Brief Survey Of Minimal Submanifolds II

Manfredo P. do Carmo

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 77-91 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-3-642-25588-5_7

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