Fredholm theory and transversality for the parametrized and for the -invariant symplectic action
Frédéric BourgeoisUniversité Libre de Bruxelles, Belgium
Alexandru OanceaUniversité Pierre et Marie Paris, France
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.
Cite this article
Frédéric Bourgeois, Alexandru Oancea, Fredholm theory and transversality for the parametrized and for the -invariant symplectic action. J. Eur. Math. Soc. 12 (2010), no. 5, pp. 1181–1229DOI 10.4171/JEMS/227