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Real forms of complex surfaces of constant mean curvature


Author: Shimpei Kobayashi
Journal: Trans. Amer. Math. Soc. 363 (2011), 1765-1788
MSC (2000): Primary 53A10
Published electronically: November 8, 2010
MathSciNet review: 2746664
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Abstract: It is known that complex constant mean curvature (CMC for short) immersions in $ \mathbb{C}^3$ are natural complexifications of CMC-immersions in $ \mathbb{R}^3$. In this paper, conversely we consider real form surfaces of a complex CMC-immersion, which are defined from real forms of the twisted $ \mathfrak{sl}(2, \mathbb{C})$ loop algebra $ \Lambda \mathfrak{sl}(2, \mathbb{C})_\sigma$, and classify all such surfaces according to the classification of real forms of $ \Lambda \mathfrak{sl}(2, \mathbb{C})_\sigma$. There are seven classes of surfaces, which are called integrable surfaces, and all integrable surfaces will be characterized by the (Lorentz) harmonicities of their Gauss maps into the symmetric spaces $ S^2$, $ H^2$, $ S^{1,1}$ or the $ 4$-symmetric space $ SL(2, \mathbb{C})/U(1)$. We also give a unification to all integrable surfaces via the generalized Weierstrass type representation.


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Additional Information

Shimpei Kobayashi
Affiliation: Graduate School of Science and Technology, Hirosaki University, Bunkyocho 3 Aomori 036-8561 Japan
Email: shimpei@cc.hirosaki-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2010-05164-5
Received by editor(s): June 3, 2008
Published electronically: November 8, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.