Generic regularity of free boundaries for the obstacle problem

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in R-n. By classical results of Caffarelli, the free boundary is C-infinity outside a set of singular points. Explicit examples show that the singular set could be in general(n-1)-dimensional-that is, as large as the regular set. Our main result establishes that, generically, the singular set has zeroHn-4 measure (in particular, it has codimension 3 inside the free boundary). Thus, for n <= 4, the free boundary is generically a C-infinity manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n <= 4.