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

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

2009-09-14

arxiv.org/abs/0909.2588v2

Mathematics

Symplectic Geometry

63 pages

Scientific paper

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.

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