Analytic disks with boundaries in a maximal real submanifold of ${𝐂}^2$

Forstneric, Franc

Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 1-44.

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.

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.

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