A Geometric Characterization of the Symmetrized Bidisc

Jim Agler; Zinaida Lykova; N. J. Young

doi:10.1016/j.jmaa.2019.01.027

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arXiv · 1901.03100

A Geometric Characterization of the Symmetrized Bidisc

Jim Agler·Zinaida Lykova·N. J. 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.

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BibTeX RIS

Jim Agler, Zinaida Lykova, N. J. Young. 2019-01-10. A Geometric Characterization of the Symmetrized Bidisc. https://doi.org/10.1016/j.jmaa.2019.01.027

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