Disks area-minimizing in mean convex Riemannian n-manifolds

2020/06/21 by Ezequiel Barbosa, Barbosa, Ezequiel, Franciele Conrado +1 · 1 citation
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2006.11788

openalex publication_date 2020/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbD²× Tⁿ with nonzero degree, where \mathbbD² is a disk and Tⁿ 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 Ambrózio in \citeAMB to higher dimensions.

Citations

• Free boundary minimal surfaces in the unit ball : recent advances and open questions

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