A properly embedded simply connected minimal surface in R^3 is either a plane or a helicoid

Geometry and Topology 1 Minute

You need to know: Euclidean space {\mathbb R}^3, surface in {\mathbb R}^3, surface area, neighbourhood of a point in the surface, boundary, path, endpoints of a path, surface with and without self-intersections, compact subset of {\mathbb R}^3, compact subset of a surface.

Background: A surface M \subset {\mathbb R}^3 is called a minimal surface if every point p \in M 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 M \subset {\mathbb R}^3 is called simply connected if for every 2 points x\in M and y\in M there is a path in M with endpoints x and y, and every 2 such paths can be continuously transformed into each other within M while keeping both endpoints x and y fixed. A surface M \subset {\mathbb R}^3 is called properly embedded in {\mathbb R}^3, if it has no boundary, no self-intersections, and its intersection with any compact subset of {\mathbb R}^3 is compact. A helicoid is the surface defined by equations x = s \cos (\alpha t), \,\, y = s \sin (\alpha t), \,\, z=t, where \alpha is a constant, and s,t are real parameters, ranging from -\infty to \infty.

The Theorem: On 13th March 2001, William Meeks III and Harold Rosenberg submitted to Annals of Mathematics a paper in which they proved that any properly embedded simply connected minimal surface in {\mathbb R}^3 is either a plane or a helicoid.

Short context: Minimal surfaces, ones that locally minimises their areas, are among the most studied objects in geometry and topology. A trivial example of minimal surface is the plane. One of the simplest non-trivial examples, described by Euler in 1774, is a helicoid. Since 1774, many other minimal surfaces has been discovered, but none of them are properly embedded and simply connected. The Theorem proves that in fact there are no other minimal surfaces with these natural properties.

Links: The original paper is available here. See also Section 5.3 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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