A Proof of the Lawson Conjecture for Minimal Tori Embedded in S

A peculiarity of the geometry of the euclidean 3-sphere S 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 S 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 S. 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 ([MR]) and the assumption of a “two-piece property” to these tori: every equator divides a torus minimally embedded in S in exactly two connected components ([R]).