Example 2.15 . [03N9]

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Example 2.15.

In dimension $m=1$ , the unique connected Lagrangian MCF translator in ${\mathbin{\mathbb{C}}}$ , up to rigid motions and rescalings, is the ‘grim reaper’

\bigl\{x+iy\in{\mathbin{\mathbb{C}}}:y\in(-\pi/2,\pi/2),\quad x=-\log\cos y\bigr\},

with translating vector $v=1\in{\mathbin{\mathbb{C}}}$ , which is sketched in Figure 2.1.

Figure 2.1: ‘Grim reaper’ Lagrangian MCF translating soliton in

{\mathbin{\mathbb{C}}}