Localization of Floer homology of engulfed topological Hamiltonian loop (Q2253766)

In [Commun. Math. Phys. 120, No.4, 575--611 (1989; Zbl 0755.58022)], \textit{A. Floer} introduced the localization of Floer homology in the context of Hamiltonian Floer homology. In this paper, for a given Hamiltonian \(H\) which is \(C^1\) small, the author creates the so-called thick-thin dichotomy for Floer trajectories bounding a Lagrangian \(L\) and its Hamiltonian perturbation \(\phi_H^1 (L)\). This singles out thin trajectories that are by definition contained in a small tubular neighborhood \(U\) of the Lagrangian in a Darboux-Weinstein neighborhood. Then, the local Floer homology \(HF_\ast (H,L;J;U)\) can be defined by counting thin trajectories. Let \(L\) be a compact Lagrangian submanifold of a symplectic manifold \((M,\omega)\) and let \(V\subset \overline V\subset U\) be a pair of Darboux neighborhoods of \(L\) with \(\omega=-d\Theta\) on \(U\), where \(\Theta\) is the Liouville \(1\)-form on \(U\) regarded as an open neighborhood of the zero section of \(T^*L\). If \(J_g\) is is the canonical Sasakian almost complex structure on \(V\) as a subset \(T^*L\) which is induced by a Riemannian metric \(g\) on \(L\), then the set of such almost complex structures is \({\mathcal J}_\omega(V,J_g)\). The author proves that if \(L\subset (M,\omega)\) is a compact Lagrangian submanifold, \(V\subset \overline V\subset U\) is a pair of Darboux neighborhoods of \(L\), and \(\phi_H\) is a \(V\)-engulfed Hamiltonian path, then whenever \(\bar d(\phi^1_H,\text{id})\leq \delta\) for any \(\delta<d(V,\Theta)\), any solution \(v\) of the perturbed Cauchy-Riemannian equation \(\frac{\partial v}{\partial \tau}+J_0\frac{\partial v}{\partial t}=0\) with \(v(\tau,0)\in\phi^1_H(L)\), \(v(\tau,1)\in L\) satisfies either (i)\,\(\text{Image}\,v\subset D_\delta(L)\subset V\), where \(D_\delta(L)\) is the \(\delta\)-neighborhood of \(L\), or (ii)\,\(\text{Image}\,v\not\subset V\). Also, it is proven that if \(L\subset M\) and \(U\) is a Darboux neighborhood of \(L\) and \({\mathcal H}:s\mapsto{\mathcal H}(s)\) is a family of \(U\)-engulfed Hamiltonians with \({\mathcal H}(0)=0\), then if \(\max_{s\in[0,1]}\bar d(\phi^1_H,\text{id})<\delta\) and \(|J_t-J_0|_{C^1}<\delta\) for some time independent \(J_0\) and if \(J\) is \((L,\phi^1_H(L))\)-regular, then \(HF_*(H,L;J;U)\cong H_*(L,\mathbb Z)\).