On the number of minimal two-spheres of small area in manifolds with curvature bounded above
type: Item
instance of: scholarly article
title: On the number of minimal two-spheres of small area in manifolds with curvature bounded above
zbMATH Open document ID: 0694.53053
DOI: 10.1007/BF01444536
author: John Douglas Moore
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164738
review text: This article establishes the equivariant Morse inequalities for minimal two-spheres of area \(<8\pi\) in a compact Riemannian manifold M whose sectional curvatures K(\(\sigma)\) satisfy the inequality K(\(\sigma)\)\(\leq 1\). The Morse inequalities are useful for proving the existence of minimal two-spheres of small area. For example, they yield the following corollary: Let M be a compact manifold with Riemannian metric g whose sectional curvatures satisfy the inequality K(\(\sigma)\)\(\leq 1\), and such that all nonconstant harmonic two-spheres of energy \(<8\pi\) are nondegenerate. Let \(h: S^ n\to M\) be a diffeomorphism such that \(g_ 0\leq h^*g<2g_ 0,\) where \(g_ 0\) is the standard metric of constant curvature one on \(S^ n\). If M has no minimal projective planes of area \(<4\pi\), then M possesses at least \(\left( \begin{matrix} n+1\\ 3\end{matrix} \right)\) distinct prime minimal two-spheres of area \(<8\pi\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 4136934
zbMATH Keywords: equivariant Morse inequalities
zbMATH Keywords: minimal two-spheres
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4136934