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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1295–1335 | Cite as

Minimal two-spheres of low index in manifolds with positive complex sectional curvature

  • John Douglas MooreEmail author
  • Robert Ream
Article
  • 75 Downloads

Abstract

Suppose that \(S^n\) is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures \(K_r(\sigma )\) satisfy
$$\begin{aligned} 1/2 < K_r(\sigma ) \le 1. \end{aligned}$$
Then the number of minimal two spheres of Morse index \(\lambda \), for \(n-2 \le \lambda \le 2n-5\), is at least \(p_{3}(\lambda -n+2)\), where \(p_{3}(k)\) is the number of k-cells in the Schubert cell decomposition for \(G_3({\mathbb {R}}^{n+1})\).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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