Hessian estimates for the sigma-2 equation in dimension four

Abstract

We derive a priori interior Hessian estimates and interior regularity for the ${\sigma }_2$ equation in dimension four. Our method provides respectively a new proof for the corresponding three dimensional results and a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher $n\ge 5$ dimensions.

Ann. of Math, to appear