Embedded minimal ends of finite type

We prove that the end of a complete embedded minimal surface in $\mathbb{R} ^3$ with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end $E$ provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is $C^0$-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in $\mathbb{R} ^3$.