Minimal surface systems, maximal surface systems and special Lagrangian equations

Abstract:

We extend Calabi’s correspondence between minimal graphs in Euclidean space ${\mathbb {R}}^{3}$ and maximal graphs in Lorentz-Minkowski space ${\mathbb {L}}^{3}$. We establish the twin correspondence between $2$-dimensional minimal graphs in Euclidean space ${\mathbb {R}}^{n+2}$ carrying a positive area-angle function and $2$-dimension- al maximal graphs in pseudo-Euclidean space ${\mathbb {R}}^{n+2}_{n}$ carrying the same positive area-angle function.

We generalize Osserman’s Lemma on degenerate Gauss maps of entire $2$-dimensional minimal graphs in ${\mathbb {R}}^{n+2}$ 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 Jörgens’ Theorem on the $2$-variable unimodular Hessian equation.

We introduce the correspondence from $2$-dimensional minimal graphs in ${\mathbb {R}}^{n+2}$ to special Lagrangian graphs in ${\mathbb {C}}^{2}$, which induces an explicit correspondence from $2$-variable symplectic Monge-Ampère equations to the $2$-variable unimodular Hessian equation.