Let $m>2$, $\alpha\geqslant 0$ and $a_{1},\ldots,a_{m}>0$, and define a smooth function $P:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{R}}}$ by $P(0)=\alpha+a_{1}+\cdots+a_{m}$ and

$$P(x)=\textstyle\frac{1}{x^{2}}\bigl(e^{\alpha x^{2}}\prod_{k=1}^{m}(1+a_{k}x^{2})-1\bigl),\quad x\neq 0.$$

(2.9)

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

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

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

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

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

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

Then $L_{\boldsymbol{\phi}}^{\alpha}$ is a closed, embedded Lagrangian diffeomorphic to ${\mathbin{\cal S}}^{m-1}\times{\mathbin{\mathbb{R}}}$ and satisfying $H=\alpha F^{\perp}$. If $\alpha>0$ it is an LMCF expander, and if $\alpha=0$ it is one of the Lawlor necks $L_{{\boldsymbol{\phi}},A}$ from Example 2.5. It is graded, with Lagrangian angle

$$\theta_{L_{\boldsymbol{\phi}}^{\alpha}}\bigl((z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m})\bigr)=\textstyle\sum_{k=1}^{m}\psi_{k}(y)+\arg\bigl(-y-iP(y)^{-1/2}\bigr).$$

Note that the only difference between the constructions of $L_{{\boldsymbol{\phi}},A}$ in Example 2.5 and $L_{\boldsymbol{\phi}}^{\alpha}$ above is the term $e^{\alpha x^{2}}$ in (2.9), which does not appear in (2.3). If $\alpha=0$ then $e^{\alpha x^{2}}=1$, and the two constructions agree.

As in [43, Th. D], $L_{\boldsymbol{\phi}}^{\alpha}$ is asymptotically conical, with cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two Lagrangian $m$-planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

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

But in contrast to Example 2.5, for $\alpha>0$ we do not have $\phi_{1}+\cdots+\phi_{m}=\pi$, so $\Pi_{\boldsymbol{\phi}}$ and $C$ are not special Lagrangian.

In [43, Th. D] we prove that for fixed $\alpha>0$, the map $\Phi^{\alpha}:(a_{1},\ldots,a_{m})\mapsto(\phi_{1},\ldots,\phi_{m})$ gives a diffeomorphism

$$\Phi^{\alpha}:(0,\infty)^{m}\longrightarrow\bigl\{(\phi_{1},\ldots,\phi_{m})\in(0,\pi)^{m}:0<\phi_{1}+\cdots+\phi_{m}<\pi\bigr\}.$$

That is, for all $\alpha>0$ and ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ with $0<\phi_{1},\ldots,\phi_{m}<\pi$ and $0<\phi_{1}+\cdots+\phi_{m}<\pi$, the above construction gives a unique LMCF expander $L_{\boldsymbol{\phi}}^{\alpha}$ asymptotic to $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$.