Nicholas Schmitt

New constant mean curvature surfaces

We use the Dorfmeister–Pedit–Wu construction to present three new classesof immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the third author, describes the DPW potentials that appear to give CMC punctured spheres with k Delaunay ends (k-noids): the evidence is experimental at present. These can have both unduloidal and nodoidal ends.

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

FLOWS OF CONSTANT MEAN CURVATURE TORI IN THE 3-SPHERE: THE EQUIVARIANT CASE

\R^3

Deformations of Symmetric CMC Surfaces in the 3-Sphere

,

Constant Mean Curvature Surfaces of any Positive Genus

\bbS^3

The spectral curve theory for (k, l)-symmetric CMC surfaces

and

Dressing CMC n-Noids

\bbH^3

Constant Mean Curvature Trinoids

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

Constant Mean Curvature n-noids with Platonic Symmetries

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.