Brief Survey Of Minimal Submanifolds II

  • Manfredo P. do Carmo


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.


Unit Sphere Spherical Harmonic Fundamental Form Homogeneous Polynomial Isometric Immersion 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manfredo P. do Carmo
    • 1
  1. 1.Rio de JaneiroBrasil

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