Minimal surface systems, maximal surface systems and special Lagrangian equations (Q2846983)

Let \(\mathbb R^{n+2}\) be the Euclidean space endowed with the metric \(dx_1^2+dx_2^2+\dots+dx_{n+2}^2\) and let \(\mathbb R^{n+2}_n\) be the pseudo-Euclidean space equipped with the pseudo-Euclidean metric \(dx_1^2+dx_2^2-dx_3^2-\dots-dx_{n+2}^2\). S.~N.~Bernstein proved that the only entire solutions of the minimal surface equation \(0=(1+f^2_y)f_{xx}-2\,f_x\,f_y\,f_{xy}+(1+f^2_x)f_{yy}\) in \(\mathbb R^3\) are affine functions. E.~Calabi extended this result and showed that the only entire maximal graphs in the space-time \(\mathbb L^3\) are spacelike planes. Also, he introduced an interesting duality between the minimal surface equation in \(\mathbb R^3\) and the maximal surface equation in \(\mathbb L^3\).NEWLINENEWLINEIn this paper, the author extends the geometric integrability in the Calabi correspondence to higher codimensions \(n\geq 2\) by constructing the twin correspondence between \(2\)-dimensional minimal graphs having a positive area-angle function in the Euclidean space \(\mathbb R^{n+2}\) and \(2\)-dimensional maximal graphs having the same positive area-angle function in the pseudo-Euclidean space \(\mathbb R^{n+2}_n\). It is shown that this twin correspondence shows that the minimal surface system in \(\mathbb R^{n+2}\) becomes the integrability condition for the maximal surface system in \(\mathbb R^{n+2}_n\). The Bernstein problem for surfaces in higher codimension is to find conditions under which extremal graphs of functions from \(\mathbb R^2\) to \(\mathbb R^n\) are affine functions. Unlike Bernstein's Theorem in \(\mathbb R^3\), for the higher codimensions \(n\geq 2\), there exist plenty of entire \(2\)-dimensional minimal non-planar graphs in \(\mathbb R^{n+2}\). The author studies the generalized Gauss map of minimal surfaces in \(\mathbb R^{n+2}\) to investigate their global properties and to generalize Osserman's Lemma on degenerate Gauss maps of entire \(2\)-dimensional minimal graphs in \(\mathbb R^{n+2}\) which indicates that pairs of their height functions having nowhere zero Jacobian determinant on the whole plane contribute to the degeneracy of their Gauss maps. Also, it allows to prove several Bernstein type theorems for minimal surfaces with codimensions \(n\geq 2\). A simultaneous application of the Harvey-Lawson Theorem on special Lagrangian equations and the generalized Osserman's Lemma yield a geometric proof of Jörgens' Theorem on the \(2\)-variable unimodular Hessian equation. Next, the author constructs the twin correspondence between \(2\)-dimensional minimal graphs in \(\mathbb R^{n+2}\) with a positive area-angle function and \(2\)-dimensional maximal graphs in \(\mathbb R^{n+2}_n\) with the same positive area-angle function. This generalizes the classical Calabi correspondence between minimal graphs in \(\mathbb R^3\) and maximal graphs in \(\mathbb L^3\). The application of this twin correspondence yields a higher codimension extension of Calabi's Theorem in \(\mathbb L^3\). Also, the author introduces the special Lagrangian equation and the split special Lagrangian equation to see why they are geometrically equivalent to each other. Making use of Jörgens' Theorem on the unimodular Hessian equation, the author classifies all entire solutions of these two symplectic Monge-Ampère equations. Finally, the existence of simultaneous conformal coordinate transformations for twin graphs in \(\mathbb R^{n+2}\) and \(\mathbb R^{n+2}_n\) is proved.