Properties of Pseudoholomorphic Curves in Symplectizations Iii: Fredholm Theory
نویسنده
چکیده
We shall study smooth maps ~ u : _ S ! RM of nite energy deened on the punctured Riemann surface _ S = S n? and satisfying a Cauchy-Riemann type equation T ~ u j = J(~ u) T ~ u for special almost complex structures J, related to contact forms on the compact three manifold M. Neither the domain nor the target space are compact. This diiculty leads to an asymptotic analysis near the punctures. A Fredholm theory determines the dimension of the solution space in terms of the asymptotic data deened by non-degenerate periodic solutions of the Reeb vector eld associated with on M, the Euler characteristic of S, and the number of punctures. Furthermore, some transversality results are established. Contents 1. Introduction 2 2. Linear Fredholm theory 17 3. The Cauchy-Riemann operator for unparametrized curves 27 4. The asymptotic model problem and special coordinates 31 5. The implicit function theorem for the local model 43 6. Fredholm theory with exponential weights 52 7.
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