the boundary were totally geodesic, then $\partial C_\pm$ would be parametrised by $c$ which is impossible for $c$ is contractible. We can thus apply Theorem \[thm:Bangert\] in this case.

We now assume that $M\setminus c(S^1)$ has only one unbounded component $C$. Let us identify $M$ with $S^1\times{{\mathbb{R}}}$ in the remaining of this proof in order to fix notations. Let $\pi : {{\mathbb{R}}}^2 \to S^1\times{{\mathbb{R}}}$ be the universal cover of $S^1\times{{\mathbb{R}}}$. By compactness of $c(S^1)$, there exists $A>0$ such that $c(S^1)\subset S^1\times
(-A,A)$. Let $y_0 > A$, since $S^1\times (-\infty,-A)$ and $S^1\times (A,+\infty)$ belong to the same component of $M\setminus c(S^1)$, there exists a smooth path $\alpha:[0,1]\to M\setminus c(S^1)$ such that $\alpha(0)=(0,-y_0)$ and $\alpha(1)=(0,y_0)$. Let $\beta_0$ be the smooth lift of $\alpha$ in ${{\mathbb{R}}}^2$ such that $\beta_0(0) = (0,-y_0)$ and $\beta_0(1) = (n_0,y_0)$ for some $n_0 \in {{\mathbb{Z}}}$ that we can take equal to $n_0=0$ by chaining $\alpha$ with $t\mapsto (tn_0\bmod 1,y_0)$. Let $\delta_{n,\pm}:[0,1]\to{{\mathbb{R}}}^2$ be the path $t\mapsto (nt,\pm y_0)$ and $\beta_n:[0,1]\to{{\mathbb{R}}}^2$ be the family of lifts $\beta_n := (n,0) + \beta_0$, $n\in{{\mathbb{N}}}$. We define the family of loops $\tilde{\gamma}_n\in\Lambda{{\mathbb{R}}}^2$ by $$\tilde{\gamma}_n := \beta_0\cdot \delta_{n,+} \cdot \beta_n^{-1} \cdot
    \delta_{n,-}^{-1}.$$ They project to $\gamma_n := \pi\circ\tilde{\gamma}_n$ in $M\setminus c(S^1)$. Let $q_0\in{{\mathbb{R}}}^2$ be 