We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R 2,1 . The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group S U 2 with S U 1,1 . The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form S U 1,1 , and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R 2,1 . In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.

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.

No abstract

The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation (ODE) with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.

We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.