Hyperbolic Unfoldings of Minimal Hypersurfaces

We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Sigma. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H \ Sigma. These are canonical conformal deformations of H \ Sigma into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Sigma. These new concepts and results naturally extend to the larger class of almost minimizers.