I got stuck by the 3.2 Lemma of An isoperimetric estimate for the Ricci flow on the two-sphere, which is collected in Collected papers on Ricci flow.
As picture below, $L$ is the length of $\Lambda^t$. I think the contrary of Lemma 3.2 is that:
There is a surface and $\eta>0$ such that $\forall E >0$ there is $\Lambda^0$ such that $$ C_S(\Lambda^t)\le C_S(\Lambda^0) \tag{1} $$ and $$ \int_{\Lambda^t} k^2 dt \le E \tag{2} $$ cannot be met simultaneously.
I can't understand the proof of red 2. The Grayson's theorem should be the Theorem 0.1 of Shortening embedded curves. Which state that under curve shortening (shrinking) flow, if the flow has finit time solution, then the curve converges to a point. Otherwise (namely the flow has infinite time solution), the curve converges to geodesic.
But, in Grayson's paper, there is not theory about the decrease of $C_S(\Lambda^t)$ and the curve shortening flow.
Besides, the Lemma 3.2 seemly be contrary with the red 1.
And, I also try to prove the Lemma 3.2. What I try:
case 1: $(2)$ false, namely, for any $E>0$, there is $$ \int_{\Lambda^t} k^2 ds >E. $$ Since the length of $\Lambda^t$ decrease, we have $k^2$ is unbounded on $\Lambda^t$. Namely, the flow has finite time solution. By the Theorem 0.1 of Grayson, the curve convergence to a point, which is contrary with $L\ge \lambda$, where $L$ is the length of $\Lambda^t$.
case 2: $(1)$ false. Since from the red 1, when $(1)$ false, $(2)$ is right. So I don't know how to get the contradiction. (In fact, I spend much time, but fail. I doubt whether my understand is right, since the proof seemly not be complex.)
