Wayne Rossman

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

Abstract 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 SU 2 with SU 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 SU 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.

Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

We present a theorem on the unitarizability of loop group valued monodromy representations and apply this 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

The first bifurcation point for Delaunay nodoids

\R^3

Zero mean curvature surfaces in Lorentz--Minkowski 3-space which change type across a light-like line

,

Discrete linear Weingarten surfaces

\bbS^3

The Morse Index of Wente Tori

and

Lower Bounds for Morse Index of Constant Mean Curvature Tori

\bbH^3

Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space

. Additionally, we compute the extended frame for any associated family of Delaunay surfaces.

Constant Mean Curvature Surfaces of any Positive Genus

We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo- Pacard is the same as a limiting point encountered in the integrable systems method of Dorfmeister-Pedit-Wu.

Embedded, doubly periodic minimal surfaces

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across the light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in R^{n+1}_1 that change type across an (n-1)-dimensional light-like plane.