A Hamiltonian characterization of the three-ball
Let \(M\) be an oriented compact 3-dimensional manifold with the boundary \(\partial M\) diffeomorphic to \(S^ 2\). If \(\lambda\) is a contact form on \(M\), then \(\lambda\) is standard near \(\partial M\) if there exists a diffeomorphism \(\Psi : U \to V\) where \(U\) is an open neighborhood of \(\partial M\) in \(M\) and \(V\) is an open neighborhood of \(S^ 2 = \partial \overline{B}^ 3\), such that \(\lambda_{| U} = \Psi^*((dz + xdy)_{| V})\). Theorem. Let \(M\) be as above. Then the following holds. 1. For every \(\lambda\) on \(M\), which is standard near \(\partial M\), we have \(\inf(\lambda) \in (0,\pi] \cup \{+\infty\}\); 2. \(M\) is diffeomorphic to \(\overline{B}^ 3\) if and only if \(\sup(M) = +\infty\) where \(\sup(M) = \sup\{\sup(\xi)\mid \xi (=\text{ the Reeb vector field of }\lambda\)) is standard near \(\partial M\)\}.
scientific article; zbMATH DE number 597519