Levi-flat filling of real two-spheres in symplectic manifolds (I)

Gaussier, Hervé; Sukhov, Alexandre

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 515-539.

Soit $(M,J,\omega )$ une variété dont la structure presque complexe $J$ est contrôlée par la forme symplectique $\omega$. On suppose $M$ de dimension complexe deux, Levi-convexe et à géométrie bornée. On démontre que toute 2-sphère possédant deux points elliptiques et plongée dans le bord de $M$ est feuilletée par des bords de disques pseudoholomorphes.

Let $(M,J,\omega )$ be a manifold with an almost complex structure $J$ tamed by a symplectic form $\omega$. We suppose that $M$ has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of $M$ can be foliated by the boundaries of pseudoholomorphic discs.

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