Poly cylinder

In multi-dimensional function theory , the poly cylinder or poly circle is the Cartesian product of circular disks .

If one denotes more precisely with an open circular disk in the complex plane , then the poly cylinder around the point with the multi-radius is given as ${\displaystyle \Delta (z,r)=\{w\in \mathbb {C} \mid |z-w|<r\}}$ ${\displaystyle z=(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}}$ ${\displaystyle r=(r_{1},\dots ,r_{n})}$

${\displaystyle \Delta (z_{1},\ldots ,z_{n};r_{1},\ldots ,r_{n}):=\Delta (z_{1},r_{1})\times \dots \times \Delta (z_{n},r_{n})}$

or equivalent as

${\displaystyle \{w=(w_{1},\dots ,w_{n})\in \mathbb {C} ^{n}\mid |z_{k}-w_{k}|<r_{k},\,k=1,\dots ,n\}.}$

The closed poly cylinder is defined by replacing the <symbol with : ${\displaystyle \leq }$

${\displaystyle {\overline {\Delta }}(z_{1},\ldots ,z_{n};r_{1},\ldots ,r_{n}):=\{w=(w_{1},\dots ,w_{n})\in \mathbb {C} ^{n}\mid |z_{k}-w_{k}|\leq r_{k},\,k=1,\dots ,n\}.}$

The poly-cylinder, like the Euclidean sphere, is a generalization of the one-dimensional circular disk . For these two sets are not biholomorphically equivalent. This statement was proven by Poincaré in 1907 by showing that the automorphism groups of the two sets as Lie groups have different dimensions. ${\displaystyle \{w\in \mathbb {C} ^{n}\mid \sum _{j=1}^{n}|w_{j}-z_{j}|^{2}<r^{2}\}}$ ${\displaystyle n>1}$

literature

• Steven G Krantz: Function Theory of Several Complex Variables , American Mathematical Society, 2002, ISBN 0-8218-2724-3
• Walter Rudin: Function theory in polydiscs , Benjamin, New York 1969

Individual evidence

1. ^ Wolfgang Ebeling: Theory of functions, differential topology and singularities , Vieweg-Verlag 2001, ISBN 978-3-528-03174-9 , page 43
2. ^ Joseph Wloka: Basic Spaces and Generalized Functions , Springer Lecture Notes in Mathematics 82 (1969), page 3