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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The maximum volume of hyperbolic polyhedra


Author: Giulio Belletti
Journal: Trans. Amer. Math. Soc. 374 (2021), 1125-1153
MSC (2010): Primary 52B10
DOI: https://doi.org/10.1090/tran/8215
Published electronically: November 3, 2020
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Abstract: We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal, or hyperideal). We find that the supremum is always equal to the volume of the rectification of the $ 1$-skeleton.

The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lower-dimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron.


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Additional Information

Giulio Belletti
Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
Email: giulio.belletti@sns.it

DOI: https://doi.org/10.1090/tran/8215
Received by editor(s): February 18, 2020
Received by editor(s) in revised form: May 21, 2020
Published electronically: November 3, 2020
Article copyright: © Copyright 2020 American Mathematical Society