Symplectic rigidity for Anosov hypersurfaces.

*(English)*Zbl 1109.37025First, the authors show how to derive from the results by C. Croke [Comment. Math. Helv. 65, 150–169 (1990; Zbl 0704.53035)] and J. P. Otal [Ann. Math. (2) 131, 151–162 (1990; Zbl 0699.58018)] the following statement: if the interiors of the unit cotangent bundles of two compact Riemann surfaces of strictly negative curvature are exact symplectomorphic then the underlying Riemann surfaces are isometric. Then, they generalize this theorem for some deformations of such open domains.

Let us denote by \(W^0\) the open set (in the topology of smooth convergence) which is formed by open domains in the cotangent bundle such that every domain \(U\) from \(W^0\) contains the zero-section and is star-shaped with respect to this section, the canonical Liouville form \(\lambda\) restricts to a contact form on the boundary of \(U\) and its Reeb field generates an Anosov flow. Moreover it is assumed that \(W^0\) is connected and contains the domains bounded by constant energy surfaces on which the Reeb fields generate the geodesic flows of negatively curved metrics. The main theorem reads that any exact symplectomorphism between two domains from \(W^0\) is homotopic to one which is smooth up to the boundary. Although the first statement concerning the interiors of the unit cotangent bundles is generalized to higher dimensions the main theorem is proved only for surfaces and the proof is based on the properties of Anosov-Reeb flows in three dimensions.

Let us denote by \(W^0\) the open set (in the topology of smooth convergence) which is formed by open domains in the cotangent bundle such that every domain \(U\) from \(W^0\) contains the zero-section and is star-shaped with respect to this section, the canonical Liouville form \(\lambda\) restricts to a contact form on the boundary of \(U\) and its Reeb field generates an Anosov flow. Moreover it is assumed that \(W^0\) is connected and contains the domains bounded by constant energy surfaces on which the Reeb fields generate the geodesic flows of negatively curved metrics. The main theorem reads that any exact symplectomorphism between two domains from \(W^0\) is homotopic to one which is smooth up to the boundary. Although the first statement concerning the interiors of the unit cotangent bundles is generalized to higher dimensions the main theorem is proved only for surfaces and the proof is based on the properties of Anosov-Reeb flows in three dimensions.

Reviewer: Iskander A. Taimanov (Novosibirsk)