HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

In this work, we introduce the complete Riemannian manifold 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 chara cterizes the helicoidal minimal surfaces in F-3. We show that the helicoid is a complete minimal surface in 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 F-3.