A notation and a question from Colding-Minicozzi's paper

Question

Hi, of late I am trying to read the Colding-Minicozzi paper "Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman". I have a couple of questions (I am somewhat of a novice in this Ricci flow business):

1. They use a notation like $L^{2}_{1} (S^2, M)$. I cannot find a definition of this anywhere.

2. On page 4, Prop 3.1, they write: "Given a metric $g$ on $M$ and a non-trivial $[\beta]\in \pi_{1}(C^{0}\cap L^{2}_{1}(S^2, M))$, there exists a sequence of sweep-outs $\gamma^{j} : [0,1] \rightarrow C^{0}\cap L^{2}_{1} (S^2, M)$ with $\gamma^j \in [\beta]$ so that \[ W(g) = \lim_{j \to \infty}max_{s \in [0,1]} E(\gamma^{j}_{s}) \] Furthermore, there exist $s_{j} \in [0,1]$ and branched conformal minimal immersions $u_0, ..., u_m : S^2 \rightarrow M$ with index at most one so that, as $j \rightarrow \infty$, the maps $\gamma^{j}_{s_{j}}$ converge to $u_0$ weakly in $L^{2}_{1}$ and uniformly on compact subsets of $S^2 - \{x_1,....,x_k\}$ and....."

The question is: why index at most one?

Answer 1

1. This is the Sobolev space of $L^2$ functions with one derivative in $L^2$. A more common notation for that space is $W^{1,2}$.

2. Let me give an example first: On $S^3$ the min-max is attained by the equatorial $S^2$. This $S^2$ has index 1, meaning that there is exactly one unstable deformation decreasing the area. In general, the idea is that the max is taken over a 1-parameter family, causing one unstable deformation. To turn this into a proof you have to understand the min-max / sweep-out construction quite well (see the two papers by CM and references therein). Also, as far as I remember, you actually don't need the index bound to prove the finite extinction result.