above.

A technical issue is given by the fact that the gradient flow of $-\nabla E$ may not be defined for all times as the sublevel sets of $E|_\Omega$ are not compact. Ultimately we are only going to be interested in curves intersecting the compact set $M\setminus(C_-\cup
        C_+)$, i.e. the subset $$\begin{aligned}
            K:=\{\gamma\in\Omega \,|\, \gamma(S^1)\cap (M\setminus(C_-\cup
            C_+))\neq\varnothing\}
        \end{aligned}$$ of $\Omega$. We introduce a smooth function $g:\Omega\to[0,1]$ with the property that $$\begin{cases}
                g(\gamma) = 1 &\text{if } \operatorname{dist}(\gamma(S^1), K)\le \frac 12
                \kappa^{\frac 12}, \\
                g(\gamma) = 0 &\text{if } \operatorname{dist}(\gamma(S^1), K)> \frac 32 \kappa^{\frac 12}.
            \end{cases}$$ Then the flow $\phi_t$ of $-g\nabla E$ is defined for all times $t\ge 0$ and coincides with the negative gradient flow for curves in $K$. Two crucial observations about the set $K$ are the following: firstly, for all $\bar\kappa<\kappa$ the set $K \cap
        E^{-1}([0,\bar\kappa])$ is compact. Secondly, if $\phi_t(\gamma) \in K$ for some $\gamma\in\Omega$ and some time $t\ge 0$, we already have $\gamma\in K$ as the flow $\phi_t$ respects the convex sets $C_\pm$. From this it follows:

ii) \[it:existsT\] Let $0<\kappa_0<\kappa_0+\varepsilon<\kappa$. Let $V$ denote a neighborhood of the closed geodesics in $K$ of energy $\kappa_0$. Suppose there is no closed geodesic in $K \cap
                E^{-1}((\kappa_0,\kappa_0+\varepsilon])$. 