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

TL;DR

This paper establishes an inequality relating area and boundary length for immersed disks in certain mean convex Riemannian manifolds, with a rigidity result characterizing the equality case, generalizing previous work to higher dimensions.

Contribution

It extends area-minimizing disk inequalities and rigidity results to higher-dimensional mean convex Riemannian manifolds with specific curvature and topological conditions.

Findings

Proves an inequality involving area and boundary length of immersed disks.

Establishes a rigidity result for the equality case when the boundary is totally geodesic.

Generalizes a known result to higher dimensions.

Abstract

We prove the validity of 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 $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambr\'ozio in \cite{AMB} to higher dimensions.