Korean J. Math. Vol. 30 No. 4 (2022) pp.643-652
DOI: https://doi.org/10.11568/kjm.2022.30.4.643

Affine homogeneous domains in the complex plane

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Kang-Hyurk Lee


In this paper, we will describe affine homogeneous domains in the complex plane. For this study, we deal with the Lie algebra of infinitesimal affine transformations, a structure of the hyperbolic metric involved with affine automorphisms. As a consequence, an affine homogeneous domain is affine equivalent to the complex plane, the punctured plane or the half plane.

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Samsung Science and Technology Foundation (SSTF-BA2201-01)


[1] L. V. Ahlfors. An extension of Schwarz’s lemma. Trans. Amer. Math. Soc., 43(3):359–364, 1938. Google Scholar

[2] L. V. Ahlfors. Complex analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable. Google Scholar

[3] Y.-J. Choi and K.-H. Lee. Existence of a complete holomorphic vector field via the K ̈ahler- Einstein metric. Ann. Global Anal. Geom., 60(1):97–109, 2021. Google Scholar

[4] Y.-J. Choi, K.-H. Lee, and S. Yoo. A certain K ̈ahler potential of the Poincar ́e metric and its characterization. J. Korean Math. Soc., 57(6):1335–1345, 2020. Google Scholar

[5] C. Kai and T. Ohsawa. A note on the Bergman metric of bounded homogeneous domains. Nagoya Math. J., 186:157–163, 2007. Google Scholar

[6] P. Koebe. U ̈ber die Uniformisierung reeller algebraischer Kurven. Nachr. Ges. Wiss. Go ̈ttingen, Math.-Phys. Kl., 1907:177–190, 1907. Google Scholar

[7] K.-H. Lee. A method of potential scaling in the study of pseudoconvex domains with noncompact automorphism group. J. Math. Anal. Appl., 499(1):Paper No. 124997, 15, 2021. Google Scholar

[8] H. Poincar ́e. Sur l’uniformisation des fonctions analytiques. Acta Math., 31(1):1–63, 1908. Google Scholar