The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces
Sbornik. Mathematics, Tome 206 (2015) no. 11, pp. 1564-1609
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A flexible polyhedron in an $n$-dimensional space $\mathbb{X}^n$ of constant curvature is a polyhedron with rigid $(n-1)$-dimensional faces and hinges at $(n-2)$-dimensional faces. The Bellows conjecture claims that, for $n\geqslant 3$, the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces $\mathbb{E}^n$ was proved by Sabitov for $n=3$ (1996) and by the author for $n\geqslant 4$ (2012). Counterexamples to the Bellows conjecture in open hemispheres $\mathbb{S}^n_+$ were constructed by Alexandrov for $n=3$ (1997) and by the author for $n\geqslant 4$ (2015). In this paper we prove the Bellows conjecture for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces. The proof is based on the study of the analytic continuation of the volume of a simplex in Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths.
Bibliography: 37 titles.
Keywords:
flexible polyhedron, Bellows conjecture, Lobachevsky space, Schläfli's formula, analytic continuation.
@article{SM_2015_206_11_a2,
author = {A. A. Gaifullin},
title = {The analytic continuation of volume and the {Bellows} conjecture in {Lobachevsky} spaces},
journal = {Sbornik. Mathematics},
pages = {1564--1609},
publisher = {mathdoc},
volume = {206},
number = {11},
year = {2015},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2015_206_11_a2/}
}
A. A. Gaifullin. The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces. Sbornik. Mathematics, Tome 206 (2015) no. 11, pp. 1564-1609. http://geodesic.mathdoc.fr/item/SM_2015_206_11_a2/