  We study partial collapsing degeneration of Hamiltonian-perturbed Floer
trajectories for an adiabatic $\varepsilon$-family and its reversal adiabatic
gluing, as the prototype of the partial collapsing degeneration of
$2$-dimensional (perturbed) $J$-holomorphic maps to $1$-dimensional gradient
segments. We consider the case when the Floer equations are $S^1$-invariant on
parts of their domains whose adiabatic limits have positive lengths as
$\varepsilon \to 0$, which we call thimble-flow-thimble configurations. The
main gluing theorem we prove also applies to the case with Lagrangian
boundaries such as in the problem of recovering holomorphic disks out of pearly
configurations. In particular, our gluing theorem gives rise to a new direct
proof of the chain isomorphism property between the Morse-Bott version of
Lagrangian intersection Floer complex of $L$ by Fukaya-Oh-Ohta-Ono and the
pearly complex of $L$ by Lalonde and Biran-Cornea (for monotone Lagrangian
submanifolds). It also provides another proof of the present authors' earlier
proof of the isomorphism property of the PSS map without involving the target
rescaling and the scale-dependent gluing. (This is a rewritten version of our
previous arXiv posting, arXiv:1103.3525.)

Symplectic Geometry;Differential Geometry