Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

Constant mean curvature surfaces with cylindrical ends

Abstract. We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends. Surfaces which minimize area under a volume constraint have constant mean curvature (cmc); this condition can be expressed as a nonlinear partial differential equation. We are interested in complete cmc surfaces properly embedded in R 3; we rescale them to have mean curvature one. For technical reasons, we consider a slight generalization of embeddedness (introduced by Alexandrov [2]): An immersed surface is almost embedded if it bounds a properly immersed three-manifold. Alexandrov [1, 2] showed that the round sphere is the only compact almost embedded cmc surface. The next case to consider is that of finite-topology surfaces, homeomorphic to a compact surface with a finite number of points removed. A neighborhood of any of these punctures is called an end of the surface. The unduloids, cmc surfaces of revolution described by Delaunay [4], are genus-zero example