= Special Lagrangian graph equation
{title2=$\Delta f=\det D^2f\quad\text{in dimension three at phase zero}$}
For a graph $z=x+i\nabla f$ in $\mathbb C^3$, symmetry of the <Hessian matrix> makes the graph Lagrangian. Pulling back $\Omega=dz^1\wedge dz^2\wedge dz^3$ gives $\det(I+iD^2f)$. Its imaginary part is $\operatorname{tr}D^2f-\det D^2f$, while its real part is $1-\sigma_2(D^2f)$. 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
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