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?

  • $\begingroup$ I strongly suggest you look at this much more detailed exposition of the same result by Colding and Minicozzi arxiv.org/pdf/0707.0108 $\endgroup$
    – YangMills
    Mar 9 '12 at 21:38
  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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.