Let $a,b,c\in\mathbin{\mathbb{R}}$. Then there exist unique functions $u,v:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$ such that

$$\begin{split}N=\Bigl\{(z_{1}&,z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:\mathop{\rm Re}(z_{1}z_{2})=u\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr),\\ &\mathop{\rm Im}(z_{3})=v\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr),\quad|z_{1}|^{2}-|z_{2}|^{2}=a\Bigr\}\end{split}$$

(40)

is the special Lagrangian $3$-fold $N_{a,b+ic}$ of Definition 5. Furthermore:

• (a)

  $u,v$ are smooth on $\mathbin{\mathbb{R}}^{2}$ and satisfy (33), except at $(b,0)$ when $a=0$, where they are only continuous.

• (b)

  $u(x,y)>0$ when $x>b$ for all $y$, and $u(b,y)=0$ for all $y$, and $u(x,y)<0$ when $x<b$ for all $y$.

• (c)

  $v(x,y)<c$ when $y>0$ for all $x$, and $v(x,0)=c$ for all $x$, and $v(x,y)>c$ when $y<0$ for all $x$.

• (d)

  $u(x,0)=(x-b)\bigl((x-b)^{2}+|a|\bigr)^{1/2}$ for all $x$.

• (e)

  $v(b,y)=c-y\Bigl(\frac{1}{2}|a|+\sqrt{y^{2}+\frac{1}{4}a^{2}}\,\,\Bigr)^{-1/2}$ for all $y$.