Constructions of Helicoidal Surfaces in Euclidean Space with Density

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 ϕ , where ϕ = - x 2 - y 2 , ( x , y , z ) ∈ 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.