Proof of lemma 1.2

3.1 In this section we prove the following statement

Let $M$ be a closed three-manifold, and let $(M,g^t)$ be a smooth solution to the Ricci flow on a finite time interval $[t_0,t_1].$ Suppose that $\Gamma\subset\Lambda M$ is a compact family. Then for any $\xi>0$ one can construct a continuous deformation $\Gamma ^t, t\in [t_0,t_1], \Gamma ^{t_0}=\Gamma ,$ such that for each curve $c\in\Gamma$ either the value $A(c^{t_1},g^{t_1})$ is bounded from above by $\xi$ plus the value at $t=t_1$ of the solution to the ODE $\frac{d}{dt}w(t)=-2\pi-\frac{1}{2}R^t_{\mathrm{min}}w(t)$ with the initial data $\ \ w(t_0)=A(c^{t_0},g^{t_0}),\ \$ or $L(c^{t_1})\le \xi;$ moreover, if $c$ was a constant map, then all $c^t$ are constant maps.

It is clear that our statement implies lemma 1.2, because a family consisting of very short loops can not represent a nontrivial relative homotopy class.

3.2 As a first step of the proof of the statement we can replace $\Gamma$ by a family, which consists of piecewise geodesic loops with some large fixed number of vertices and with each segment reparametrized in some standard way to make the parametrizations of the whole curves twice continuously differentiable.

Now consider the manifold $M_{\lambda }=M\times\mathbb{S}^1_{\lambda }, 0<\lambda <1,$ and for each $c\in\Gamma$ consider the smooth embedded closed curve $c_{\lambda }$ such that $p_1c_{\lambda }(x)=c(x)$ and $p_2c_{\lambda }(x)=\lambda x\ \mathrm{mod}\ \lambda ,$ where $p_1$ and $p_2$ are projections of $M_{\lambda }$ to the first and second factor respectively, and $x$ is the parameter of the curve $c$ on the standard circle of length one. Using 2.3 we can construct a solution $c_{\lambda }^t, t\in [t_0,t_1]$ to the curve shortening flow with initial data $c_{\lambda }.$ The required deformation will be obtained as $\Gamma ^t=p_1\Gamma ^t_{\lambda }$ (where $\Gamma ^t_{\lambda }$ denotes the family consisting of $c^t_{\lambda }$) for certain sufficiently small $\lambda >0.$ We'll verify that an appropriate $\lambda$ can be found for each individual curve $c,$ or for any finite number of them, and then show that if our $\lambda$ works for all elements of a $\mu$-net in $\Gamma,$ for sufficiently small $\mu>0,$ then it works for all elements of $\Gamma .$

3.3 In the following estimates we shall denote by $C$ large constants that may depend on metrics $g^t,$ family $\Gamma$ and $\xi,$ but are independent of $\lambda ,\mu$ and a particular curve $c.$

The first step in 3.2 implies that the lengths and total curvatures of $c_{\lambda }$ are uniformly bounded, so by 2.1 the same is true for all $c^t_{\lambda }.$ It follows that the area swept by $c^t_{\lambda }, t\in [t',t'']\subset [t_0,t_1]$ is bounded above by $C(t''-t'),$ and therefore we have the estimates $A(p_1c^t_{\lambda },g^t)\le C, A(p_1c^{t''}_{\lambda },g^{t''})-A(p_1c^{t'}_{\lambda },g^{t'})\le C(t''-t').$

3.4 It follows from (5) that $\int_{t_0}^{t_1} \int k^2 dsdt \le C$ for any $c^t_{\lambda }.$ Fix some large constant $B,$ to be chosen later. Then there is a subset $I_B(c_{\lambda })\subset [t_0,t_1]$ of measure at least $t_1-t_0-CB^{-1}$ where $\int k^2 ds\le B,$ hence $\int kds \le \epsilon$ on any arc of length $\le \epsilon ^2 B^{-1}.$ Assuming that $c^t_{\lambda }$ are at least that long, we can apply 2.2 and construct another subset $J_B(c_{\lambda })\subset [t_0,t_1]$ of measure at least $t_1-t_0-CB^{-1},$ consisting of finitely many intervals of measure at least $C^{-1}B^{-2}$ each, such that for any $t\in J_B(c_{\lambda })$ we have pointwise estimates on $c^t_{\lambda }$ for curvature and higher derivatives, of the form $k\le CB,...$

