The Topology, Geometry and Conformal Structure of Properly Embedded Minimal Surfaces

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 topology, geometry and conformal structure of properly embedded minimal surfaces

Let M denote the set of connected properly embedded minimal surfaces in R with at least two ends. At the beginning of the past decade, there were two outstanding conjectures on the asymptotic geometry of the ends of an M ∈ M that were known to lead to topological restrictions on M . The first of these conjectures, the generalized Nitsche conjecture, stated that an annular end of such a M ∈ M is...

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

Complete Properly Embedded Minimal Surfaces in R3

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface 2 ⊂ R3 with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature. In this short paper, we apply estimates and ideas from [CM4] to study the en...

Conformal Structure of Minimal Surfaces with Finite Topology

In this paper, we show that a complete embedded minimal surface in R with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if g is the stereographic projection of the ...