Abstract
We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in {mathbb {R}}^3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.
Highlights
A hypersurface M in a given ambient manifold (M, g) is said a soliton to the mean curvature flow if there is a one-parameter group G = {φt | t ∈ R } of isometries of gsuch that the evolution by mean curvature flow starting from M is given at any time t by
Let V be the Killing vector field associated to the group G, it turns out that the property of being a soliton can be translated in a prescribed mean curvature problem: H = g(ν, V ), (1.1)
Special solitons are the translators in the Euclidean space: V is a constant vector field and M evolves translating with constant speed in the direction of V
Summary
A hypersurface M in a given ambient manifold (M, g) is said a soliton to the mean curvature flow if there is a one-parameter group G = {φt | t ∈ R } of isometries of gsuch that the evolution by mean curvature flow starting from M is given at any time t by. Mt = φt (M)
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