Weighted Alexandrov-Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu
Authors:
Frederico Girão, Diego Pinheiro, Neilha M. Pinheiro and Diego Rodrigues
Journal:
Proc. Amer. Math. Soc. 149 (2021), 369-382
MSC (2020):
Primary 51M16; Secondary 53E10, 53A35
DOI:
https://doi.org/10.1090/proc/15127
Published electronically:
October 16, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when
is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.
- [1] A. Alexandroff, Über die Oberflächenfunktion eines konvexen Körpers. (Bemerkung zur Arbeit “Zur Theorie der gemischten Volumina von konvexen Körpern”), Rec. Math. N.S. [Mat. Sbornik] 6(48) (1939), 167–174 (Russian, with German summary). MR 0001597
- [2] A. Alexandroff, Zur Theorie der gemischten Volumina von konvexen Körpern III. Die Erweiterung zweier Lehrsätze Minkowskis über die konvexen Polyeder auf die beliebigen konvexen Körper., Rec. Math. Moscou, n. Ser. 3 (1938), 27-46 (Russian).
- [3] 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, https://doi.org/10.1002/cpa.21556
- [4] Mattias Dahl, Romain Gicquaud, and Anna Sakovich, Penrose type inequalities for asymptotically hyperbolic graphs, Ann. Henri Poincaré 14 (2013), no. 5, 1135–1168. MR 3070749, https://doi.org/10.1007/s00023-012-0218-4
- [5] Levi Lopes de Lima and Frederico Girão, The ADM mass of asymptotically flat hypersurfaces, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6247–6266. MR 3356936, https://doi.org/10.1090/S0002-9947-2014-05902-3
- [6] 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, https://doi.org/10.1007/s00023-015-0414-0
- [7] Alexandre de Sousa and Frederico Girão, The Gauss-Bonnet-Chern mass of higher-codimension graphs, Pacific J. Math. 298 (2019), no. 1, 201–216. MR 3910054, https://doi.org/10.2140/pjm.2019.298.201
- [8] Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I, arXiv e-prints (2013), arXiv:1303.1714.
- [9] Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237–260. MR 3263518
- [10] Yuxin Ge, Guofang Wang, and Jie Wu, A new mass for asymptotically flat manifolds, Adv. Math. 266 (2014), 84–119. MR 3262355, https://doi.org/10.1016/j.aim.2014.08.006
- [11] Yuxin Ge, Guofang Wang, and Jie Wu, The GBC mass for asymptotically hyperbolic manifolds, Math. Z. 281 (2015), no. 1-2, 257–297. MR 3384870, https://doi.org/10.1007/s00209-015-1483-y
- [12] Frederico Girão and Diego Rodrigues, Weighted geometric inequalities for hypersurfaces in sub-static manifolds, Bulletin of the London Mathematical Society 52 (2020), no. 1, 121-136.
- [13] Pengfei Guan and Junfang Li, The quermassintegral inequalities for 𝑘-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. MR 2522433, https://doi.org/10.1016/j.aim.2009.03.005
- [14] R. R. Hall, A class of isoperimetric inequalities, J. Analyse Math. 45 (1985), 169–180. MR 833410, https://doi.org/10.1007/BF02792548
- [15] Yingxiang Hu, Haizhong Li, and Yong Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, arXiv e-prints (2020), arXiv:2002.10643.
- [16] Gerhard Huisken, Evolution of hypersurfaces by their curvature in Riemannian manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 349–360. MR 1648085
- [17] Mau-Kwong George Lam, The Graph Cases of the Riemannian Positive Mass and Penrose Inequalities in All Dimensions, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Duke University. MR 2873434
- [18] Haizhong Li, Yong Wei, and Changwei Xiong, The Gauss-Bonnet-Chern mass for graphic manifolds, Ann. Global Anal. Geom. 45 (2014), no. 4, 251–266. MR 3180948, https://doi.org/10.1007/s10455-013-9399-4
- [19] Haizhong Li, Yong Wei, and Changwei Xiong, A geometric inequality on hypersurface in hyperbolic space, Adv. Math. 253 (2014), 152–162. MR 3148549, https://doi.org/10.1016/j.aim.2013.12.003
- [20] Carlo Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949
- [21] H. Mirandola and F. Vitório, The positive mass theorem and Penrose inequality for graphical manifolds, Comm. Anal. Geom. 23 (2015), no. 2, 273–292. MR 3298670, https://doi.org/10.4310/CAG.2015.v23.n2.a2
- [22] Horst Sachs, Über eine Klasse isoperimetrischer Probleme. I, II, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 8 (1958/59), 121–126, 127–134 (German). MR 108770
- [23] Horst Sachs, Ungleichungen für Umfang, Flächeninhalt und Trägheitsmoment konvexer Kurven, Acta Math. Acad. Sci. Hungar. 11 (1960), 103–115 (German, with Russian summary). MR 140002, https://doi.org/10.1007/BF02020628
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2020): 51M16, 53E10, 53A35
Retrieve articles in all journals with MSC (2020): 51M16, 53E10, 53A35
Additional Information
Frederico Girão
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza 60455-760, Brazil
Email:
fred@mat.ufc.br
Diego Pinheiro
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza 60455-760, Brazil
Email:
diegodsp01@gmail.com
Neilha M. Pinheiro
Affiliation:
Departamento de Matemática, Universidade Federal do Amazonas, Manaus 69067-005, Brazil
Email:
neilha@ufam.edu.br
Diego Rodrigues
Affiliation:
Campus de Quixadá, Instituto Federal do Ceará, Quixadá 63902-580, Brazil
Email:
diego.sousa.ismart@gmail.com
DOI:
https://doi.org/10.1090/proc/15127
Keywords:
Alexandrov--Fenchel inequalities,
horospherical convexity,
support function flow
Received by editor(s):
June 24, 2019
Received by editor(s) in revised form:
April 6, 2020
Published electronically:
October 16, 2020
Additional Notes:
The first author was partially supported by CNPq, grant number 306196/2016-6, and by FUNCAP, grant number 00068.01.00/15. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
Communicated by:
Jia-Ping Wang
Article copyright:
© Copyright 2020
American Mathematical Society