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A geometric characterization of the symmetrized bidisc

Lookup NU author(s): Professor Jim Agler, Dr Zinaida Lykova, Emeritus Professor Nicholas Young

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Abstract

The symmetrized bidisc \[G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}\]has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer.We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.

Publication metadata

Author(s): Agler J, Lykova ZA, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Mathematical Analysis and Applications

Year: 2019

Volume: 473

Issue: 2

Pages: 1377-1413

Print publication date: 15/05/2019

Online publication date: 16/01/2019

Acceptance date: 09/01/2019

Date deposited: 10/01/2019

ISSN (print): 0022-247X

Publisher: Elsevier

URL: https://doi.org/10.1016/j.jmaa.2019.01.027

DOI: 10.1016/j.jmaa.2019.01.027

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