Martin Kilian

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.

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

EQUIVARIANT HARMONIC CYLINDERS

,

Dressing preserving the fundamental group

\bbS^3

Constant Mean Curvature Surfaces of any Positive Genus

and

On the associated family of Delaunay surfaces

\bbH^3

Delaunay ends of constant mean curvature surfaces

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

On mean-convex Alexandrov embedded surfaces in the 3-sphere

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.

Mean-convex Alexandrov embedded constant mean curvature tori in the 3-sphere

We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries. In this paper, we study the simplest case of this theory: that of equivariant harmonic maps where the underlying PDE reduces to an ODE. We show that such maps are characterised as those with a holomorphic potential of the simplest kind: the 1-form has simple poles at zero and infinity and satisfies a natural reality condition. Along the way, we show that equivariance is a property which is preserved under spectral deformation. A pleasant application of the foregoing theory lies in the fact that several types of surface of classical geometric interest are characterised by harmonicity of an appropriate Gauss map (3, 7, 17). In particular, the classical theory of constant mean curvature surfaces in R 3 amounts to the study of harmonic maps to a 2-sphere with the link between the loop group approach and the classical surfaces being provided by the Sym-Bobenko formula. We apply our general theory to this case which means that we study constant mean curva- ture surfaces with screw motion symmetry. We find a very simple proof of a result of Do Carmo-Dajczer (8) which asserts that these surfaces are precisely those in the associated family of a Delaunay surface (that is, a constant mean curvature surface of revolution). For this, we provide an interpretation of the Sym-Bobenko formula in terms of a homomorphism from a loop group to the Euclidean group which may be of independent interest. We then turn to a detailed study of the period problem for constant mean curvature surfaces with screw motion symmetry. Armed with our knowledge of the holomorphic potential, we are able to explicitly compute, in terms of elliptic functions, the corresponding map into the loop group (usually, this involves solving a Riemann-Hilbert problem). With this in hand, we can prove the existence of infinitely many non-congruent cylinders in the associated family of each Delaunay surface. Otherwise said, in each associated family of equivariant