HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

Abstract

Abstract. In this work, we introduce the complete Riemannian man-ifold F 3 which is a three-dimensional real vector space endowed with aconformally flat metric that is a solution of the Einstein equation. Weobtain a second order nonlinear ordinary differential equation that char-acterizes the helicoidal minimal surfaces in F 3 . We show that the helicoidis a complete minimal surface in F 3 . Moreover we obtain a local solutionof this differential equation which is a two-parameter family of functionsλ h,K 2 explicitly given by an integral and defined on an open interval.Consequently, we show that the helicoidal motion applied on the curvedefined from λ h,K 2 gives a two-parameter family of helicoidal minimalsurfaces in F 3 . 1. IntroductionIn the monograph entitled “Ricci Curvature of Seminar on Differential Ge-ometry” edited by Yau (see Problem Section in [13]), there is a problem whichis to find the necessary and sufficient conditions for a symmetric tensor T ij ona compact manifold so that one can find a metric g