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\(\mathcal C^0\)-rigidity of characteristics in symplectic geometry. (English) Zbl 1186.53054
The main result (Theorem 1) says that a symplectic homeomorphism in the sense of Eliashberg and Gromov [e. g., Y. M. Eliashberg, Funct. Anal. Appl. 21, 227-232 (1987); translation from Funkts. Anal. Prilozh. 21, No. 3, 65–72 (1987; Zbl 0655.58015)] which preserves a smooth hypersurface also preserves its characteristic foliation. Let \(B\) be a symplectic embedding of a Euclidean ball centered at origin into a symplectic manifold \(M\) which sends \(\mathbb R^{2n-1}:= \mathbb R\times\mathbb C^{n-1}\subset \mathbb C^n\) to a hyperspace \(S\subset M\). \(B\) is divided into two components \(S_+\) and \(S_-\). For a real number \(\varepsilon>0\) and \(x, y\in S\cap B\), the author uses a notion of continuous path of symplectomorphisms \(\Phi_t(t\in[0,1])\) with common supports in \(B\) for which there exists two open sets \(U_{\varepsilon}(x)\) and \(U_{\varepsilon}(y)\) contained in the \(\varepsilon\)-balls around \(x\) and \(y\) respectively, such that:
1. \(\Phi_0=Id\),
2. \(\Phi_1(z)\in S_+\) for all \(t\in ]0,1]\) and \(z\in S\cap U_{\varepsilon}(x)\),
3. \(\Phi_{1}(z)\in S_{-}\) { for all} \(t\in ]0,1]\) and \(z\in S\cap U_{\varepsilon}(y)\),
4. \(\Phi_{t}(y)\in S\) { for all} \( t\in [0,1]\) and \(z\in S\backslash(U_{\varepsilon}(x)\cap U_{\varepsilon}(y))\).
By using Gromov’s non-squeezing theorem it is shown (Proposition 1.3) that the path \(\Phi_t\) exists if and only if \(x\) and \(y\) are on the same characteristic leaf. The author calls the path \(\Phi_t\) an \(\varepsilon\)-symplectic hammer. By consecutive applications of \(\varepsilon\)-symplectic hammers, the main theorem is proved. The author adds a one-sided version (Theorem 1.2) on a smoothly bounded open set \(U\subset \mathbb R^4\) symplectomorphic to \(B^4(1)\) and its boundary \(\partial U\).

53C24 Rigidity results
53D05 Symplectic manifolds (general theory)
57R17 Symplectic and contact topology in high or arbitrary dimension
53C12 Foliations (differential geometric aspects)
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