Regularity of subelliptic Monge–Ampère equations

In dimension n⩾3, for k≈|x|2m that can be written as a sum of squares of smooth functions, we prove that a C2 convex solution u to a subelliptic Monge–Ampère equation detD2u=k(x,u,Du) is itself smooth if the elementary (n−1)st symmetric curvature kn−1 of u is positive (the case m⩾2 uses an additional nondegeneracy condition on the sum of squares). Our proof uses the partial Legendre transform, Calabi's identity for ∑uijσij where σ is the square of the third order derivatives of u, the Campanato method Xu and Zuily use to obtain regularity for systems of sums of squares of Hörmander vector fields, and our earlier work using Guan's subelliptic methods.