Now fix $c, B,$ and consider any sequence of $\lambda \to 0.$ Assume again that the lengths of $c^t_{\lambda }$ are bounded below by $\epsilon ^2 B^{-1},$ at least for $t\in [t_0,t_2],$ where $t_2=t_1-B^{-1}.$ Then an elementary argument shows that we can find a subsequence $\Lambda_c$ and a subset $J_B(c)\subset [t_0,t_2]$ of measure at least $t_1-t_0-CB^{-1},$ consisting of finitely many intervals, such that $J_B(c)\subset J_B(c_{\lambda })$ for all $\lambda \in \Lambda_c.$ It follows that on every interval of $J_B(c)$ the curve shortening flows $c^t_{\lambda }$ smoothly converge (as $\lambda \to 0$ in some subsequence of $\Lambda_c$ ) to a curve shortening flow in $M.$

Let $w_c(t)$ be the solution of the ODE $\frac{d}{dt}w_c(t)=-2\pi-\frac{1}{2}R^t_{\mathrm{min}}w_c(t)$ with initial data $w_c(t_0)=A(c,g^{t_0}).$ Then for sufficiently small $\lambda \in\Lambda_c$ we have $A(p_1c^t_{\lambda },g^t)\le w_c(t)+\frac{1}{2}\xi$ provided that $B>C\xi^{-1}.$ Indeed, on the intervals of $J_B(c)$ we can estimate the change of $A$ for the limit flow using the minimal disk argument as in 1.2, and this implies the corresponding estimate for $p_1c^t_{\lambda }$ if $\lambda \in\Lambda_c$ is small enough, whereas for the intervals of the complement of $J_B(c)$ we can use the estimate in 3.3.

On the other hand, if our assumption on the lower bound for lengths does not hold, then it follows from (5) that $L(c^{t_2}_{\lambda })\le CB^{-1}\le\frac{1}{2}\xi.$

3.5 Now apply the previous argument to all elements of some finite $\mu$-net $\hat{\Gamma }\subset \Gamma$ for small $\mu>0$ to be determined later. We get a $\lambda >0$ such that for each $\hat{c}\in\hat{\Gamma }$ either $A(p_1\hat{c}_{\lambda }^{t_1},g^{t_1})\le w_{\hat{c}}(t_1)+\frac{1}{2}\xi$ or $L(\hat{c}_{\lambda }^{t_2})\le \frac{1}{2}\xi.$ Now for any curve $c\in\Gamma$ pick a curve $\hat{c}\in\hat{\Gamma },$ $\mu$-close to $c,$ and apply the result of 2.4. It follows that if $A(p_1\hat{c}_{\lambda }^{t_1},g^{t_1})\le w_{\hat{c}}(t_1)+\frac{1}{2}\xi$ and $\mu \le C^{-1}\xi,$ then $A(p_1c_{\lambda }^{t_1},g^{t_1})\le w_c(t_1)+\xi.$ On the other hand, if $L(\hat{c}_{\lambda }^{t_2})\le \frac{1}{2}\xi,$ then we can conclude that $L(c_{\lambda }^{t_1})\le \xi$ provided that $\mu>0$ is small enough in comparison with $\xi$ and $B^{-1}.$ Indeed, if $L(c_{\lambda }^{t_1})>\xi,$ then $L(c_{\lambda }^t)>\frac{3}{4}\xi$ for all $t\in [t_2,t_1];$ on the other hand, using (5) we can find a $t\in [t_2,t_1],$ such that $\int k^2ds \le CB$ for $c^t_{\lambda };$ hence, applying 2.5, we get $L(\hat{c}_{\lambda }^t)>\frac{2}{3}\xi$ for this $t,$ which is incompatible with $L(\hat{c}_{\lambda }^{t_2})\le \frac{1}{2}\xi.$ The proof of the statement 3.1 is complete.