Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds

TL;DR

This paper develops a min-max theory for finding free boundary minimal hypersurfaces in locally wedge-shaped manifolds, allowing for boundaries and edges, and proves existence results in certain geometric configurations.

Contribution

It introduces a min-max framework for area minimization in wedge-shaped manifolds and establishes existence of smooth free boundary minimal hypersurfaces with specific wedge angles.

Findings

Existence of free boundary minimal hypersurfaces in wedge-shaped manifolds.

Smoothness of hypersurfaces up to edges and faces.

Results for acute and 90-degree wedge angles.

Abstract

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges. Fix a dimension $3 \le n+1 \le 6$. As our main theorem, we prove that every compact locally wedge-shaped manifold $M^{n+1}$ with acute wedge angles contains a locally wedge-shaped free boundary minimal hypersurface $\Sigma^n$ which is smooth in its interior and on its faces and is $C^{2,\alpha}$ up to and including its edge. We can also handle the case of 90 degree wedge angles under an additional assumption.