A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

In dimension n⩾3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation detD2u=k(x,u,Du) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n⩾3. A corollary is that if k⩾0 vanishes only at nondegenerate critical points, then a C2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be C3,1-2n+ɛ when not smooth.