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Brief Survey Of Minimal Submanifolds II

  • Manfredo P. do Carmo
Chapter

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.

Keywords

Unit Sphere Spherical Harmonic Fundamental Form Homogeneous Polynomial Isometric Immersion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Calabi, E., Minimal immersions of surfaces in euclidean spheres, J. Diff. Geom. 1 (1967) 111-125.Google Scholar
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    Chern, s.s., On the minimal immersions of the two-sphere in a space of constant curvature, to appear.Google Scholar
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    do Carmo, M. and Wallach, N., Representations of compact groups and minimal immersions into spheres, J. Diff. Geom., to appear.Google Scholar
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    do Carmo, M. and Wallach, N., Minimal immersions of spheres into spheres, to appear.Google Scholar
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    do Carmo, M. and Wallach, N., Minimal immersions of sphere bundles over spheres, An. Acad. Brasil. Cienc. to appear.Google Scholar
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    Wallach, N., Extensions of locally defined minimal immersions into spheres, Archiv der Mathematik, to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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