Real forms of complex surfaces of constant mean curvature
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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.References
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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
- Received by editor(s): June 3, 2008
- Published electronically: November 8, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1765-1788
- MSC (2000): Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9947-2010-05164-5
- MathSciNet review: 2746664