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The uniqueness of the helicoid. (English) Zbl 1102.53005

From the introduction: We discuss the geometry of finite topology properly embedded minimal surfaces \(M\) in \(\mathbb R^3\). For \(M\) to be of finite topology means \(M\) is homeomorphic to a compact surface \(\widehat{M}\) (of genus \(k\) and empty boundary) minus a finite number of points \(p_1,\dots,p_j\in\widehat {M}\), called the punctures.
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.
In 1993, D. Hoffman, H. Karcher and F. Wei [Uhlenbeck, Karen (ed.), Global analysis in modern mathematics. Proceedings of the symposium in honor of Richard Palais’ sixtieth birthday, University of Maine, Orono, ME, USA, August 8–10, 1991, and at Brandeis University, Waltham, MA, USA, August 12, 1992. Houston, TX: Publish or Perish, Inc. 119–170 (1993; Zbl 1049.53502)] 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. D. Hoffman, M. Weber and M. Wolf [The existence of the genus-one helicoid (preprint)] 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 prove that there are no such examples.
Theorem 0.1. A properly embedded simply-connected minimal surface in \(\mathbb R^3\) is either a plane or a helicoid.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 1049.53502
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