MINIMAL SURFACE SYSTEMS, MAXIMAL SURFACE SYSTEMS AND SPECIAL LAGRANGIAN EQUATIONS

We extend Calabi's correspondence between minimal graphs in Euclidean space R-3 and maximal graphs in Lorentz-Minkowski space L-3. We establish the twin correspondence between 2-dimensional minimal graphs in Euclidean space Rn+2 carrying a positive area-angle function and 2-dimensional maximal graphs in pseudo-Euclidean space R-n(n+2) carrying the same positive area-angle function. We generalize Osserman's Lemma on degenerate Gauss maps of entire 2-dimensional minimal graphs in Rn+2 and offer several Bernstein-Calabi type theorems. A simultaneous application of the Harvey-Lawson Theorem on special Lagrangian equations and our extended Osserman's Lemma yield a geometric proof of Jorgens' Theorem on the 2-variable unimodular Hessian equation. We introduce the correspondence from 2-dimensional minimal graphs in Rn+2 to special Lagrangian graphs in C-2, which induces an explicit correspondence from 2-variable symplectic Monge-Ampere equations to the 2-variable unimodular Hessian equation.