Analytic disks with boundaries in a maximal real submanifold of 𝐂 2

Franc Forstneric

Annales de l'institut Fourier (1987)

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

Abstract

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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 b Δ 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 Δ ) 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.

How to cite

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Forstneric, 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 -

References

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