Paper · Aug 24, 2025

Invariant $λ$ -translators for the Gauss curvature flow in Euclidean space

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Abstract

A $λ$ -translator is a surface in Euclidean space $R^{3}$ whose Gauss curvature $K$ satisfies $K = ⟨ N, v ⟩ + λ$ , where $N$ is the Gauss map, $v$ is a fixed direction, and $λ \in R$ . In this paper, we classify all $λ$ -translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

Comments: 5 pages, 20 figures

Other curvature flows Gauss curvature flow Self-shrinkers & translators

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Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) 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)

Invariant $λ$ -translators for the Gauss curvature flow in Euclidean space

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Invariant λ-translators for the Gauss curvature flow in Euclidean space

Invariant $λ$ -translators for the Gauss curvature flow in Euclidean space