Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds

TL;DR

This paper develops curvature estimates and a compactness theorem for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds, advancing understanding of their geometric properties and potential for min-max theory applications.

Contribution

It introduces new curvature estimates and a compactness theorem for free boundary minimal hypersurfaces in wedge-shaped manifolds, extending classical results to these singular spaces.

Findings

Proved a compactness theorem for hypersurfaces with curvature and area bounds.

Established a Bernstein-type theorem for stable free boundary minimal hypersurfaces in Euclidean wedges.

Derived curvature estimates under wedge angle conditions.

Abstract

In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen-Simon-Yau's estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold with certain wedge angle assumptions. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.