Paper · May 24, 2024

Translators of the mean curvature flow in the special linear group $S L (2, R)$

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Abstract

Translators in the special linear group $S L (2, R)$ are surfaces whose mean curvature $H$ and unit normal vector $N$ satisfy $H = ⟨ N, X ⟩$ , where $X$ is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is $4$ and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.

Comments: 16 pages, 2 figures

Mean curvature flow Self-shrinkers & translators

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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)

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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.

Translators of the mean curvature flow in the special linear group $S L (2, R)$

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Translators of the mean curvature flow in the special linear group SL(2,R)

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Translators of the mean curvature flow in the special linear group $S L (2, R)$