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 C^1, 1 across their singular set, and pointed out that, if correct, this regularity statement could not be improved to C² in view of a certain example. In this paper, we prove Leon Simon’s conjecture.
Klaus Ecker (Thu,) studied this question.