analytic disc

Definition.

Let be theopen unit disc. A non-constant holomorphic mapping $\phi :D\to {ℂ}^n$ is called an analytic disc in ${ℂ}^n$. 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 discand $\phi (\partial D)$ is called the boundary of a closed analyticdisc.

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

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 attachedto a set $M\subset {ℂ}^n$ 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.