FLOER TRAJECTORIES WITH IMMERSED NODES AND SCALE-DEPENDENT GLUING

Abstract

Development of pseudo-holomorphic curves and Floer homology in symplectic topology has led to moduli spaces of pseudo-holomorphic curves consisting of both "smooth elements" and "spiked elements", where the latter are combinations of J-holomorphic curves (or Floer trajectories) and gradient flow line segments. In many cases the "spiked elements" naturally arise under adiabatic degeneration of "smooth elements" which gradually go through thick-thin decomposition. The reversed process, the recovering problem of the "smooth elements" from "spiked elements" is recently of much interest. In this paper, we define an enhanced compactification of the moduli space of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the one-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of nonzero length will be treated elsewhere. An immediate application of our result is a complete proof of the isomorphism property of the PSS map: a proof of this isomorphism property was outlined by Piunikhin-Salamon-Schwarz [PSS] in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in [FOOO2] in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves 'adiabatic' parameters or rescaling of the targets.