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

TL;DR

This paper investigates the existence and quantity of minimal two-spheres with low Morse index in manifolds with positive complex sectional curvature, establishing lower bounds based on topological cell decompositions.

Contribution

It introduces a new pinching condition related to complex sectional curvatures and derives lower bounds on the number of minimal two-spheres of certain Morse indices.

Findings

Lower bounds on minimal two-spheres for specified Morse indices

Pinching condition involving complex sectional curvatures

Relation to Schubert cell decomposition in real Grassmannians

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 $$1/2 < K_r(\sigma ) \leq 1.$$ Then the number of minimal two spheres of Morse index $\lambda $, for $n-2 \leq \lambda \leq 2n-5$, is at least $p_{3}(\lambda -n+2)$, where $p_{3}(k)$ is the number of $k$-cells in the Schubert cell decomposition for $G_3({\mathbb R}^{n+1})$.