2002/07/24 by Yong-Geun Oh, Yong‐Geun Oh, Oh, Yong-Geun
Mathematics · #53D35 #53D40 #53D45 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53D35 #msc:53D40 #msc:53D45
paper · pdf · doi:10.48550/arxiv.math/0207214
a sequel to the paper math.SG/0206092, the title slightly changed, the abstract and the introduction partly rewritten, and some details added and improved
openalex publication_date 2002/07/24 · arxiv created 2003/11/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.