Then there exists a time $\tau>0$, such that $$\begin{aligned}
                    \phi_\tau\left(E^{-1}\left([0,\kappa_0+\varepsilon]\right)\right)\cap
                    K \subset
                    E^{-1}([0,\kappa_0)) \cup V.
                \end{aligned}$$

This is just the deformation lemma; for a proof see for instance [@Str90 Lemma 3.4]. We are now set to complete the proof of the theorem. Define the set of homotopies $$\begin{aligned}
 \Pi := \left\{\beta \,|\,
    \beta:[0,1]\to\Omega,\, \beta_0\in P_n^-, \, \beta_1\in P_n^+\right\}.\end{aligned}$$ Note that $\Pi$ is not empty, as $h\in\Pi$. Furthermore, $\phi_t\circ\beta \in \Pi$ for all $\beta\in\Pi$ and all $t\ge 0$ as the flow respects the convex sets $C_\pm$ and therefore $\phi_t(\beta_0) \in P_n^-$ and $\phi_t(\beta_1) \in P_n^+$ for all $t\ge 0$. Define now $$\begin{aligned}
    \kappa_0 := 
    \inf\limits_{\beta\in\Pi}\max\limits_{\substack{t\in[0,1] \\ \beta_t\in K}}
E(\beta_t) \, .  \end{aligned}$$

By definition of $\kappa$, one has $\kappa_0<\kappa$. For every $\beta\in\Pi$ for time $t_0:=\min\{t\in[0,1]\,|\,\beta_t\notin P_n^-\}$ it holds that $\beta_{t_0}\in
K$ and $E(\beta_{t_0})\ge A$ (as $E$ and $\beta$ are continuous and there exists a sequence $(t_k) \nearrow t_0$ such that $\beta_{t_k}\in P_n^-$ and $\operatorname{dist}(\beta_{t_k},M\setminus(C_- \cup C_+))<\delta$). Consequently, we get $\kappa_0\ge A$. Since $\kappa_0<\kappa$, for $\varepsilon>0$ small enough, the subset $K\cap E^{-1}([0,\kappa_0+\varepsilon])$ is compact and there are only finitely many $S^1$-orbits of closed geodesics inside (we assumed every orbit to be isolated). Let $\{S^1\cdot d_j\}_{1\le j\le l}$ denote the critical circles of 