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Poisson brackets and symplectic invariants. (English) Zbl 1242.53099
Let \((M^{2n},\omega)\) be a symplectic manifold, \({\mathcal F}\) the function space \(C^\infty_c(M)\times C^\infty_c(M)\), \((X,Y,Z)\) (resp. \((X_0, X_1, Y_0, Y_1)\)) a triple (resp. quadruple) of compact subsets of \(M\), and \[ {\mathcal F}_3(X,Y,Z)= \{(F,G)|F|_X\leq 0,\,G|_Y\leq 0,\,(F+ G)|_Z\geq 1\}, \]
\[ {\mathcal F}_4(X_0, X_1, Y_0, Y_1)= \{(F,G)|F|_{X_0}\leq 0,\,F|_{X_1}\geq 1,\,G|_{Y_0}\leq 0,\,G|_{Y_1}\geq 1\}, \] where \(F\), \(G\) are compactly supported smooth function on \(M\), be pairs of functions from \({\mathcal F}\). Then the authors introduce the following Poisson bracket invariants \[ pb_3(X,Y,Z)= \text{inf}_{(F,G)\in{\mathcal F}_3(X,Y,Z)}\|\{F, G\}\|, \]
\[ pb_4(X_0, X_1, Y_0, Y_1)= \text{inf}_{(F,G)\in{\mathcal F}_4(X_0, X_1, Y_0, Y_1)}\|\{F, G\}\|. \] Here \(\{F, G\}=\omega(\text{sgrad\,}G,\text{sgrad\,}F)\), \(i_{\text{sgrad\,}F\omega}= -dF\) is the Poisson bracket of two Hamiltonians \(F\), \(G\) and \(\| F\|= \max_{x\in M}|F(x)|\). One has \({\mathcal F}_3\neq\emptyset\) if \(X\cap Y\cap Z= \emptyset\) (cf. §1, Fig.1); if \({\mathcal F}_3=\emptyset\), the authors put \(pb_3=\infty\).
In this paper, applications of these invariants to symplectic approximation (§1.3 and §3), Hamiltonian chord (§1.4 and §4), Lagrangian Floer theory (§1.7 and §5) and related topics together with examples are presented. Some of them are as follows:
Let \({\mathcal K}_s\) be \(\{(H, K)\in{\mathcal F}: \|\{H,K\}\|\leq s\}\). Then the profile function \(\rho_{F,G}:[0,\infty)\to\mathbb{R}\) is defined by \[ \rho_{F,G}(s)= d((F,G),{\mathcal K}_s), \text{where}\;d((F,G),(H,K))=\| F-H\|+\| G-K\| \] (M. Entov, L. Polterovich and D. Rosen, Discrete Contin. Dyn. Syst. 28, No. 4, 1455–1468 (2010; Zbl 1200.53068)). The value \(\rho_{F,G}(0)\) is responsible for the optimal uniform approximation of \((F, G)\) by a pair of Poisson-commuting functions. Let \({\mathcal F}^\flat_k\) be the subclasses of \({\mathcal F}_k\) consisting of all pairs \((F,G)\) such that at least one of the functions \(F\), \(G\) has its range in \([0,1]\) (\(k=3\) or \(4\)). Let \(p\) be \(pb_k\), \(k= 3,4\). Then if \(p=0\), for every \(s>0\), there exists \((F,G)\in{\mathcal F}^\flat_k\) with \(\rho_{F,G}(s)= 0\). If \(p> 0\), then for every \((F,G)\in{\mathcal F}^\flat_k\), \(\rho_{F,G}(s)\) is continuous, \(\rho_{F,G}(0)={1\over 2}\) and \[ {1\over 2}- {s\over 2\|\{F, G\}\|}\geq \rho_{F,G}(s),\quad s\in [0,\|\{F, G\}\|], \]
\[ \rho_{F,G}(s)\geq {1\over 2}- {\sqrt{s}\over 2\sqrt{p}},\quad (F,G)\in{\mathcal F}^\flat_3,\;s> 0, \]
\[ \rho_{F,G}(s)\geq {1\over 2}-{s\over 2p},\quad (F,G)\in{\mathcal F}^\flat_4,\;s> 0 \] (Th.1.4. proved in §3.2).
Let \(g_t\) be a Hamiltonian flow of a Hamiltonian \(G\) such that \(g_Tx\in X_1\) for some \(x\in X_0\), then the curve \(\{g_tx\}_{t\in[0,T]}\) is referred to as Hamiltonian chord of \(G\). Let \(T(X_0, X_1; G)\) be the minimal time-length of a Hamiltonian \(G\) which connects \(X_0\) and \(X_1\) and let \[ T(X_0, X_1, Y_0, Y_1)= \sup\{T(X_0, X_1; G); g\in C^\infty_c(M), G|_{Y_0}\leq 0, G|_{Y_1}\geq 1\}. \] Then \(pb_4(X_0, X_1,Y_0, Y_1)= T(X_0, X_1,Y_0, Y_1)^{-1}\) (Th.1.11. proved in §4.1).
