Regularity of area-minimizing surfaces in 3D polytopes and of invariant   hypersurfaces in R^n
  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 nu and soap films. 
Frank Morgan