zbMATH — the first resource for mathematics

Constant energy surfaces of Hamiltonian systems, enumeration of three- dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds. (English. Russian original) Zbl 0671.58008
Russ. Math. Surv. 43, No. 1, 3-24 (1988); translation from Usp. Mat. Nauk 43, No. 1, 5-22 (1988).
The authors calculate volumes of closed, orientable three-dimensional manifolds having the hyperbolic structures. The method used in the paper is based on Fomenko’s topological theory of isoenergetic surfaces of Hamiltonian systems and Matveev’s theory of complexity of three- dimensional manifolds.
Let H be a class of three-dimensional isoenergetic surfaces. The authors prove that all three-dimensional orientable of the order of complexity not greater than 8 belong to H and they are not hyperbolic manifolds. They give 11 manifolds having hyperbolic structures not belonging to H. In the last pages they give the result of computer calculations, and give some hypotheses about minimal volumes of some manifolds, about relation between the complexity of manifold and its volume.
Reviewer: Nguyen Huu Duc

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI