Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces inR n

Abstract

In (the surface of) a convex polytope P 3 inR 4,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films.