It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

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.

After Gálvez, Martínez and Milán discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space H 3 , the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H 3 . In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces.This more general class contains the caustics of flat fronts -shown also to be flat by Roitman (who gave a holomorphic representation formula for them) -which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.

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In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces.Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete Ω-nets, a discrete analogue of Demoulin's Ω-surfaces. It is shown that the Lie-geometric deformation of Ω-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory.