The uniqueness of the helicoid

Abstract

In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R 3 . M of finite topology means M is homeomorphic to a compact surface M (of genus k and empty boundary) minus a finite number of points p 1 , ..., p j ∈ M , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S 1 × [0, 1) and hence, annular. We remark that M is orientable since M is properly embedded in R 3 .The simplest examples (discovered by Meusnier in 1776) are the helicoid and catenoid (and a plane of course). It was only in 1982 that another example was discovered. In his thesis at Impa, Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3-punctured torus. He observed the three ends of this surface were embedded: one top catenoidtype end 1 , one bottom catenoid-type end, and a middle planar-type end 2 [8]. Subsequently, Hoffman and Meeks [15] proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus k and three ends.In 1993, Hoffman, Karcher and Wei [14] discovered the Weierstrass data of a complete minimal surface of genus one and one annular end. Computer generated pictures suggested this surface is embedded and the end is asymptotic to an end of a helicoid. Hoffman, Weber and Wolf [17] have now given a proof that there is such an embedded surface. Moreover, computer evidence suggests that one can add an arbitrary finite number k of handles to a helicoid to obtain a properly embedded genus k minimal surface asymptotic to a helicoid.For many years, the search went on for simply connected examples other than the plane and helicoid. We shall prove that there are no such examples.

The rigidity of embedded constant mean curvature surfaces

“…The main theorem of Meeks and Rosenberg in [12] then implies that M is a plane or is asymptotic to a helicoid Σ. In the latter case, M can be thought of as being a helicoid with a finite positive number of handles attached 4 .…”

On curvature estimates for constant mean curvature surfaces

“…In the case of embedded minimal disks such a description was given by Colding and Minicozzi in [7]; see also [32,33] for related results. By rescaling arguments this description can be improved upon once one knows that the helicoid is the unique complete, embedded, non-flat minimal surface in R 3 as explained below; see [17] and also [1] for proofs of the uniqueness of the helicoid which are based in part on results in [6,7,8,9,10].…”

Final Report: Doe-Fg03-95er25250

“…Recently, Meeks and Rosenberg (4) showed that the helicoid is the unique simply connected, properly embedded (nonplanar) minimal surface R 3 with one end. The method of proof uses in an essential manner the work of Colding and Minicozzi (5-8) concerning curvature estimates for embedded minimal disks and geometric limits of those disks.…”