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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 $\mathbb{F}_3$.
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참고문헌 (13)
참고문헌 신청
C. Baikoussis / 1998 / Helicoidal surfaces with prescribed mean or gaussian curvature / J. Geom 63(1-2):25~29
Chr. C. Beneki / 2002 / Helicoidal surfaces in three-dimensional Minkowski space / J. Math. Anal. Appl 275(2):586~614
M. P. do Carmo / 1982 / Helicoidal surfaces with constant mean curvature / Tohoku Math. J 34(3):425~435
A. V. Corro / 2011 / Surfaces of rotation with constant extrinsic cur-vature in a conformally flat 3-space / Results. Math 60(1-4):225~234
N. Cui / 2011 / Minimal rotational hypersurfaces in Minkowski (α, β)-space / Geom. Dedicata 151:27~39
Chr. C. Beneki / 2002 / Helicoidal surfaces in three-dimensional Minkowski space / J. Math. Anal. Appl 275(2):586~614
M. P. do Carmo / 1982 / Helicoidal surfaces with constant mean curvature / Tohoku Math. J 34(3):425~435
A. V. Corro / 2011 / Surfaces of rotation with constant extrinsic cur-vature in a conformally flat 3-space / Results. Math 60(1-4):225~234
N. Cui / 2011 / Minimal rotational hypersurfaces in Minkowski (α, β)-space / Geom. Dedicata 151:27~39
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