analytic disc

Definition.

Let $D:=\\{z\\in{\\mathbb{C}}\\mid\\lvert z\\rvert<1\\}$ be theopen 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 discand $\\varphi(\\partial D)$ is called the boundary of a closed analyticdisc.

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 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 $\\varphi$ is said to be attachedto 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.

单词	AnalyticDisc
释义	analytic disc Definition. Let $D:=\\{z\\in{\\mathbb{C}}\\mid\\lvert z\\rvert<1\\}$ be theopen 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 discand $\\varphi(\\partial D)$ is called the boundary of a closed analyticdisc. 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 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 $\\varphi$ is said to be attachedto 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.
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