Periodic geodesic conjecture for manifolds with locally convex ends

Let $M$ be a complete non-compact Riemannian manifold with locally convex ends, meaning that each end can be cut off by a bounded hypersurface that is locally convex toward infinity. Periodic geodesic conjecture. There exists a non-constant periodic geodesic on $M$. This extends the Fet–Lyusternik theorem, which guarantees a periodic geodesic on every closed Riemannian manifold; the claim concerns the corresponding existence problem in the non-compact setting with locally convex ends.