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

Forstneric, Franc

Annales de l'Institut Fourier, Tome 37 (1987), p. 1-44 / Harvested from Numdam

Résumé

Soit $M$ une sous-variété totalement réelle de dimension 2 et classe $C^2$ dans ${\bf C}^2$. Une application continue $F:\bar{\Delta }\rightarrow {\bf C}^2$ de disque-unité fermé $\bar{\Delta }\subset {\bf C}$ dans ${\bf C}^2$, 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 $F^0(b\Delta )$ 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 ${\bf C}^3$ avec une étrange enveloppe convexe polynomiale.

Abstract

Let $M$ be a two dimensional totally real submanifold of class $C^2$ in ${\bf C}^2$. A continuous map $F:\bar{\Delta }\rightarrow {\bf C}^ 2$ of the closed unit disk $\bar{\Delta }\subset {\bf C}$ into ${\bf C}^2$ 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 $F^0(b\Delta )$ 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 ${\bf C}^3$ with a bizzare polynomially convex hull.

Détail

Publié le : 1987-01-01

MR 88j:32019 | Zbl 0583.32038 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1076

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