UC Santa Barbara

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 $S^n$ can be used to compute the low degree $O(3)$-equivariant cohomology of $Map(S^2,S^n)$.

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 $S^2$ of low index.