Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

TL;DR

This paper constructs families of constant mean curvature hypersurfaces in a domain that concentrate along a minimal submanifold of the boundary as their mean curvature increases, intersecting the boundary perpendicularly.

Contribution

It demonstrates the existence of large constant mean curvature hypersurfaces concentrating along boundary submanifolds, extending understanding of geometric concentration phenomena.

Findings

Existence of hypersurfaces with large constant mean curvature concentrating along submanifolds.

Hypersurfaces intersect the boundary orthogonally.

Concentration behavior as mean curvature tends to infinity.

Abstract

Given a domain $\Omega$ of $\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\pa \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly.