Fredholm theory and transversality for the parametrized and for the $S^1$-invariant symplectic action

TL;DR

This paper proves Fredholm properties and transversality for parametrized and $S^1$-invariant Hamiltonian action functionals, foundational for defining $S^1$-equivariant Floer homology, and generalizes unique continuation theorems.

Contribution

It establishes Fredholm and transversality results for parametrized and $S^1$-invariant Hamiltonian families, enabling the development of $S^1$-equivariant Floer homology.

Findings

Fredholm property for linearized operators of gradient lines

Transversality for generic $S^1$-invariant Hamiltonian families

Generalization of Aronszajn's unique continuation theorem

Abstract

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^1$-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^1$-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn's unique continuation theorem to a class of elliptic integro-differential inequalities of order two.