Optimal boundary regularity for a singular Monge-Ampère equation

In this paper we study the optimal global regularity for a singular Monge-Ampère type equation which arises from a few geometric problems. We find that the global regularity does not depend on the smoothness of domain, but it does depend on the convexity of the domain. We introduce (a,η) type to describe the convexity. As a result, we show that the more convex is the domain, the better is the regularity of the solution. In particular, the regularity is the best near angular points.