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

In (the surface of) a convex polytope P-3 in R-4, an area-minimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an area-minimizing 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.