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Helicoidal minimal surfaces in a conformally flat 3-space
Kellcio Oliveira Ara\'ujo,Ningwei Cui,Romildo da Silva Pina 대한수학회 2016 대한수학회보 Vol.53 No.2
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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HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE
Araujo, Kellcio Oliveira,Cui, Ningwei,Pina, Romildo da Silva Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
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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