Optimal Regularity of Area Maximizing Hypersurfaces in Minkowski Space

In 1982, Bartnik and Simon proved that area maximizing hypersurfaces are smooth, strictly spacelike and maximal (mean curvature zero) away from a singular set which consists of light rays connecting boundary points. Around the same time, Leon Simon conjectured that area maximizing hypersurfaces ought to be of class C1,1 across their singular set, and pointed out that, if correct, this regularity statement could not be improved to C^2 in view of a certain example. In this paper, we prove Leon Simon’s conjecture.