PLANE METRICS AND THE UNIFORMIZATION THEOREM ON MINIMAL THE THREE DIMENSIONAL SPHERE

Abstract

Let (M, g) be a torus with a riemannian metric g. The Uniformization theorem implies the existence of a covering conforma! map <:p : R2 ----> /1.1 such that the group H,t, of diffeo­morphisms p: R2 ---+ R2 such that <Pop= <j,, is a subgroup of the group of translations in R2 and is isomorph to Z2• The classical proof of his theorem uses tools from algebraic topology and PDE's in Sobolev spaces. In this paper we will give an alternative and easier proof of this theorem in the case that M is a minimal immersed torus in the unit three dimensional sphere, S3. We will achieve this theorem by studying the shape operator of minimal torus on the sphere. One of the main ideas of the this paper is to induce the reader to the study of minimal torus in S3• Recall that one important conjecture in differential geometry is Law­son's conjecture. This conjecture states that the only minimal embedded tori in S3 are the Clifford tori.