# Curvature estimates for stable free boundary minimal hypersurfaces

Research paper by Martin Li, Xin Zhou

Indexed on: 08 Nov '16Published on: 08 Nov '16Published in: arXiv - Mathematics - Differential Geometry

#### 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 famous Schoen-Simon-Yau interior curvature estimates \cite{SSY75} 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 in \cite{LiZ16}. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension.