Constructions of Helicoidal Surfaces in Euclidean Space with Density

Dae Yoon,Young Kim,Dong-Soo Kim,Jae Lee

doi:10.3390/sym9090173

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 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.

Highlights

Differential geometers have been of interest in studying surfaces of constant mean curvature and constant Gaussian curvature in space forms for a long time
As a generalization of surfaces with constant Gaussian curvature or mean curvature, Kenmotsu [1], who generalized an old result of Delaunay [2], constructed surfaces of revolution with the mean curvature as a smooth function
A helicoidal surface in the Euclidean 3-space R3 is defined as the orbit of a plane curve under a helicoidal motion

Summary

Introduction

Differential geometers have been of interest in studying surfaces of constant mean curvature and constant Gaussian curvature in space forms for a long time. By using the first variation of the weighted area, the mean curvature Hφ of a surface in the Euclidean 3-space R3 with density Φ = eφ can be defined. In [11], authors introduced a generalized Gaussian curvature of a surface in a manifold with density eφ and it is defined by. Lopez [15] considered a linear density e ax+by+cz , a, b, c ∈ R, and he classified the weighted minimal translation surfaces and cyclic surfaces in a Euclidean 3-space R3. We construct all helicoidal surfaces in the space, in terms of the weighted Gaussian curvature and mean curvature, as smooth functions

Preliminaries

Main Theorems and Examples