Paper · Jun 30, 2026

Invariant $λ$ -translators in the Heisenberg group

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Abstract

We study oriented surfaces in the Heisenberg space $Nil_{3}$ whose mean curvature $H$ at each point is $H = ⟨ N, \partial_{z} ⟩ + λ$ , where $N$ is the unit normal, $\partial_{z}$ is the vertical Killing vector field and $λ \in R$ . These surfaces are known as $λ$ -translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify $λ$ -translators invariant by the following one-parameter groups of isometries of $Nil_{3}$ : left-translations, rotations and helicoidal motions.

Mean curvature flow Self-shrinkers & translators Minimal & CMC surfaces

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Invariant $λ$ -translators in the Heisenberg group

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Invariant λ-translators in the Heisenberg group

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Invariant $λ$ -translators in the Heisenberg group