Constructions of Helicoidal Surfaces in Euclidean Space with Density

Abstract

Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space R-3 with a positive density function e(phi), where phi = -x(2-) y(2), ( x, y, z) is an element of R-3 and construct all the helicoidal surfaces in the space by solving the second- order non- linear ordinary differential equation with the weighted Gaussian curvature and the mean curvature functions. As a result, we give a classification of weighted minimal helicoidal surfaces as well as examples of helicoidal surfaces with some weighted Gaussian curvature and mean curvature functions in the space.