There exists an embedded genus-one helicoid in R^3

You need to know: Surface, boundary of a surface, area of a surface, compact surface, total curvature of a surface, homeomorphic surfaces.

Background: A surface is called a minimal surface if every point has a neighbourhood S, with boundary B, such that S has a minimal area out of all surfaces S’ with the same boundary B. A surface is called properly embedded in , if it has no boundary, no self-intersections, and its intersection with any compact subset of is compact. A helicoid is the surface defined by equations where is a constant, and are real parameters, ranging from to . A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed.

The Theorem: On 8th January 2004, Matthias Weber, David Hoffman, and Michael Wolf submitted to arxiv a paper in which they proved the existence of a properly embedded minimal surface in with finite topology and infinite total curvature, which is not a helicoid.

Short context: Helicoid, described by Euler in 1774, is one of the simplest non-trivial examples of minimal surfaces, and it is unique in many ways. In particular, Meeks III and Rosenberg proved that it, together with a plane, are the only examples of properly embedded simply connected minimal surfaces in . Before 2004, there was a possibility that helicoid is also the only properly embedded minimal surface in with finite topology and infinite total curvature. The Theorem, however, proves the existence of at least one more such surface. It looks like a helicoid with a hole and have got a name “embedded genus one helicoid”.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.3 of this book for an accessible description of the Theorem.

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