On the number of minimal two-spheres of small area in manifolds with curvature bounded above
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 \(<8\pi\).
scientific article; zbMATH DE number 4136934
scientific article