Constructions of helicoidal surfaces in Euclidean space with density (Q2333561)

Summary: Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space \(\mathbb{R}^3\) with a positive density function \(e^\phi\), where \(\phi = - x^2 - y^2\), \((x, y, z) \in \mathbb{R}^3\) 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.