A proof of the Lawson conjecture for minimal tori embedded in $\S3$

Details A proof of the Lawson conjecture for minimal tori embedded in $\S3$ A proof of the Lawson conjecture for minimal tori embedded in $\S3$

2007-03-05

arxiv.org/abs/math/0703136v2

Mathematics

Differential Geometry

20 pages, 8 figures, typos corrected (see esp. Lem. 4.2) and clarifications added. We refer esp. to a more straightforward pro

Scientific paper

A peculiarity of the geometry of the euclidean 3-sphere $\S3$ is that it allows for the existence of compact without boundary minimally immersed surfaces. Despite a wealthy of examples of such surfaces, the only known tori minimally embedded in $\S3$ are the ones congruent to the Clifford torus. In 1970 Lawson conjectured that the Clifford torus is, up to congruences, the only torus minimally embedded in $\S3$. We prove here Lawson conjecture to be true. Two results are instrumental to this work, namely, a characterization of the Clifford torus in terms of its first eingenfunctions (\cite{MR}) and the assumption of a "two-piece property" to these tori: every equator divides a torus minimally embedded in $\S3$ in exactly two connected components (\cite{Rs}).

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