Real forms of complex surfaces of constant mean curvature

Kobayashi, Shimpei

doi:10.1090/S0002-9947-2010-05164-5

Real forms of complex surfaces of constant mean curvature
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Trans. Amer. Math. Soc. 363 (2011), 1765-1788 Request permission

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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