Conformal Structure of Minimal Surfaces with Finite Topology

The Geometry and Conformal Structure of Properly Embedded Minimal Surfaces of Finite Topology in R

In this paper we study the conformal structure and the asymptotic behavior of properly embedded minimal surfaces of finite topology in . One consequence of our study is that when such a surface has at least two ends, then it has finite conformal type, i.e., it is conformally diffeomorphic to a compact Riemann surface punctured in a finite number of points. Except for the helicoid and a recently...

The Geometry and Conformal Structure of ProperlyEmbedded Minimal Surfaces of Finite Topology inR 3

Compact Complete Minimal Immersions In

ANTONIO ALARC´ON ABSTRACT. In this paper we construct complete minimal surfaces with boundary in R 3 of arbitrary finite topology. For any arbitrary finite topological type we find a compact Riemann surface M, an open domain M ⊂ M with the fixed topological type, and a conformal complete minimal immersion X : M → R 3 which can be extended to a continuous map X : M → R 3 , such that X |∂M is an ...

On the Existence of a Proper Minimal Surface in R with a Conformal Type of Disk

Proper minimal surfaces in R3 have peculiar properties that are not shared by general minimal surfaces; especially in the embedded case. It has been proved that, under additional conditions, this family of surfaces has strong restrictions on their conformal structures. For instance, Huber and Osserman proved that if M is a complete minimal surface with finite total curvature, then M has the con...

ar X iv : 0 90 3 . 32 09 v 1 [ m at h . D G ] 1 8 M ar 2 00 9 Uniform Approximation by Algebraic Minimal Surfaces in R 3

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in R is obtained. This result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case. As a consequence, complete minimal surfaces in...