Analytic disks with boundaries in a maximal real submanifold of 𝐂 2
Annales de l'Institut Fourier, Volume 37 (1987) no. 1, p. 1-44

Let M be a two dimensional totally real submanifold of class C 2 in C 2 . A continuous map F:Δ ¯→C 2 of the closed unit disk Δ ¯⊂C into C 2 that is holomorphic on the open disk Δ and maps its boundary bΔ into M is called an analytic disk with boundary in M. Given an initial immersed analytic disk F 0 with boundary in M, we describe the existence and behavior of analytic disks near F 0 with boundaries in small perturbations of M in terms of the homology class of the closed curve F 0 (bΔ) in M. We also prove a regularity theorem for immersed families of analytic disks, consider several examples, and construct a totally real three torus in C 3 with a bizzare polynomially convex hull.

Soit M une sous-variété totalement réelle de dimension 2 et classe C 2 dans C 2 . Une application continue F:Δ ¯→C 2 de disque-unité fermé Δ ¯⊂C dans C 2 , qui est holomorphe sur Δ et applique sa frontière bΔ dans M, est appelée un disque analytique avec frontière dans M. Etant donné un disque initial F 0 avec frontière dans M, on détermine l’existence des disques près de F 0 avec les frontières dans les petites perturbations de M à l’aide de la classe d’homologie de courbe F 0 (bΔ) dans M. On démontre aussi un théorème de régularité pour des familles des disques et on construit un tore totalement réel de dimension 3 dans C 3 avec une étrange enveloppe convexe polynomiale.

@article{AIF_1987__37_1_1_0,
     author = {Forstneric, Franc},
     title = {Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {37},
     number = {1},
     year = {1987},
     pages = {1-44},
     doi = {10.5802/aif.1076},
     zbl = {0583.32038},
     mrnumber = {88j:32019},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_1_1_0}
}
Forstneric, Franc. Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$. Annales de l'Institut Fourier, Volume 37 (1987) no. 1, pp. 1-44. doi : 10.5802/aif.1076. http://www.numdam.org/item/AIF_1987__37_1_1_0/

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