Geometry & Topology

We prove a Michael-Simon inequality for varifolds of every dimension and codimension with respect to anisotropic energies. In particular, this resolves the problem for every convex even hypersurface anisotropy. Using work of Allard, this yields the regularity of hypersurfaces with bounded anisotropic mean curvature in every dimension. The proof relies on a projection method for anisotropic stress measures as well as the recent resolution of the vanishing mass conjecture. This enables one to conclude density bounds and compactness properties for rectifiable varifolds with uniformly bounded anisotropic first variation without the use of a monotonicity formula. This is joint work with R. Tsiamis and A. De Rosa and R. Tsiamis.