Every embedded minimal torus in S^3 is congruent to the Clifford torus

Geometry and Topology 1 Minute

You need to know: Euclidean space {\mathbb R}^4, unit sphere {\mathbb S}^3 \subset {\mathbb R}^4, surface embedded in {\mathbb S}^3, congruent surfaces, mean curvature of a surface (see here), torus.

Background: Let \Sigma be a two-dimensional surface embedded in {\mathbb S}^3. We say that \Sigma is a minimal surface if the mean curvature of \Sigma vanishes identically. The Clifford torus is defined by \{(x_1, x_2, x_3, x_4)\in {\mathbb S}^3 : x_1^2+x_2^2 = x_3^2+x_4^2 = \frac{1}{2}\}.

The Theorem: On 29th March 2012, Simon Brendle submitted to arxiv a paper in which he proved that every embedded minimal torus in {\mathbb S}^3 is congruent to the Clifford torus.

Short context: Minimal surfaces are among the most studied objects in geometry and topology. Of particular interest are minimal surfaces embedded in Euclidean space {\mathbb R}^3 or in 3-sphere {\mathbb S}^3. The simplest examples of minimal surfaces in {\mathbb S}^3 are equator and the Clifford torus. In 1970, Lawson conjectured that the Clifford torus is in fact the only embedded minimal torus in {\mathbb S}^3. The Theorem confirms this conjecture.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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