Curvature estimates for stable free boundary minimal hypersurfaces

TL;DR

This paper establishes uniform curvature bounds for stable free boundary minimal hypersurfaces under area constraints, extending classical interior estimates to free boundary cases and enabling compactness results crucial for min-max theory.

Contribution

It generalizes Schoen-Simon-Yau curvature estimates to free boundary minimal hypersurfaces and introduces a monotonicity formula for free boundary submanifolds.

Findings

Proves uniform curvature estimates for stable free boundary minimal hypersurfaces.

Derives a smooth compactness theorem for free boundary minimal hypersurfaces.

Establishes a monotonicity formula for free boundary minimal submanifolds.

Abstract

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For the case of $3$-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without any assumption on the area bound. This generalizes Schoen's interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.