Screw motion surfaces in $\widetilde{PSL}_2 (\mathbb{R}, \tau)$

Peñafiel, Carlos

doi:10.4310/ajm.2015.v19.n2.a4

Cited by 3 publications

(1 citation statement)

References 3 publications

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“…Perdomo [27] proved that curves, which intersect with the plane perpendicular to the helicoidal axis and the minimal helicoidal surfaces in R 3 , is characterized by TreadmillSled. Earp and Toubiana [31] dealt with a minimal or constant mean curvature surface invariant under screw motions in H 2 × R and S 2 × R. There have been several interesting results for helicoidal surface [4,19,21,22,26].…”

Section: Daehwan Kim and Juncheol Pyomentioning

confidence: 99%

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Kim¹,

Pyo²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type II singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in R 3 and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [10]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in R n+1 and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.

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Section: Daehwan Kim and Juncheol Pyomentioning

confidence: 99%

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Kim¹,

Pyo²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds

Käse

2024

Journal of Mathematical Analysis and Applications

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Constant mean curvature Isometric Immersions into 𝕊 2 ×ℝ and ℍ 2 ×ℝ and related results

Daniel¹,

Domingos²,

Vitório³

2023

Annales De l'Institut Fourier

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In this article, we study constant mean curvature isometric immersions into S 2 × R and H 2 × R and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the 3-dimensional homogenous manifolds E(κ, τ ) and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in S 2 × S 2 and H 2 × H 2 with constant intrinsic curvature. It is worthwhile to point out that these classifications provide new examples.Résumé. -Dans cet article, nous étudions les immersions à courbure moyenne constante dans S 2 × R et H 2 × R et nous classifions ces immersions quand la surface est à courbure intrinsèque constante. Comme applications, nous utilisons la correspondance des surfaces soeurs pour classifier les surfaces à courbure moyenne constante et courbure intrinsèque constante dans les variétés de dimension 3 homogènes E(κ, τ ) et nous utilisons la correspondance de Torralbo-Urbano pour classifier les surfaces à courbure moyenne parallèle et courbure intrinsèque constante dans S 2 × S 2 et H 2 × H 2 . Il est important de noter que ces classifications fournissent de nouveaux exemples.

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Screw motion surfaces in $\widetilde{PSL}_2 (\mathbb{R}, \tau)$

Abstract: A screw motion surfaces in P SL 2 (R, τ) is either a minimal or a constant mean curvature surface which is invariant by helicoidal isometries. In this paper, we study the geometric behavior of such screw motion surfaces.

Cited by 3 publications

References 3 publications

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds

Constant mean curvature Isometric Immersions into 𝕊 2 ×ℝ and ℍ 2 ×ℝ and related results

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