Suppose that $u,v$ are solutions of (32) near $(0,0)$ in $\mathbin{\mathbb{R}}^{2}$, with an isolated singularity of order $k$ at $(0,0)$, and that $u(0,0)=v(0,0)=0$. Let $N$ be the associated SL 3-fold in $\mathbin{\mathbb{C}}^{3}$, defined by (31) with $a=0$. Then $N$ has an isolated singular point at $(0,0,0)$. What can we say about it?

Well, the coordinates $x,y$ give a natural map from $N$ to $\mathbin{\mathbb{R}}^{2}$. The fibre of this map over $(0,0)$ is a point, and the fibre of the map over other points $(x,y)\neq(0,0)$ near $(0,0)$ in $\mathbin{\mathbb{R}}^{2}$ is a circle. Thus, $N$ near $(0,0,0)$ has the topology of ${\mathcal{S}}^{1}\times\mathbin{\mathbb{R}}^{2}$ with ${\mathcal{S}}^{1}\times\bigl\{(0,0)\bigr\}$ collapsed to a point. That is, topologically $N$ is a $T^{2}$-cone near $(0,0,0)$.

The author conjectures that when $k=1$, to leading order $u$ and $v$ should agree with the functions $u,v$ of Proposition 6.7 or 6.8 with $a=b=c=0$, at least in the generic case, and therefore that the tangent cone to $N$ at $(0,0,0)$ should be one of the $T^{2}$-cones $L_{0}^{\pm}$ from (16). This gives a good description of the local geometry of $N$ when $k=1$.

When $k>1$, the author conjectures that $u,v$ satisfy

$$u(x,y)=o\bigl(x^{2}+|y|\bigr)\quad\text{and}\quad v(x,y)=o\bigl(|x|+|y|^{1/2}\bigr)\quad\text{for small $x,y$.}$$

(48)

This is because $u(x,y)=O(x^{2}+|y|)$ and $v(x,y)=O(|x|+|y|^{1/2})$ when $k=1$, and by analogy with the zeros of holomorphic functions we expect zeros of higher order to decrease more quickly near $(0,0)$.

Let $t>0$, and define $t^{-1}N=\{t^{-1}{\bf z}:{\bf z}\in N\}$. Then $t^{-1}N$ is also an SL 3-fold, and may be written in the form (30) with $u,v$ replaced by

$$u^{t}(x,y)=t^{-2}u(tx,t^{2}y)\quad\text{and}\quad v^{t}(x,y)=t^{-1}v(tx,t^{2}y).$$

Now equation (48) implies that $u^{t}(x,y)\rightarrow 0$ and $v^{t}(x,y)\rightarrow 0$ as $t\rightarrow 0$ for fixed $x,y$. It follows that $t^{-1}N$ converges to

$$N_{0}=\bigl\{(z_{1},z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:\mathop{\rm Re}(z_{1}z_{2})=0,\quad\mathop{\rm Im}(z_{3})=0,\quad|z_{1}|=|z_{2}|\bigr\}$$

as $t\rightarrow 0$, and this is the tangent cone to $N$ at $(0,0,0)$.

But this is just the union of the two special Lagrangian 3-planes

$$\Pi_{+}=\bigl\{(z,i\bar{z},x):z\in\mathbin{\mathbb{C}},\quad x\in\mathbin{\mathbb{R}}\bigr\},\quad\Pi_{-}=\bigl\{(z,-i\bar{z},x):z\in\mathbin{\mathbb{C}},\quad x\in\mathbin{\mathbb{R}}\bigr\},$$

which intersect in the real line $\bigl\{(0,0,x):x\in\mathbin{\mathbb{R}}\bigr\}$. Thus, when $k>1$ we expect $N$ to resemble the union of two SL 3-planes $\Pi_{\pm}$ intersecting in a line, to leading order near $(0,0,0)$.

As this tangent cone is singular not just at $(0,0,0)$ but all along the line $\Pi_{+}\cap\Pi_{-}$, to have a good picture of $N$ near $(0,0,0)$ we need to include the next nonzero terms in $u$ and $v$ as well. Unfortunately, the author does not know what these terms are. But here is a rather crude approximation, which illustrates the kind of behaviour we expect.

For $k>1$ even, suppose that $u(x,y)\approx|x|^{\alpha}$ for some $\alpha>2$, and $v(x,y)\approx 0$ for small $x,y$. Then we have

$$\begin{split}N\approx\bigl\{(z_{1},z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:&\mathop{\rm Re}(z_{1}z_{2})=|\mathop{\rm Re}(z_{3})|^{\alpha},\\ &\mathop{\rm Im}(z_{3})=0,\quad|z_{1}|=|z_{2}|\bigr\}.\end{split}$$

(49)

This may be written more nicely in different coordinates on $\mathbin{\mathbb{C}}^{3}$. Define new coordinates $(w_{1},w_{2},x_{1},x_{2})$ on $\mathbin{\mathbb{C}}^{3}$ by

$$w_{1}=z_{1}-i\bar{z}_{2},\quad w_{2}=z_{2}+i\bar{z}_{1},\quad x_{1}=\mathop{\rm Re}(z_{3}),\quad x_{2}=\mathop{\rm Im}(z_{3}).$$

Then $w_{1}w_{2}=2\mathop{\rm Re}(z_{1}z_{2})+i\bigl(|z_{1}|^{2}-|z_{2}|^{2}\bigr)$. Therefore, in these new coordinates, (49) becomes

$$N\approx\bigl\{(w_{1},w_{2},x_{1},x_{2})\in\mathbin{\mathbb{C}}^{2}\times\mathbin{\mathbb{R}}^{2}:w_{1}w_{2}=2|x_{1}|^{\alpha},\quad x_{2}=0\bigr\}.$$

(50)

So $N$ may be thought of as approximating a slowly varying 1-parameter family of complex quadratics $w_{1}w_{2}=c$ in $\mathbin{\mathbb{C}}^{2}$, for $c\in\mathbin{\mathbb{R}}$ varying with $x_{1}$. When $x_{1}=0$ the quadratic degenerates into $w_{1}w_{2}=0$, the union of two complex lines in $\mathbin{\mathbb{C}}^{2}$. For $k>1$ odd, the appropriate approximation is $u(x,y)\approx x|x|^{\alpha-1}$ for some $\alpha>2$ and $v(x,y)\approx 0$, and then

$$N\approx\bigl\{(w_{1},w_{2},x_{1},x_{2})\in\mathbin{\mathbb{C}}^{2}\times\mathbin{\mathbb{R}}^{2}:w_{1}w_{2}=2x_{1}|x_{1}|^{\alpha-1},\quad x_{2}=0\bigr\}.$$

(51)

We stress that (50) and (51) are just very approximate guesses, and the true behaviour of $u,v$ and $N$ will be different and more complicated than this.