a lift of some point of $c(S^1)$ and define $q_n := q_0 + (n,0)$. Then the first homology group $H_1({{\mathbb{R}}}^2\setminus\{q_n\}_{n\in{{\mathbb{Z}}}})$ is the free abelian group with generators $(g_n)_{n\in{{\mathbb{Z}}}}$, and by construction the class of $\tilde{\gamma}_n$ is $g_1 + g_2 + \cdots + g_n$. Since the covering transformations of ${{\mathbb{R}}}^2\setminus\{q_n\}_{n\in{{\mathbb{Z}}}}\to S^1\times{{\mathbb{R}}}\setminus \pi(q_0)$, which form a group isomorphic to ${{\mathbb{Z}}}$, act on the first homology group by $k\cdot g_i = g_{i+k}$, we see that, if $n\neq m$ and $k,l\in{{\mathbb{Z}}}^*$, the iterated loops $\gamma_n^k$ and $\gamma_m^l$ are not freely homotopic in $M\setminus \pi(q_0)$ and hence in the unbounded component $C$ of $M\setminus c(S^1)$. We want to apply the $(-\nabla E)$-flow to $\gamma_n$, $n\in{{\mathbb{N}}}$. Since $c$ is a closed geodesic, the unbounded component $C$ of $M\setminus
c(S^1)$ is a locally convex neighborhood. Hence $\Lambda C$ is preserved by the $(-\nabla E)$-flow. Since $M$ is complete, the set of points at distance less than $\ell >0$ from $c(S^1)$ is compact. Moreover, the image of $\gamma_n$ by the flow (when defined) is kept inside this compact set for $\ell = \frac{1}{2}\sqrt{E(\gamma_n)}$. Thus one can apply the $(-\nabla E)$-flow to $\gamma_n$ at all time $t>0$ and ultimately get a closed geodesic of $C$ homotopic to $\gamma_n$. We 