Paper · Dec 20, 2024

$K^{α}$ -translators of offset surfaces

Burcu Bektaş Demirci, Ferdağ Kahraman Aksoyak, Murat Babaarslan

As on arXiv

Abstract

In this paper, we study $K^{α}$ –translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space $E^{3}$ . First, we investigate the condition under which two parallel surfaces can become $K^{α}$ –translators moving with the same speed $w$ . Then, we examine $K^{α}$ –translators on canal surfaces and we show that if a canal surface is $K^{α}$ –translator, then it must be a surface of revolution in $E^{3}$ . We also provide examples for moving a surface of revolution under $K$ –flow (Gauss curvature flow) and $K^{- 1/2}$ –flow (inverse Gauss curvature flow) along a direction $w = (0, 0, 1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^{α}$ –translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$ , while the such rotational surfaces itself is a $K^{α}$ –translator with speed $w$ .

Comments: 13 pages, 4 figures

Other curvature flows Gauss curvature flow Self-shrinkers & translators 3-manifolds Rotational symmetry & warped products

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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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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$K^{α}$ -translators of offset surfaces

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Kα-translators of offset surfaces

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$K^{α}$ -translators of offset surfaces