Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, open set $U$ with $C^0$ boundary. Moreover, we assume that $M\setminus U$ is connected and either the relative homotopy set $\pi_1(M,M\setminus U)=0$ or the image of the homomorphism $\pi_1(M\setminus U)\rightarrow \pi_1(M)$ (induced by the inclusion $M\setminus U\hookrightarrow M$) is a finite index subgroup or a normal subgroup of $\pi_1(M)$ or the relative homology group $H_1(M,M\setminus U)\neq 0$. Then there exists a non-trivial closed geodesic on $M$. This partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman.PDF: Existence of closed geodesics on certain non-compact Riemannian manifolds.pdf
