
  • [32] Guofang Wang and Chao Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532-556. MR 3197666
  • [33] Kui Wang, Singularities of mean curvature flow and isoperimetric inequalities in $ \mathbb{H}^3$, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2651-2660. MR 3326044

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Additional Information

Yingxiang Hu
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing 100086, People’s Republic of China
Email: huyingxiang10@163.com

DOI: https://doi.org/10.1090/proc/13968
Keywords: Willmore inequality, inverse curvature flow, hyperbolic space
Received by editor(s): November 27, 2016
Received by editor(s) in revised form: September 13, 2017
Published electronically: February 1, 2018
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society

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