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

Annales de l'institut Fourier (1987)

- Volume: 37, Issue: 1, page 1-44
- ISSN: 0373-0956

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topForstneric, Franc. "Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$." Annales de l'institut Fourier 37.1 (1987): 1-44. <http://eudml.org/doc/74744>.

@article{Forstneric1987,

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.},

author = {Forstneric, Franc},

journal = {Annales de l'institut Fourier},

keywords = {real submanifold; nearby disks; small deformations; regularity theorem for immersed families of analytic disks; polynomially convex hull},

language = {eng},

number = {1},

pages = {1-44},

publisher = {Association des Annales de l'Institut Fourier},

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

url = {http://eudml.org/doc/74744},

volume = {37},

year = {1987},

}

TY - JOUR

AU - Forstneric, Franc

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

JO - Annales de l'institut Fourier

PY - 1987

PB - Association des Annales de l'Institut Fourier

VL - 37

IS - 1

SP - 1

EP - 44

AB - 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.

LA - eng

KW - real submanifold; nearby disks; small deformations; regularity theorem for immersed families of analytic disks; polynomially convex hull

UR - http://eudml.org/doc/74744

ER -

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