Let $m>2$ and $a_{1},\ldots,a_{m}>0$, and define polynomials $p,P$ by

$$p(x)=(1+a_{1}x^{2})\cdots(1+a_{m}x^{2})-1\quad\text{and}\quad P(x)=\frac{p(x)}{x^{2}}.$$

(2.3)

Define real numbers $\phi_{1},\ldots,\phi_{m}$ and $A$ by

$$\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\quad\text{and}\quad A=\int_{-\infty}^{\infty}\frac{{\rm d}x}{2\sqrt{P(x)}}\,.$$

Clearly $\phi_{k},A>0$. But writing $\phi_{1}+\cdots+\phi_{m}$ as one integral gives

$$\phi_{1}+\cdots+\phi_{m}=\int_{0}^{\infty}\frac{p^{\prime}(x){\rm d}x}{(p(x)+1)\sqrt{p(x)}}=2\int_{0}^{\infty}\frac{{\rm d}w}{w^{2}+1}=\pi,$$

making the substitution $w=\sqrt{p(x)}$. So $\phi_{k}\in(0,\pi)$ and $\phi_{1}+\cdots+\phi_{m}=\pi$. This yields a 1-1 correspondence between $m$-tuples $(a_{1},\ldots,a_{m})$ with $a_{k}>0$, and $(m\!+\!1)$-tuples $(\phi_{1},\ldots,\phi_{m},A)$ with $\phi_{k}\in(0,\pi)$, $\phi_{1}+\cdots+\phi_{m}=\pi$ and $A>0$.

For $k=1,\ldots,m$, define a function $z_{k}:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{C}}}$ by

$$z_{k}(y)={\rm e}^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\quad\text{where}\quad\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,.$$

Now write ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$, and define a submanifold $L_{{\boldsymbol{\phi}},A}$ in ${\mathbin{\mathbb{C}}}^{m}$ by

$$L_{{\boldsymbol{\phi}},A}=\bigl\{(z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m}):y\in{\mathbin{\mathbb{R}}},\;x_{k}\in{\mathbin{\mathbb{R}}},\;x_{1}^{2}+\cdots+x_{m}^{2}=1\bigr\}.$$

Then $L_{{\boldsymbol{\phi}},A}$ is closed, embedded, and diffeomorphic to ${\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$, and Harvey [26, Th. 7.78] shows that $L_{{\boldsymbol{\phi}},A}$ is special Lagrangian. Also $L_{{\boldsymbol{\phi}},A}$ is asymptotically conical, with rate $\rho=2-m$ and cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two special Lagrangian $m$-planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

$$\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.$$

Apply Theorem 2.4 with $L=L_{{\boldsymbol{\phi}},A}$ and $\rho\in(2-m,0)$. As $L\cong{\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$ we have $b^{1}_{\rm cs}(L)=1$, so Theorem 2.4 shows that $\mathop{\rm dim}\nolimits{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}=1$. This is consistent with the fact that when $\boldsymbol{\phi}$ is fixed, $L_{{\boldsymbol{\phi}},A}$ depends on one real parameter $A>0$. Here $\boldsymbol{\phi}$ is fixed in ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ as the cone $C=\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of $L$ depends on $\boldsymbol{\phi}$, and all $\hat{L}\in{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ have the same cone $C$, by definition.