Convex functions with symplectic Hessian

We prove a third-order derivative estimate for convex solutions to the real Monge-Ampère equation $\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(u)=1$ on an open set in ${\mathbb{R}}^{2m}$ under the additional assumption that $\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(u)$ lies in $\mathrm{S}\mathrm{p}(2m,\mathbb{R})$ at every point. Our method is a geometric interpretation and extension to higher dimensions of Nitsche’s classical proof of the Bernstein theorem for the real Monge-Ampère equation on ${\mathbb{R}}^2$. For $m=1$ we also improve Nitsche’s constant as well as some estimates due to Calabi, and we construct examples of solutions with interesting geometric behavior.