Rigidity for general semiconvex entire solutions to the sigma-2 equation

Abstract

We show that every general semiconvex entire solution to the sigma-2 equation is a quadratic polynomial. A decade ago, this result was shown for almost convex solutions. Two decades ago, this result was obtained in three dimensions, as a by-product of the work on special Lagrangian equations. Warren’s rare saddle entire solution in 2014 confirmed the necessity of the semiconvexity assumption.