ncts

Given a compact Riemannian manifold $M^{n+1}$ with smooth boundary $\partial M$ , a free boundary minimal hypersurface (FBMH) in M is a critical point for the area functional with respect to the variations that constrain its boundary to lie in $\partial M$ but is otherwise free to vary. When the ambient manifold $M$ has a stratified singular structure (e.g. a polyhedron), a natural question is whether there is a FBMH in $M$ with a compatible stratified singular structure (e.g. a minimal polygon whose k-skeleton lies in $M$ 's (k+1)-skeleton). In this talk, I will introduce related concepts of FBMHs in a class of spaces we call locally wedge-shaped manifolds, whose boundaries are formed by faces and edges. By extending Almgren-Pitts min-max theory, we show the existence of a $C^{2,\alpha}$ FBMH in any locally wedge-shaped manifolds of dimension $3 \leq n+1 \leq 6$ with either acute wedge angle or right wedge angle coupled with a certain additional assumption. This talk is based on the joint work with Liam Mazurowski.