A Frankel type theorem in Euclidean and hyperbolic spaces.

We prove that a connected mean convex region in R^(n+1) with at least two boundary components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature must decay at infinity, and generalize the results to hyperbolic space. The strategy is a stablility argument on minimizing geodesics inspired by Frankel.