Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space

Abstract

In this paper, we study a conformally flat 3-space. F-3 which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1/F-2, where F(x) = e(-x12-x22) for x = (x(1), x(2), x(3)) is an element of R-3. In particular, we construct all helicoidal surfaces in F-3 by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in F-3.