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A theorem on the unitarizability of loop group valued monodromy representations is presented and applied to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply connected 3-dimensional space forms R 3 , S 3 and H 3 . Additionally, the extended frame for any associated family of Delaunay surfaces is computed.We identify Euclidean three-space R 3 with the matrix Lie algebra su 2 . The double cover of the isometry group under this identification is SU 2 su 2 . Let T denote the stabilizer of ∈ su 2 under the adjoint action of SU 2 on su 2 . We shall view the two-sphere as S 2 = SU 2 /T. Lemma 1. The mean curvature H of a conformal immersion f : M → su 2 is given byProof. Let U ⊂ M be an open simply connected set with coordinate z : U → C. Writing df = f z dz and df = fz dz, conformality is equivalent to f z , f z = fz, fz = 0 and the existence of a function v ∈ C ∞ (U, R + ) such that 2 f z , fz = v 2 . Let N : U → SU 2 /T be the Gauss map with lift F : U → SU 2 such that N = F F −1 and df = vF ( − dz + + dz)F −1 . The mean curvature is H = 2v −2 f zz , N and the Hopf differential is Q dz 2 with Q = f zz , N . Hence [df ∧ df ] = 2iv 2 N dz ∧ dz. Then F −1 dF = (1/2v)((−v 2 Hdz − 2Qdz)i − + (2Qdz + v 2 Hdz)i + − (v z dz − vzdz)i ). This allows us to compute d * df = iv 2 HN dz ∧ dz and proves the claim.

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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.

Abstract. We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle D × GL 2 C over a simply connected domain D in the complex plane. In particular for minimal surfaces, we give an immersion formula, the so-called Sym-formula, and a generalized Weierstrass type representation via the loop group method.

Abstract. We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus g ≥ 1, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

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