analytic disc

Definition.

Let $D:=\{z\in{\mathbb{C}}\mid\lvert z\rvert<1\}$ be the open unit disc. A non-constant holomorphic mapping $\varphi\colon D\to{\mathbb{C}}^{n}$ is called an analytic disc in ${\mathbb{C}}^{n}$ . The really refers to both the embedding and the image. If the mapping $\varphi$ extends continuously to the closed unit disc $\bar{D}$ , then $\varphi(\bar{D})$ is called a closed analytic disc and $\varphi(\partial D)$ is called the boundary of a closed analytic disc.

Analytic discs play in some sense a role of line segments in ${\mathbb{C}}^{n}$ . For example they give another way to see that a domain $G\subset{\mathbb{C}}^{n}$ 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 $\varphi$ is said to be attached to a set $M\subset{\mathbb{C}}^{n}$ if $\varphi(\partial D)\subset M$ , that is if $\varphi$ 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