Let \(M\) be a closed connected symplectic manifold. A functional \(\zeta: C(M)\to\mathbb{R}\) is said to be a symplectic quasi state, if \(\zeta(1)= 1\), \(\zeta(F)\geq 0\) if \(F\geq 0\) and linear on every Poisson-commutative subspace of \(C(M)\). A closed subset \(X\) of \(M\) is said to be superheavy with respect to \(\zeta\) whenever \(\zeta(F)\geq c\) for any \(F\), \(F|_X\geq c\). If there exists \(K>0\) such that \[ |\zeta(F+G)- \zeta(F)- \zeta(G)|\leq\sqrt{K\|\{F,G\}\|}, \] then \(\zeta\) is said to satisfy the PB-inequality. Assuming that \(M\) admits a symplectic quasi state \(\zeta\) which satisfies the PB-inequality and applying results from M. Entov, I. Polterovich and F. Zapolsky [Pure Appl. Math. Q. 3, No. 4, 1037–1055 (2007; Zbl 1143.53070)], it is shown that if \(X\), \(Y\), \(Z\) are a triple of superheavy closed sets with \(X\cap Y\cap Z= \emptyset\), then \[ pb_3(X, Y,Z)\geq{1\over K}. \] If \(X_0\), \(X_1\), \(Y_0\), \(Y_1\) are a quadruple of closed subsets of \(M\) such that \(X_0\cap X_1= Y_0\cap Y_1=\emptyset\) and \(X_0\cup Y_0\), \(Y_0\cup X_1\), \(X_1\cup Y_1\), \(Y_1\cup X_0\) are all superheavy, then \[ pb_4(X_0, X_1, Y_0, Y_1)\geq {1\over 4k}. \] If \(X_0\cup Y_0\), \(Y_0\cup X_1\), \(Y_1\) are all superheavy, then \[ pb_4(X_0, X_1, Y_0, Y_1)\geq {1\over K} \] (Th.1.15. proved in §1.5).
Let \({\mathcal L}= (L_0, L_1,\dots, L_{k-1})\) be a collection of Lagrangian submanifolds in \(M\) in general position, \({\mathcal T}_k\) the set of homotopy classes of \(k\)-gons in \(M\) whose sides lie in \(L_0,L_1,\dots, L_{k-1}\), \(m(\alpha)\) and \(\omega(\alpha)\) are Maslov index and symplectic area of \(\alpha\in{\mathcal T}_k\). Then \({\mathcal L}\) is said to be of finite type if \[ A(L_0,\dots, L_{k-1}; N)= \sup\{\omega(\alpha): \alpha\in{\mathcal T}_k, m(\alpha)= N\}<+\infty, \] for all \(N\in\mathbb{Z}\). Then, assuming the Lagrangian Floer homology \(HF(L_0,L_1)\), etc. and the \(\mu^k\)-operations in the Donaldson-Fukaya category are well defined (for \(k=2\) and \(k=3\)), the estimates \[ pb_3(L_0, L_1, L_2)\geq {1\over 2A(L_0, L_1, L_2; 2n)}, \]
\[ pb_4(L_0, L_1, L_2, L_3)\geq {1\over A(L_0, L_1, L_2, L_3; 3n-1)} \] hold (Th.1.27, Th.1.30. The proof of Th.1.30 is given in §5.7. Review on Lagarangian Floer homology and the Donaldson-Fukaya \(\mu^k\)-operations are also given in §5. A sketch of the proof of Th.1.27 is given in §1.7 using deformation of the symplectic form studied in §1.6).
There are alternative sets \({\mathcal F}_k'\) of \({\mathcal F}_k\) to define \(pb_k\); \[ pb_k(X_1,\dots, X_k)= \text{inf}_{(F,G)\in{\mathcal F}_k'}\|\{F, G\}\| \] (\(k=3,4\). Prop. 1.3). This is useful in later discussions and proved in §2 together with other properties of \(pb_k\). If \(M\) is a symplectic surface with finite symplectic area, then there is a simplified formula of \(pb_4\) (Th.1.20). This formula is simplified for \(M=S^2\) (Prop.1.22). This is proved by using symplectic field theory in §6. In §7, a vanishing theorem for \(pb_4\) in the case \(M=T^*S^1\) is proved. In the last section, an attempt to define \(pb_N\), \(N\geq 5\) and some other problems are presented.

MSC:
53D05 Symplectic manifolds (general theory)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
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