Conformal structure of minimal surfaces with finite topology

Abstract

In this paper we show that a complete, embedded minimal surface in ${\mathbb{R}}^3$, with finite topology and one end, is conformal to a once-punctured compact Riemann surface. Moreover, using this conformal structure and the embeddedness of the surface, 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 Gauss map, then in a neighborhood of the puncture, $g(p)=\exp (i\alpha z(p)+F(p))$, where $\alpha \in \mathbb{R}$, $z=x_3+i{x}_3^{\ast }$ is a holomorphic coordinate defined in this neighborhood and $F(p)$ is holomorphic in the neighborhood and extends over the puncture with a zero there. As a consequence, the end is asymptotic to a helicoid. This completes the understanding of the conformal and geometric structure of the ends of complete, embedded minimal surfaces in ${\mathbb{R}}^3$ with finite topology.