Conformal and asymptotic properties of embedded genus-g minimal surfaces with one end

Abstract

Using the tools developed by Colding and Minicozzi in their study of the structure of embedded minimal surfaces in R3 [12, 19-22], we investigate the conformal and asymptotic properties of complete, embedded minimal surfaces of finite genus and one end. We first present a more geometric proof of the uniqueness of the helicoid than the original, due to Meeks and Rosenberg [45]. That is, the only properly embedded and complete minimal disks in R3 are the plane and the helicoid. We then extend these techniques to show that any complete, embedded minimal surface with one end and finite topology is conformal to a once -punctured compact Riemann surface. This completes the classification of the conformal type of embedded finite topology minimal surfaces in R3. Moreover, we show that such s surface has Weierstrass data asymptotic to that of the helicoid, and as a consequence is asymptotic to a helicoid (in a Hausdorff sense). As such, we call such surfaces genus-g helicoids. In addition, we sharpen results of Colding and Minicozzi on the shapes of embedded minimal disks in R3, giving a more precise scale on which minimal disks with large curvature are "helicoidal". Finally, we begin to study the finer properties of the structure of genus-g helicoids, in particular showing that the space of genus-one helicoids is compact (after a suitably normalization).