By Gromov M.

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D. D2. this local family of disks 342 M, Gromov admits the desired global extension, since the areas of the disks are a priori bounded by const area W and since the transversality of dD2C W to O n T ( W ) prevents the disks from degeneration to cusp-curves (see [Gr02] for more general results). D'2. Examples and corollaries. (a) The above theorem applies to the image WCS3CB4CI~ 2 of the standard sphere S 2 Q S 3 under an arbitrary contact transformation of S 3 (compare [B-G]). D 2. the field O n T ( W ) has not closed orbits, as the curves OD2 ( W cover W\{w o, wl} and they are everywhere transversal to O n T(W).

The group of symplectic diffeomorphisms contracts to the subgroup of isometries. A'~. Let us give a criterion for a closed manifold (V, to) to split as (S 2, eOo)@(S2, COo). We assume the existence of two embedded spheres $1 and $2 with trivial normal bundles in V, such that the form to does not vanish on $1 and on $2 and such that $1 and $2 have a single intersection point where they meet transversally with the intersection number + 1 for the orientations induced by colS1 and t o l S 2. I f every smoothly mapped sphere S ~ V has S to = k ~ to = k ~ to for some integer S $1 $2 k = k ( S ~ V), then the manifold (V, to) splits, (V, to) = (S 2, too)O(S 2, COo).

Insure in the present case holomorphic disks(D2, OD2)~(V, W). ). That is a limit f : (S, ~S)~( V, W) of regular J-curves f j : (S, ~S)~( V, W) is regular unless the map f is nowhere one-to-one on some connected component of S. B]. in case the structure J is complex near W and W is real analytic. Indeed, V lies in a slightly greater manifold V + 3 V, such that a small neighborhood U+CV + of W admits an anti-holomorphic involution I : U + ~ U +. Since OVis J-convex, the intersection VnI(U +n V) equals W.

### Pseudo holomorphic curves in symplectic manifolds by Gromov M.

by Kenneth

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