Minimal surface system in Euclidean four-space

Abstract

Generalizing the Cauchy-Riemann equations, we construct the Osserman system of the first order for a pair $\left(f(x,y),g(x,y)\right)$ of two $\mathbb{R}$-valued functions on the domain $\Omega \subset {\mathbb{R}}^2$. The graph $\left\{\left(x,y,f(x,y),g(x,y)\right)\in {\mathbb{R}}^4|(x,y)\in \Omega \right\}$ becomes a minimal surface in ${\mathbb{R}}^4$, whose generalized Gauss map lies on the intersection of a hyperplane of the complex projective space ${\mathbb{CP}}^3$ and the complex cone ${z_1}^2+{z_2}^2+{z_3}^2+{z_4}^2=0$. We present two applications of the Lagrangian potential on minimal graphs in ${\mathbb{R}}^3$. First, we deform a minimal graph ${\Sigma }_0$ in ${\mathbb{R}}^3$ to the one parameter family of the two dimensional minimal graph ${\Sigma }_{\lambda }$ in ${\mathbb{R}}^4$ with the invariance of the metric ${\left(\det \left({\mathbf{g}}_{{\Sigma }_{\lambda }}\right)\right)}^{-\frac{1}{2}}{\mathbf{g}}_{{\Sigma }_{\lambda }}$. Second, we construct the three dimensional special Lagrangian graphs in ${\mathbb{R}}^6={\mathbb{C}}^3$.