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M. Kilian

Publications and source records attributed to M. Kilian.

6 recordsLinked to original sources
arXiv2015

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

We introduce the moduli space of spectral curves of constant mean curvature (\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly determined using a combination of integrable systems and geometric analysis techniques. We prove that these cylinders are surfaces of revolution. As a consequence all mean-convex Alexandrov embedded {\sc{cmc}} tori in the 3-sphere are surfaces of revolution.

math.DG↗
arXiv2015

Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic harmonic maps $G : \mathbb{C} \to \mathbb{S}^2$ of finite type. Due to the algebraic geometric correspondence of Hitchin [14], these harmonic maps are parametrized by hyperelliptic algebraic curves together with Abelian differentials with prescribed poles. We deform annuli by deforming spectral data in the corresponding moduli space. Along this deformation we control the flux and we preserve embeddedness. The center of the theory concerns the study of singularities of the flow. In particular we open and close nodes of singular spectral curves. This approach applies also to mean convex Alexandrov embedded cmc annuli in $\mathbb{S}^3$ [12].

math.DG↗
arXiv2013

Finite type minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

We study minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set. We consider polynomial Killing fields with zeroes and the corresponding singular spectral curves, bubbletons and simple factors. We investigate the differentiable structure on the isospectral set of any finite type minimal annulus. We apply the theory to a 2-parameter family of embedded minimal annuli foliated by horizontal circles.

math.DG↗
arXiv2007

Delaunay Ends of Constant Mean Curvature Surfaces

The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.

math.DG↗
arXiv2005

Equivariant harmonic cylinders

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.

math.DG↗