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 $\left\{\, \left(x, y, f(x, y), g(x,y) \right) \in {\mathbb{R}}^{4} \, \vert \, (x,y) \in \Omega \, \right\}$ 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 ${\mathbb{R}}^{6}={\mathbb{C}}^{3}$.