The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

Abstract

A flexible polyhedron in an n-dimensional space of constant curvature, namely, in the Euclidean space, or in the Lobachevsky space, or in the sphere, is a polyhedron with rigid (n-1)-dimensional faces and hinges at (n-2)-dimensional faces. The Bellows conjecture claims that, for n greater than or equal to 3, the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov in dimension 3 (1996) and by the author in dimensions 4 and higher (2012). Counterexamples to the Bellows conjecture in open hemispheres were constructed by Alexandrov in dimension 3 (1997) and by the author in dimensions 4 and higher (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 the Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths.

41 pages, two errors corrected, in Lemma 3.6 (former Lemma 3.5) and in the proof of Theorem 1.2

References in corpus (2)

• Flexible cross-polytopes in spaces of constant curvature
• Embedded flexible spherical cross-polytopes with non-constant volumes

Cited by in corpus (3)

• Dehn invariant of flexible polyhedra
• A sufficient condition for a polyhedron to be rigid
• The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in ${\mathbb{R}}^d$ does not always remain unaltered during the flex