Minimal two-spheres of low index in manifolds with positive complex sectional curvature. (English) Zbl 1428.53074
Authors’ 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 \(1/2 < K_r(\sigma ) \le 1.\) 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 \(G_3({\mathbb {R}}^{n+1})\).
Reviewer: Liviu Popescu (Craiova)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C20 | Global Riemannian geometry, including pinching |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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