Paper · Feb 11, 2024

Invariant $λ$ -translators in Lorentz-Minkowski space

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Abstract

Given $λ \in R$ and $v \in L^{3}$ , a $λ$ -translator with velocity $v$ is an immersed surface in $L^{3}$ whose mean curvature satisfies $H = ⟨ N, v ⟩ + λ$ , where $N$ is a unit normal vector field. When $λ = 0$ , we fall into the class of translating solitons of the mean curvature flow. In this paper we study $λ$ -translators in $L^{3}$ that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the $λ$ -translators. In the case of rotational $λ$ -translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational $λ$ -translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.

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6A.10
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Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Invariant $λ$ -translators in Lorentz-Minkowski space

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Invariant λ-translators in Lorentz-Minkowski space

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Invariant $λ$ -translators in Lorentz-Minkowski space