A contact condition for \(p\)-codimensional submanifolds of a symplectic manifold \((2\leq p\leq n)\)
Let \((M, \omega)\) be a \(2n\)-dimensional symplectic manifold. A smooth submanifold \(S_p\) of \(M\) of codimension \(p\) \((1\leq p\leq n)\) is said to be coisotropic if \(\text{Ker} \omega |_{S_p}\) is \(p\)-dimensional everywhere on \(S_p\). The author defines a \(p\)-contact condition for any coisotropic submanifold \(S_p\) of a symplectic manifold \((M,\omega)\). \(S_p\) is of \(p\)-contact type if there exist \(p\) 1-forms \(\alpha_i\) on \(S_p\), such that: (i) \(d\alpha_i = \omega |_{S_p}\) for all \(i\); (ii) at each point \(x \in S_p\), the map \(\phi(x) : \text{Ker} \omega|_{S_p} (x) \to \mathbb{R}^p\), \(X \mapsto (\alpha_1(x) X, \cdots, \alpha_p (x) X)\) is an isomorphism. The author proves that for \(M=\mathbb{R}^{2n}\), with its standard symplectic structure \(\omega= \sum^n_{i=1}dx_i\wedge dy_i\), a compact coisotropic submanifold \(S_p\) carries at least one periodic path \(\gamma\) with positive action on leaves of a foliation of \(S_p\) determined by \(\text{Ker} \omega |_{S_p}\) and \(\gamma'\in \text{Ker} \omega |_{S_p}\). This result generalizes the theorems of \textit{C. Viterbo} that solve the Weinstein conjecture for hypersurfaces (\(p =1\)) in \(\mathbb{R}^{2n}\) [Ann. Inst. H. Poincaré, Anal. Non Linéaire 4, 337-356 (1987; Zbl 0631.58013)] and on Lagrangian tori in \(\mathbb{R}^{2n}\) [Invent. Math. 100, 301-320 (1990; Zbl 0727.58015)]. The variational methods used in the proof of the main theorem allow to give a new proof of the result by \textit{H. Hofer} and \textit{C. Viterbo} [Commun. Pure Appl. Math. 45, 583-622 (1992; Zbl 0773.58021)] that the Weinstein conjecture holds in \(\mathbb{C} P^{n-1}\).
scientific article; zbMATH DE number 1142975