Definition 1   Let be the open unit disc. A non-constant holomorphic mapping is called an analytic disc in . The term really refers to both the embedding and the image. If the mapping extends continuously to the closed unit disc , then is called a closed analytic disc and is called the boundary of a closed analytic disc.

Analytic discs play in some sense a role of line segments in . For example they give another way to see that a domain 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 2   A closed analytic disc is said to be attached to a set if , that is if maps the boundary of the unit disc to .

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

Bibliography

1

M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.

2

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.