For complete **Riemannian** manifolds of bounded sectional curvature the answer is YES by
L.F. Tam. *Exhaustion functions on complete manifolds* MR2648946. Then the answer for
the **Hilbert metric** on a properly convex manifold is also YES since such a manifold admits
a Riemannian metric of bounded curvature that is bi-Lipshitz to the Hilbert metric