In [37, 38, 39] we study SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ invariant under the ${\rm U}(1)$-action

$${\rm e}^{i\theta}:(z_{1},z_{2},z_{3})\longmapsto({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3})\quad\text{for ${\rm e}^{i\theta}\in{\rm U}(1)$.}$$

The three papers are surveyed in [40]. A ${\rm U}(1)$-invariant SL 3-fold $N$ may locally be written in the form

$$\begin{split}N=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:\,&z_{1}z_{2}=v(x,y)+iy,\quad z_{3}=x+iu(x,y),\\ &|z_{1}|^{2}-|z_{2}|^{2}=2a,\quad(x,y)\in S\bigr\},\end{split}$$

(2.6)

where $S$ is a domain in ${\mathbin{\mathbb{R}}}^{2}$, $a\in{\mathbin{\mathbb{R}}}$ and $u,v:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfy (in a weak sense if $a=0$) the nonlinear Cauchy–Riemann equations

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial v}{\partial x}=-2\bigl(v^{2}+y^{2}+a^{2}\bigr)^{1/2}\frac{\partial u}{\partial y}.$$

(2.7)

If $S$ is simply-connected, as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ there exists a potential $f$ for $u,v$ with $\frac{\partial f}{\partial y}=u$, $\frac{\partial f}{\partial x}=v$, satisfying

$$\Bigl(\Bigl(\frac{\partial f}{\partial x}\Bigr)^{2}+y^{2}+a^{2}\Bigr)^{-1/2}\frac{\partial^{2}f}{\partial x^{2}}+2\,\frac{\partial^{2}f}{\partial y^{2}}=0.$$

(2.8)

In [37, 38], for suitable strictly convex domains $S\subset{\mathbin{\mathbb{R}}}^{2}$ and boundary data $\phi:\partial S\rightarrow{\mathbin{\mathbb{R}}}$, we prove the existence of a unique $f:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfying (2.8) and $f|_{\partial S}=\phi$, and then $u=\frac{\partial f}{\partial y}$, $v=\frac{\partial f}{\partial x}$ satisfy (2.7) (possibly in a weak sense if $a=0$), and $N$ in (2.6) is special Lagrangian.

When $v=y=a=0$, equations (2.7)–(2.8) become singular, and the SL 3-fold $N$ in (2.6) has a singularity at $(0,0,z_{3})=\bigl(0,0,x+iu(x,0)\bigr)$ in ${\mathbin{\mathbb{C}}}^{3}$. In the simplest cases $N$ is locally modelled on the cone $C$ in (2.4) near $(0,0,z_{3})$, but there are also infinitely many other topological types of singularities not locally modelled on cones. Note that the existence and uniqueness results for $N$ are entirely independent of the singularities appearing in the interior of $N$.

The following will be important in §3.6. Using the results of [37, 38, 39, 40], by choosing a suitable family $\phi^{t}:t\in(-\epsilon,\epsilon)$ of boundary conditions for the potential $f^{t}$, we can construct a family $N^{t}:t\in(-\epsilon,\epsilon)$ of exact ${\rm U}(1)$-invariant SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ of the form (2.6) with $a=0$, with the following properties:

• (i)

  $N^{t}$ depends continuously on $t\in(-\epsilon,\epsilon)$ in a suitable sense, for instance as special Lagrangian integral currents in Geometric Measure Theory.

• (ii)

  $N^{t}$ is nonsingular for $t<0$.

• (iii)

  $N^{0}$ has one singular point at $(0,0,0)\in{\mathbin{\mathbb{C}}}^{3}$, which has tangent cone $\Pi_{1}\cup\Pi_{2}$, where $\Pi_{1},\Pi_{2}$ are ${\rm U}(1)$-invariant special Lagrangian planes in ${\mathbin{\mathbb{C}}}^{3}$ intersecting non-transversely with $\Pi_{1}\cap\Pi_{2}={\mathbin{\mathbb{R}}}$.

• (iv)

  $N^{t}$ for $t>0$ has two singular points at $(0,0,\pm z(t))$, where $z(t)$ depends smoothly on $t$ and $z(t)\rightarrow 0$ as $t\rightarrow 0$. Each singular point is locally modelled on the special Lagrangian $T^{2}$-cone $C$ in (2.4).

Thus, isolated singular points of SL $3$-folds modelled on the $T^{2}$-cone $C$ in (2.4) can appear or disappear in pairs under continuous deformation.