Brief Survey Of Minimal Submanifolds II

Abstract

This is a continuation of the survey by S.S. Chern, which will be refered as I. We will consider immersions $x:{\text{M}^{n}}\rightarrow {\text{S}}^{\text{m}}_{1} \subset{\text{R}}^{\text{m+1}}$ of an n-dimensional manifold ${\text{m}}^{\text{n}}$ into 9 the unit sphere ${\text{S}}^{\text{m}}_{1}$ of the euclidean space ${\text{R}}^{\text{m+1}}$, which are minimal in the sense that small pieces minimize area relative to boundary-preserving variations. In the particular case for which ${\text{M}}^{\text{n}} = {\text{S}}^{\text{n}}$ and the induced metric has constant sectional curvature, a qualitative description of these immersions can be obtained. This part II aims to give the details of such a description (section 2) and an idea of the methods which lead to it (section J). In section 4 we mention some related results, and in section 5 a few. open questions are discussed.