• Title/Summary/Keyword: helicoidal surfaces

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HELICOIDAL SURFACES OF THE THIRD FUNDAMENTAL FORM IN MINKOWSKI 3-SPACE

  • CHOI, MIEKYUNG;YOON, DAE WON
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1569-1578
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    • 2015
  • We study helicoidal surfaces with the non-degenerate third fundamental form in Minkowski 3-space. In particular, we mainly focus on the study of helicoidal surfaces with light-like axis in Minkowski 3-space. As a result, we classify helicoidal surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form on the surface.

HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE

  • Choi, Mie-Kyung;Kim, Young-Ho;Liu, Huili;Yoon, Dae-Won
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.859-881
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    • 2010
  • The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

ON THE GAUSS MAP OF HELICOIDAL SURFACES

  • Kim, Dong-Soo;Kim, Wonyong;Kim, Young Ho
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.715-724
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    • 2017
  • We study the Gauss map G of helicoidal surfaces in the 3-dimensional Euclidean space ${\mathbb{E}}^3$ with respect to the so called Cheng-Yau operator ${\square}$ acting on the functions defined on the surfaces. As a result, we completely classify the helicoidal surfaces with Gauss map G satisfying ${\square}G=AG$ for some $3{\times}3$ matrix A.

HELICOIDAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

  • Choi, Mie-Kyung;Kim, Dong-Soo;Kim, Young-Ho
    • Journal of the Korean Mathematical Society
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    • v.46 no.1
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    • pp.215-223
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    • 2009
  • The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1-type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine helicoidal surface of polynomial kind with pointwise 1-type Gauss map of the first kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a characterization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

BOUR'S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE

  • Hieu, Doan The;Thang, Nguyen Ngoc
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.2081-2089
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    • 2017
  • In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb{R}^4$ is isometric to a family of surfaces of revolution in $\mathbb{R}^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP

  • Kim, Young Ho;Turgay, Nurettin Cenk
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1345-1356
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    • 2013
  • In this paper, we study rotational and helicoidal surfaces in Euclidean 3-space in terms of their Gauss map. We obtain a complete classification of these type of surfaces whose Gauss maps G satisfy $L_1G=f(G+C)$ for some constant vector $C{\in}\mathbb{E}^3$ and smooth function $f$, where $L_1$ denotes the Cheng-Yau operator.

HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

  • Araujo, Kellcio Oliveira;Cui, Ningwei;Pina, Romildo da Silva
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.531-540
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    • 2016
  • In this work, we introduce the complete Riemannian manifold $\mathbb{F}_3$ which is a three-dimensional real vector space endowed with a conformally flat metric that is a solution of the Einstein equation. We obtain a second order nonlinear ordinary differential equation that characterizes the helicoidal minimal surfaces in $\mathbb{F}_3$. We show that the helicoid is a complete minimal surface in $\mathbb{F}_3$. Moreover we obtain a local solution of this differential equation which is a two-parameter family of functions ${\lambda}_h,K_2$ explicitly given by an integral and defined on an open interval. Consequently, we show that the helicoidal motion applied on the curve defined from ${\lambda}_h,K_2$ gives a two-parameter family of helicoidal minimal surfaces in $\mathbb{F}_3$.

HELICOIDAL KILLING FIELDS, HELICOIDS AND RULED MINIMAL SURFACES IN HOMOGENEOUS THREE-MANIFOLDS

  • Kim, Young Wook;Koh, Sung-Eun;Lee, Hyung Yong;Shin, Heayong;Yang, Seong-Deog
    • Journal of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1235-1255
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    • 2018
  • We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.