Special Lagrangian graph equation ($\Delta f=\det D^{2}f\text{in dimension three at phase zero}$)

For a graph $z=x+i\nabla f$ in ${\mathbb{C}}^3$, symmetry of the Hessian matrix makes the graph Lagrangian. Pulling back $\Omega =d{z}^1\wedge d{z}^2\wedge d{z}^3$ gives $\det (I+i{D}^{2}f)$. Its imaginary part is $\mathrm{tr}{D}^{2}f-\det D^{2}f$, while its real part is $1-{\sigma }_2(D^{2}f)$. Vanishing of the imaginary part and positivity of the real part give the displayed equation and oriented branch. The equation is nonlinear even though its linearization near a flat graph is the Laplace equation.

// Target: string-theory.bigb