Every two-dimensional subgroup of a non-unimodular three-dimensional Lie group with the chosen left-invariant metric has an explicit eternal mean curvature flow, translating for abelian subgroups and non-self-similar for non-abelian ones.
B\"ohm , On the long time behavior of homogeneous R icci flows , Comment
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The mean curvature flow of subgroups on Lie groups of dimension three
Every two-dimensional subgroup of a non-unimodular three-dimensional Lie group with the chosen left-invariant metric has an explicit eternal mean curvature flow, translating for abelian subgroups and non-self-similar for non-abelian ones.