Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

Girão, Frederico; Pinheiro, Diego; Pinheiro, Neilha; Rodrigues, Diego

doi:10.1090/proc/15127

Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu
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by Frederico Girão, Diego Pinheiro, Neilha M. Pinheiro and Diego Rodrigues

Proc. Amer. Math. Soc. 149 (2021), 369-382

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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Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu
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Abstract: