For given constants $\alpha>0$ and $a_{1},\ldots,a_{m-1}>0,$ define

for $j=1,\ldots,m-1$ and $y\in{\mathbin{\mathbb{R}}}$. Then

is a closed, embedded Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ diffeomorphic to ${\mathbin{\mathbb{R}}}^{m},$ which is a Lagrangian MCF translator with translating vector $(0,\ldots,0,\alpha)\in{\mathbin{\mathbb{C}}}^{m}$.

Define $\phi_{1},\ldots,\phi_{m-1}\in{\mathbin{\mathbb{R}}}$ by

Then $\phi_{1},\ldots,\phi_{m-1}\in(0,\pi)$ with $\phi_{1}+\cdots+\phi_{m-1}<\pi$, and $\psi_{j}(y)\rightarrow\phi_{j}$ as $y\rightarrow\infty$, and $\psi_{j}(y)\rightarrow 0$ as $y\rightarrow-\infty$. For fixed $\alpha>0,$ the map $(a_{1},\ldots,a_{m-1})\mapsto(\phi_{1},\ldots,\phi_{m-1})$ is a 1-1 correspondence from $(0,\infty)^{m-1}$ to $\bigl\{(\phi_{1},\ldots,\phi_{m-1})\in(0,\pi)^{m-1}:\phi_{1}+\cdots+\phi_{m-1}<\pi\bigr\}$.

The phase function $\theta_{L}$ of $L$ in (2.10) is a monotone decreasing function of $y$ only, with limits $\pi$ as $y\rightarrow-\infty$ and $\sum_{j=1}^{m-1}\phi_{j}$ as $y\rightarrow+\infty$. Thus, by choosing $\sum_{j=1}^{m-1}\phi_{j}$ close to $\pi,$ the phase variation of $L$ can be made arbitrarily small.

We can give the following heuristic description of $L$ in (2.10). If $y\gg 0$ then $\psi_{j}(y)\approx\phi_{j}$ and $\sqrt{\frac{1}{a_{j}}+y^{2}}\approx y$, and the terms $-\frac{i}{\alpha}\sum_{j=1}^{n}\psi_{j}(y)-\frac{i}{\alpha}\arg(y+iP(y)^{-1/2})$ are negligible compared to ${\textstyle\frac{1}{2}}y^{2}$ in the last coordinate. Thus, the region of $L$ with $y\!\gg\!0$ is in a weak sense approximate to

But this is just an unusual way of parametrizing

the complement of a ray in a Lagrangian plane. Similarly, the region of $L$ with $y\ll 0$ is in a weak sense approximate to

So, $L$ can be roughly described as asymptotic to the union of two Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}\cong{\mathbin{\mathbb{R}}}^{m}$ which intersect in an ${\mathbin{\mathbb{R}}}$ in ${\mathbin{\mathbb{C}}}^{m}$, the $y_{m}$-axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\in{\mathbin{\mathbb{R}}}\bigr\}$. To make $L$, we glue these Lagrangian planes by a kind of ‘connect sum’ along the negative $y_{m}$-axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\}$. Under Lagrangian mean curvature flow, $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ remain fixed, but the gluing region translates in the positive $y_{m}$ direction, as though $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ are being ‘zipped together’.

A slightly more accurate description of the ends of $L$ for large $y$ is that $L$ approximates $\tilde{\Pi}_{\boldsymbol{\phi}}$ when $y\gg 0$ and $\tilde{\Pi}_{0}$ when $y\ll 0$, where $\tilde{\Pi}_{\boldsymbol{\phi}}$ and $\tilde{\Pi}_{0}$ are the non-intersecting affine Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$

We will discuss these Lagrangian MCF translators further in Example 3.32.