$K^α$-translators of offset surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929642366042112 |
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| author | Demirci, Burcu Bektaş Aksoyak, Ferdağ Kahraman Babaarslan, Murat |
| author_facet | Demirci, Burcu Bektaş Aksoyak, Ferdağ Kahraman Babaarslan, Murat |
| contents | In this paper, we study $K^α$--translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space $\mathbb{E}^3$. 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 $\mathbb{E}^3$. 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 $w$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $K^α$-translators of offset surfaces Demirci, Burcu Bektaş Aksoyak, Ferdağ Kahraman Babaarslan, Murat Differential Geometry 53A05 In this paper, we study $K^α$--translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space $\mathbb{E}^3$. 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 $\mathbb{E}^3$. 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 $w$. |
| title | $K^α$-translators of offset surfaces |
| topic | Differential Geometry 53A05 |
| url | https://arxiv.org/abs/2412.15612 |