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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
MathSciNet review: 4172612
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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 $k$th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when $k$ is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.


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Additional Information

Frederico Girão
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Fortaleza 60455-760, Brazil
ORCID: 0000-0002-4418-2737
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
MR Author ID: 1244382
Email: neilha@ufam.edu.br

Diego Rodrigues
Affiliation: Campus de Quixadá, Instituto Federal do Ceará, Quixadá 63902-580, Brazil
ORCID: 0000-0002-1710-8013
Email: diego.sousa.ismart@gmail.com

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