Disks area-minimizing in mean convex Riemannian $n$-manifolds

Abstract

We prove an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to D 2 × T n with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in [1] to higher dimensions.