The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866910451185483776 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1504_02977 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces Gaifullin, Alexander A. Metric Geometry Algebraic Geometry Complex Variables 52C25, 51M10, 51M25, 34M35 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. |
| title | The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces |
| topic | Metric Geometry Algebraic Geometry Complex Variables 52C25, 51M10, 51M25, 34M35 |
| url | https://arxiv.org/abs/1504.02977 |