Helicoid-like Minimal Disks and Uniqueness

This paper gives a condition for an embedded minimal disk to be bi-Lipschitz to a piece of a helicoid. Namely, if such a disk is inside a ball (with boundary on the ball) and has large curvature at the center relative to nearby points, then, in a smaller ball, it is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant can be chosen as close to 1 as desired. This is a sharpening of a result of Colding and Minicozzi showing that, in a rough sense, such a disk looks like a piece of the helicoid. Our result follows from the fact that the only complete, nonflat, properly embedded minimal disk (PEMD) in R is the helicoid, for which we supply a proof. The initial proof, given by Meeks and Rosenberg in [16], depends crucially on the lamination theory and one-sided curvature estimate of Colding and Minicozzi (see [8]). Instead of appealing to the lamination theory, we make direct use of the results of Colding and Minicozzi on the existence of multivalued graphs in embedded minimal disks, as found in [5, 6, 7, 8]. For a good non-technical overview see [9]. Applying a result of [4] to these multivalued graphs, we approximate them by pieces of helicoids, giving explicit asymptotic behavior and geometric rigidity. In [1], these techniques will be further studied. In their paper, Meeks and Rosenberg first use the lamination theory to show that (after a rotation) a homothetic blow-down of a non-flat complete PEMD, Σ, is, away from some Lipschitz curve, a foliation of flat parallel planes transverse to the x3-axis. This gives, in a weak sense, that the surface is asymptotic to a helicoid, which they use to conclude that the Gauss map of Σ omits the north and south poles. The asymptotic structure combined with a result on parabolicity of Collin, Kusner, Meeks and Rosenberg [11], is then used to show that Σ is conformally equivalent to C. Finally, they look at level sets of the log of the Gauss map and use a Picard type argument to show that this holomorphic map does not have an essential singularity at∞ and in fact is linear. Using the Weierstrass representation, they conclude that Σ is the helicoid. The explicit asymptotics in our paper allow for a more direct approach. We show Σ contains a central “axis” of large curvature away from which it consists of two multivalued graphs spiraling together, one strictly upward, the other downward. This is the structure of the helicoid and more generally, away from a compact set, the structure of the (known) embedded genus one helicoid(s) i.e. the construction of Weber, Hoffman and Wolf, [13], and that of Hoffman and White, [14], and, indeed, of any symmetric genus one helicoid (see [15]). Moreover, this is the behavior of any complete, non-flat PEMD: