An existence theorem, with energy bounds, of Floer's perturbed Cauchy-Riemann equation with jumping discontinuity

TL;DR

This paper proves an existence theorem with energy bounds for specific pseudo-holomorphic sections solving Floer's perturbed Cauchy-Riemann equations with discontinuous Hamiltonian perturbations, crucial for symplectic invariants.

Contribution

It establishes a more general existence result for solutions with discontinuous Hamiltonian terms, advancing Floer theory and symplectic topology.

Findings

Proves existence of finite energy solutions with discontinuous perturbations.

Derives an energy identity for pseudo-holomorphic sections.

Uses adiabatic degeneration to analyze solution structure.

Abstract

This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the author has constructed on general symplectic manifolds [Oh4,5]. The existence theorem is also the crux of the author's recent proof of an optimal energy-capacity inequality given in [Oh5]. In this paper, we prove a more general existence result than needed in that we study Floer's perturbed Cauchy-Riemann equations with discontinous Hamiltonian perturbation terms and prove an existence theorem of certain piecewise smooth finite energy solutions of the equation. The proof relies on a careful study of the product structure in the chain level Floer homology theory and a singular degeneration (``adiabatic degeneration'') of Floer's perturbed Cauchy-Riemann equation. In the course of the proof, we also derive certain general energy identity of pseudo-holomorphic sections of the Hamiltonian fibration.