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HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

  • Araujo, Kellcio Oliveira (Departamento de Matematica, Universidade de Brasilia - UnB) ;
  • Cui, Ningwei (Departamento de Matematica, Universidade de Brasilia - UnB) ;
  • Pina, Romildo da Silva (Instituto de Matematica e Estatistica, Universidade Federal de Goias - UFG)
  • Received : 2015.03.06
  • Published : 2016.03.31

Abstract

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 $\mathbb{F}_3$.

Keywords

elicoidal minimal surfaces;conformally flat space

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Cited by

  1. On Helicoidal Surfaces in a Conformally Flat 3-Space vol.14, pp.4, 2017, https://doi.org/10.1007/s00009-017-0967-x