Real forms of complex surfaces of constant mean curvature

Abstract

Abstract. It is known that complex constant mean curvature (CMC for short) immersions in C 3 are natural complexifications of CMC-immersions in 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 sl(2, C) loop algebra Λsl(2, C) σ , and classify all such surfaces according to the classification of real forms of Λsl(2, C) σ . 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, C)/U (1). We also give a unification to all integrable surfaces via the generalized Weierstrass type representation.

Constant mean curvature surfaces in hyperbolic 3-space via loop groups

“…Remark 6.1. As pointed out above, the Maurer-Cartan form α = λ −1 α −1 + α 0 + λα 1 is a type C 4 real form of the complex CMC surface equation [40]. But it does not correspond naturally to a CMC surface in E 3 , since H is not real.…”

“…9.2. We recall that ΛSL 2 C σ,τj , j = 3, 4, defined in sections 5 and 6, are real forms of ΛSL 2 C σ , [40]. These real forms naturally induce Iwasawa decompositions of ΛSL 2 C σ : Theorem 9.2 (Iwasawa decomposition for τ 3 , [52]).…”

“…It would be interesting to characterize all surface geometries to which the DPW method is applicable. So far only partial results are known: See e.g., the survey [17] by the first named author and the recent classification, carried out by the third named author, of surfaces in 3dimensional space forms which are real forms of complex constant mean curvature surfaces [40].…”

Survey on real forms of the complex A2(2)-Toda equation and surface theory

“…One should also observe that in [26] already all real forms of the affine algebra A (1) 1 have been related to constant mean curvature/constant Gaussian curvature surfaces in the Euclidean 3-space, the Minkowski 3-space or the hyperbolic 3-space.…”

A loop group method for minimal surfaces in the three-dimensional Heisenberg group

“…For more than fifteen years now a loop group technique has been used to investigate these surfaces, see [21,36]. This 2-parameter family can be seen in the local classification of all homogeneous Riemannian metrics on R 3 due to Bianchi, [10], see also Vranceanu [46, p. 354].…”