Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. The regularity theory for its critical points, anisotropic minimal (hyper)surfaces, is significantly more challenging than the area functional case, mainly due to the lack of a monotonicity formula. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface and optimally regular minimal hypersurfaces for elliptic integrands in closed Riemannian manifolds through min-max construction. This confirms a conjecture by Allard in 1983. 

The talk is based on joint work with Guido De Philippis and Antonio De Rosa, and can be seen as Part II of the previous talk given by Antonio this November.