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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
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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
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