Min-max Minimal Hypersurfaces with Obstacle

TL;DR

This paper develops a min-max theory for hypersurfaces with obstacles in manifolds with boundary, establishing regularity and existence results for hypersurfaces with controlled singularities and prescribed mean curvature.

Contribution

It introduces a Schoen-Simon-type regularity theorem for varifolds satisfying a variational inequality and proves the existence of hypersurfaces realizing the min-max width with boundary conditions.

Findings

Existence of a $C^{1,1}$ hypersurface with singularities of codimension at least 7.

Regularity result for varifolds satisfying a variational inequality.

Construction of hypersurfaces with prescribed mean curvature in manifolds with boundary.

Abstract

We study min-max theory for area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen-Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restrict to a stable minimal hypersurface in the interior. Based on this, we show that for any admissible family of sweepouts $\Pi$ in a compact manifold with boundary, there always exists a closed $C^{1,1}$ hypersurface with codimension$\geq 7$ singular set in the interior and having mean curvature pointing outward along boundary realizing the width $\textbf{L}(\Pi)$.