When the compact manifold M has a Riemannian metric satisfying a suitable curvature condition, we show that it has many minimal two-spheres of index between n--2 and 2n--5, using Morse theory for the alpha-energy of Sacks and Uhlenbeck. The difficulty is controlling bad behavior of a sequence of alpha-energy critical points as alpha approaches one. The two bad behaviors which must be controlled are convergence toward a bubble tree and convergence to a branched cover of a minimal sphere of lower energy. We prevent these difficulties by making estimates on the index of bubble trees and branched covers.

These estimates require a new curvature condition, delta-controlled half-isotropic curvature. In order to better understand this new condition, we study the relationship between metrics with delta-controlled half-isotropic curvature and metrics satisfying the better studied conditions of pinched sectional curvature and pinched flag curvature. We are able to get a basically complete picture of the relationship between these three conditions.

If M is simply connected, then delta-controlled half-isotropic curvature implies that M is diffeomorphic to S n. In this case the constant curvature metric on Sn can be used to compute the low degree O(3)-equivariant cohomology of Map(S2, Sn). This then implies the existence of alpha-energy critical points of low index for generic metrics with delta-controlled half-isotropic curvature, when alpha is sufficiently close to one. Using index estimates to control the bad behavior of these critical points as alpha approaches 1 allows us to prove the existence of many minimal S2 of low index.