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):
They use a notation like $L^{2}_{1} (S^2, M)$. I cannot find a definition of this anywhere.
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 [
\[ W(g) = \lim_{j \to \infty}max_{s \in [0,1]} E(\gamma^{j}_{s}) \]W(g) = \lim_{j \to \infty}max_{s \in [0Furthermore,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 $S^2 there exist -$s_{j} \in [0,1]$ and branched conformal minimal immersions {x_1$u_0, ..., u_m : S^2 \rightarrow M$ with index at most one so that,.... as $j \rightarrow \infty$,x_k}$ 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?
