Minimal immersions of surfaces by the first Eigenfunctions and conformal area

Let 0: M-,S" be a minimal immersion of a compact surface into a unit sphere. Then, the linear functions of 0 are eigenfunctions for the Laplacian of M corresponding to the eigenvalue 2=2 . The main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non-zero eigenvalue of its Laplacian. This kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems, as Li and Yau have shown in [6]. The methods that we use in this paper are based, for the most part, on [6]. It is known that the only metric on a 2-dimensional sphere admitting a minimal immersion into S" by the first eigenfunctions is the standard one (this follows, for example, from the fact that the multiplicity of the first eigenvalue for such a metric is at most three, see the Cheng work [3]). Our first result shows that it is possible to extend this property for an arbitrary compact surface, in the following way: