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Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds. (English) Zbl 1090.53074

The aim of the paper under review is to prove that a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to \(M\) if the unit normal bundle of \(M\) is homologically trivial in degree \(\dim (M)\). The squeezing argument is used for the computation of the capacity, and the Floer chain group is a crucial notion to play a role in the proof which is different from the usual approach.
Let \(U_R\) be a symplectic tubular neighborhood of \(M\) with sufficiently small radius \(R\). By Weinstein’s symplectic neighborhood theorem, \(U_R\) is completely determined by the radius \(R\), the restriction and the isomorphism class of the normal bundle of \(M\) in \(W\). Is the Hofer-Zehnder capacity of \(U_R\) equal to \(\pi R^2\)?
For the weakly exact \((M, \omega)\) and the trivial symplectic tubular neighborhoods of \(M \times B^2(R)\), the capacity is \(c_{HZ}(M \times B^2(R)) =\pi R^2\) by the Floer-Hofer-Viterbo result. McDuff and Slimowitz extend these results for slightly different capacities.
The paper under review answers the above question affirmatively, provided that the ambient manifold \((W, \Omega)\) is geometrically bounded and symplectically aspherical and the homology of the unit normal bundle of \(M\) splits in degree \(2m = \dim M\). The techniques involved in the proof of the main result consists of (1) Floer’s version of Gromov’s compactness theorem for perturbed \(J\)-holomorphic spheres; (2) the general idea about using algebraic framework of filtered Floer homology to detect nontrivial orbits, Ginzburg and Gurel’s work puts this idea into an effective method to find upper bounds of Hofer-Zehnder capacities; (3) monotone continuation map nontriviality implying nontrivial Floer homology, but this is not the case in the paper under review, the author develops the Floer chain level to overcome this difficulty which is new.
Section 2 starts to give applications of Theorem 1.3 the main result on classical Hamiltonian flows and Hofer geometry. Theorem 1.3 is reduced to Theorem 3.3 on the dynamics of test Hamiltonians on these neighborhoods in section 3, then the author further reduced Theorem 3.3 to nontriviality of monotone Floer continuation maps in section 5 and prove the main result by using Floer theory described in section 4.
J. M. Schlenk [Applications of Hofer’s geometry to Hamiltonian dynamics, to appear in Comment. Math. Helvetici] proved a powerful new generation of Hofer’s energy-capacity inequality, by using Macarini’s stabilization procedure. He uses his result together with Laudenbach’s, Polterovich’s and Sikorav’s displacement of subsets of symplectic manifolds, to prove the main result of the paper under review if the dimension of \(M\) is less than equal to the codimension of \(M\), giving much broader applications to Hamiltonian dynamics.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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References:

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