# analytic disc

###### Definition.

Let $$ be the
open unit disc^{}. A non-constant holomorphic mapping $\phi :D\to {\u2102}^{n}$ is called an analytic disc in ${\u2102}^{n}$. The
really refers to both the embedding and the image.
If the mapping $\phi $ extends continuously to the closed unit disc
$\overline{D}$, then $\phi (\overline{D})$ is called a closed analytic disc
and $\phi (\partial D)$ is called the boundary of a closed analytic
disc.

Analytic discs play in some sense a role of line segments^{} in ${\u2102}^{n}$.
For example they give another way to see that a domain
$G\subset {\u2102}^{n}$ is pseudoconvex. See the Hartogs Kontinuitatssatz
theorem.

Another use of analytic discs are as a technique for extending CR functions on generic manifolds [1]. The idea here is that you can always extend a function from the boundary of a disc to the inside of the disc by solving the Dirichlet problem.

###### Definition.

A closed analytic disc $\phi $ is said to be attached to a set $M\subset {\u2102}^{n}$ if $\phi (\partial D)\subset M$, that is if $\phi $ maps the boundary of the unit disc to $M$.

Analytic discs are also used for defining the Kobayashi metric and thus plays a role in the study of invariant metrics.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | analytic disc |
---|---|

Canonical name | AnalyticDisc |

Date of creation | 2013-03-22 14:30:49 |

Last modified on | 2013-03-22 14:30:49 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32T05 |

Defines | closed analytic disc |

Defines | boundary of a closed analytic disc |

Defines | attached analytic disc |