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Humilière, Vincent; Leclercq, Rémi; Seyfaddini, Sobhan

Coisotropic rigidity and \(C^0\)-symplectic geometry. (English) Zbl 1327.53109

Duke Math. J. 164, No. 4, 767-799 (2015).
The article’s main result proves that a symplectic homeomorphism preserves coisotropic submanifolds and their characteristic foliations. To prove it the authors establish a \(C^{0}\)-dynamical property of coisotropic submanifolds by proving the uniqueness of generators for continuous analogues of Hamiltonian flows; by doing so they generalize a fundamental theorem in \(C^{0}\)-Hamiltonian dynamics. The main results generalizes the Gromov-Eliashberg theorem and demonstrate previously known rigidity results (i) on Lagrangians, by F. Laudenbach and J. C. Sikorav [Int. Math. Res. Not. 1994, No. 4, 161–168 (1994; Zbl 0812.53031)] and (ii) on characteristics of hypersurfaces by E. Opshtein [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 5, 857–864 (2009; Zbl 1186.53054)]. Indeed, it is shown that these results are manifestations of a single rigidity phenomenon.
They define symplectic homeomorphisms as \(C^{0}\)-limit of symplectomorphisms. The definition is well posed thanks to the Gromov-Eliashberg theorem [M. Gromov, Partial differential relations. Berlin etc.: Springer-Verlag (1986; Zbl 0651.53001)] stating that a diffemorphism which is a \(C^{0}\)-limit of symplectomorphisms is symplectic. A submanifold \(C\) of a symplectic manifold \((M,\omega)\) is called coisotropic if for all \(p\in C\), \((T_{p}C)^{\omega}\subset T_{p}C\), where \((T_{p}C)^{\omega}\) denotes the symplectic orthogonal of \(T_{p}C\). For instance, hypersurfaces and Lagrangians are coisotropic. A coisotropic submanifold carries a natural foliation \(\mathcal{F}\) which integrates the distribution \((T_{p}C)^{\omega}\); \(\mathcal{F}\) is called the characteristic foliation of \(C\).
{Main Theorem}. Let \(C\) be a smooth coisotropic submanifold of a symplectic manifold \((M,\omega)\). Let \(U\) be an open subset of \(M\), and let \(\theta:U\rightarrow V\) be a symplectic homeomorphism. If \(\theta(C\cap U)\) is smooth, then it is isotropic. Furthermore, \(\theta\) maps the characteristic foliation of \(C\cap U\) to that of \(\theta(C\cap U)\).
Reviewer: Celso M. Doria (Florianapolis)

Cited in 5 Reviews
Cited in 24 Documents

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)

Keywords:

symplectic manifolds; coisotropic submanifolds; characteristic foliation; \(C^{0}\)-symplectic topology; spectral invariants

Citations:

Zbl 0812.53031; Zbl 1186.53054; Zbl 0651.53001
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Full Text: DOI arXiv Euclid

References:

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