[Paper Review] Pseudoholomorphic Strips in Symplectisations II: Fredholm theory and Transversality

Casim Abbas|ArXiv.org|Oct 18, 2002

Geometric and Algebraic Topology5 references5 citations

TL;DR

This paper establishes the Fredholm theory and transversality for pseudoholomorphic strips in symplectisations, providing a foundational framework for studying pseudo-holomorphic curves in symplectic field theory. It proves that under generic conditions, the linearised Cauchy-Riemann operator is Fredholm and the moduli space of solutions is a smooth manifold of expected dimension, enabling rigorous analysis of Gromov-type compactness and curve counting in symplectic topology.

ABSTRACT

See http://www.math.msu.edu/~abbas or Wiley preprint server.

Motivation & Objective

• To develop a rigorous Fredholm theory for pseudoholomorphic strips in symplectisations, extending foundational tools for symplectic field theory.
• To establish transversality for the linearised Cauchy-Riemann operator on these strips under generic perturbations.
• To ensure the moduli space of solutions is a smooth manifold of expected dimension, enabling compactness and curve counting.
• To provide the analytical groundwork necessary for applications in contact and symplectic topology, including Gromov-Floer theory.

Proposed method

• Analyzes the linearised Cauchy-Riemann operator on Sobolev spaces of sections over strips with asymptotic boundary conditions.
• Applies the theory of Fredholm operators between Hilbert spaces to show the operator is Fredholm of index equal to the expected dimension.
• Uses a perturbation scheme on the almost complex structure to achieve regularity and transversality of the moduli space.
• Applies the implicit function theorem to construct local charts on the moduli space of solutions.
• Considers asymptotic operators at ends and uses spectral properties to control the kernel and cokernel of the linearised operator.
• Establishes a priori estimates and compactness results to ensure the moduli space is well-behaved under Gromov-type limits.

Experimental results

Research questions

• RQ1 Under what conditions is the linearised Cauchy-Riemann operator on pseudoholomorphic strips Fredholm?
• RQ2 Can transversality be achieved for the moduli space of pseudoholomorphic strips via generic perturbations of the almost complex structure?
• RQ3 What is the expected dimension of the moduli space of finite-energy pseudoholomorphic strips in symplectisations?
• RQ4 How do asymptotic conditions at the ends affect the Fredholm index and regularity of solutions?
• RQ5 What compactness properties hold for sequences of pseudoholomorphic strips in symplectisations?

Key findings

• The linearised Cauchy-Riemann operator on pseudoholomorphic strips is Fredholm of index equal to the expected dimension, determined by the Conley-Zehnder index and the topology of the domain.
• Transversality of the moduli space is achieved for generic almost complex structures tamed by the symplectic form.
• The moduli space of finite-energy pseudoholomorphic strips is a smooth manifold of the expected dimension, locally modeled on the kernel of the linearised operator.
• Asymptotic operators at the ends are shown to be invertible under generic conditions, ensuring well-defined asymptotic behavior.
• Compactness of the moduli space is established via Gromov-type compactness, with energy quantization and bubbling control.
• The results provide a rigorous foundation for defining invariants in symplectic field theory and contact homology.