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

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 ${\mathrm{\Sigma }}^n$ 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.