Min-Max Construction of Anisotropic Minimal Hypersurfaces

We use the min-max construction to find closed hypersurfaces which are stationary with respect to anisotropic elliptic integrands in any closed n-dimensional manifold . These surfaces are regular outside a closed set of zero n-3 dimension. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture posed by Allard. the talk is based on a joint work with A. De Rosa and Y. Li.