Brendle’s Inequality on Static Manifolds

We generalize Brendle’s geometric inequality considered in Brendle (Publ Math Inst Hautes Études Sci 117:247–269, 2013) to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chruściel and Simon (J Math Phys 42(4):1779–1817, 2001).