The aim of this paper is to investigate the mean curvature flow soliton solutions on the Heisenberg group $\mathcal{H}$ when the initial data is a ruled surface by straight lines. We give a family of those solutions which are generated by ${\mathfrak{Iso}}_0(\mathcal{H})$ (the isometries of $\mathcal{H}$ for which the origin is a fix point). We conclude that the function which describe the motion of these surfaces under MCF, is always a linear affine function. As an application we proof that the Grim Reaper solution evolves from a ruled surface in $\mathcal{H}$. We also provide other examples.

中文翻译：

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海森堡集团的MCF解决方案系列

本文的目的是研究Heisenberg群上的平均曲率流孤子解 $\mathcal{H}$当初始数据是直线的直纹表面时。我们给出了由${\mathfrak{矶}}_0（\mathcal{H}）$ （的等距 $\mathcal{H}$其原点是固定点）。我们得出结论，描述这些表面在MCF下运动的函数始终是线性仿射函数。作为应用，我们证明Grim Reaper解决方案是从$\mathcal{H}$。我们还提供其他示例。