analytic disc


Let D:={z|z|<1} be the open unit discPlanetmathPlanetmath. A non-constant holomorphic mapping φ:Dn is called an analytic disc in n. The really refers to both the embedding and the image. If the mapping φ extends continuously to the closed unit disc D¯, then φ(D¯) is called a closed analytic disc and φ(D) is called the boundary of a closed analytic disc.

Analytic discs play in some sense a role of line segmentsMathworldPlanetmath in n. For example they give another way to see that a domain Gn 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.


A closed analytic disc φ is said to be attached to a set Mn if φ(D)M, that is if φ 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.


  • 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