UC Santa Barbara
Index Estimates and Existence of Minimal Surfaces in Manifolds with Controlled Curvature
- Author(s): Ream, Robert
- Advisor(s): Moore, John Douglas
- et al.
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.