Conformal Structure of Minimal Surfaces with Finite Topology

In this paper, we show that a complete embedded minimal surface in $\Real^3$ 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 Gauss map, then in a neighborhood of the puncture, $g(p) = \exp(i\alpha z(p) + F(p))$, where $\alpha \in \Real$, $z=x_3+ix_3^*$ 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. This further implies that the end is actually Hausdorff close to a helicoid.