Wayne Rossman

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It is well-known that the unit cotangent bundle of any Riemann-ian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a(More)
We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution –(More)
are constant mean curvature (CMC) surfaces of revolution, and they are translationally periodic. By a rigid motion and homothety of R we may place the Delaunay surfaces so that their axis of revolution is the x1-axis and their constant mean curvature is H = 1 (henceforth we assume this). We consider the profile curve in the half-plane {(x1, 0, x3) ∈ R |x3 >(More)
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(More)
This paper presents a unified treatment of constant mean curvature (cmc) surfaces in the simply-connected 3-dimensional space forms R, S and H in terms of meromorphic loop Lie algebra valued 1-forms. We discuss global issues such as period problems and asymptotic behaviour involved in the construction of cmc surfaces with nontrivial topology. We prove(More)
1 Forward These notes are about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using(More)
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(More)
We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant’s Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some(More)