Helicoidal Minimal Surfaces in a Conformally Flat 3-Space

Abstract

Abstract. 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 characterizes 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 λ 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 λ h,K 2 gives a two-parameter family of helicoidal minimal surfaces in F 3 .

Complete surfaces with zero curvatures in conformally flat spaces

“…Proof of (b). As the torsion function f (r) = 2 1−r , r < 1, is not a constant function, we can conclude that H 2 × f R is isometric to neither H 2 × R nor H 3 .…”

“…The manifold R 3 endowed with a metric as defined above will be denoted by E 3 F . Helicoidal minimal surfaces and helicoidal surfaces with prescribed extrinsic curvature were studied in E 3 F , for a special conformal factor [2], [11]. As this space is invariant under the actions of the rotational group around x 3 -axis, it is logical to consider rotational surfaces around x 3 -axis, because they are invariant under the action of the same group.…”