# ANNALES DE L'INSTITUT FOURIER

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

Let $M$ be a two dimensional totally real submanifold of class ${C}^{2}$ in ${\mathbf{C}}^{2}$. A continuous map $F:\overline{\Delta }\to {\mathbf{C}}^{2}$ of the closed unit disk $\overline{\Delta }\subset \mathbf{C}$ into ${\mathbf{C}}^{2}$ that is holomorphic on the open disk $\Delta$ and maps its boundary $b\Delta$ 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 ${F}^{0}\left(b\Delta \right)$ 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 ${\mathbf{C}}^{3}$ with a bizzare polynomially convex hull.

Soit $M$ une sous-variété totalement réelle de dimension 2 et classe ${C}^{2}$ dans ${\mathbf{C}}^{2}$. Une application continue $F:\overline{\Delta }\to {\mathbf{C}}^{2}$ de disque-unité fermé $\overline{\Delta }\subset \mathbf{C}$ dans ${\mathbf{C}}^{2}$, qui est holomorphe sur $\Delta$ et applique sa frontière $b\Delta$ 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 ${F}^{0}\left(b\Delta \right)$ 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 ${\mathbf{C}}^{3}$ avec une étrange enveloppe convexe polynomiale.

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title = {Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1076/

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