HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

Title & Authors
HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE
Araujo, Kellcio Oliveira; Cui, Ningwei; Pina, Romildo da Silva;

Abstract
In this work, we introduce the complete Riemannian manifold $\small{\mathbb{F}_3}$ 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 $\small{\mathbb{F}_3}$. We show that the helicoid is a complete minimal surface in $\small{\mathbb{F}_3}$. Moreover we obtain a local solution of this differential equation which is a two-parameter family of functions $\small{{\lambda}_h,K_2}$ explicitly given by an integral and defined on an open interval. Consequently, we show that the helicoidal motion applied on the curve defined from $\small{{\lambda}_h,K_2}$ gives a two-parameter family of helicoidal minimal surfaces in $\small{\mathbb{F}_3}$.
Keywords
elicoidal minimal surfaces;conformally flat space;
Language
English
Cited by
1.
On Helicoidal Surfaces in a Conformally Flat 3-Space, Mediterranean Journal of Mathematics, 2017, 14, 4
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