By Gromov M.
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Extra resources for Pseudo holomorphic curves in symplectic manifolds
Sere. Sud-Rhodanien IV, Balarue, in Travaux en cours. , Zehnder, E. : The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold. Invent. math. , Desolneux-Moulis, N. ): Srminaire Sud-Rhodanien de Grom&rie. : Twistorial constructions of harmonic maps of surfaces into fourmanifolds. Warwick, 1984 (Preprint) Eliashberg, Ya. : Estimates on the number of fixed points of area preserving transformations (in Russian) Syktyvkar, 1978 (Preprint) Eliashberg, Ya. : Proof of the Arnold conjecture for surfaces and generalizations for certain K~ihler manifolds.
Then there exists an almost complex structure J on V tamed by co for which this sphere is J-holomorphic. Now, the J-simplidty of [S 2] can be insured by some topological condition. For instance, let the class IS 2] ~ H2(V; •) generate the image of the Hurewicz homomorphism. Then V is foliated by symplectic "translates" of S 2. B3. The above discussion extends up to certain degree to closed symplectic manifolds V of dimension 2n>6. For example, let (V, co) admit a symplectic embedding of the space ( ~ P " - 1, 09o) for 2n = dim V, such that the normal bundle of C P " - 1 C V is trivial.
Thus the boundary regularity reduces to the interior regularity of the extended disk. Finally, one achieves (*) for every E by applying a small perturbation to E] U, and thus one obtains (with some extra work) embeddings of D 2 x 0 D 2 and of t~O 2 x O 2 in V which send the disks 0 2 X S and s x D E, for all s e t~D2, to E-disks in V. Now, these embeddings extend to the required map F : 0 2 • D E ~ V as follows. Start with an extension F1 : D 2 x D 2--~ V which satisfies (i) (but may violate (ii) outside the boundary 0(0 2 • 2) =D2xt')DEuOD2xD 2) and consider the pull-back EI=F*(E)CGr2(DExD2).