Bulletin of the
Korean Mathematical Society BKMS

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

MSC numbers: 53C21, 53C42