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

Alexander A. Gaifullin

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.

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.

Paper Structure

This paper contains 11 sections, 43 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Unipotent filtrations on spaces of analytic functions
  3. Totally real functions
  4. The analytic continuation of volume
  5. Conditions (A) and (B)
  6. Proofs of Theorems \ref{['theorem_key']} and \ref{['theorem_key2']}
  7. Flexible polyhedra and their volumes
  8. The configuration space of polyhedra with given edge lengths
  9. Proof of Theorem \ref{['theorem_main2']} for polyhedra with non-degenerate faces
  10. Proof of Lemma \ref{['lem_log']}
  11. Proof of Theorem \ref{['theorem_main2']} for arbitrary polyhedra

Key Result

Theorem \oldthetheorem

The generalized oriented volume of any bounded flexible polyhedron in $\Lambda^n,$ where $n$ is odd and $n\ge 3,$ is constant during the flexion.

Figures (2)

  • Figure 1: Steffen's flexible polyhedron and its unfolding
  • Figure 2: The loop $\gamma$ in Lemma \ref{['slem_nu_chi']}

Theorems & Definitions (84)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 74 more