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 R3 with a positive density function eφ, where φ = −x2 − y2, (x, y, z) ∈ R3 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.


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.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 R 3 is defined as the orbit of a plane curve under a helicoidal motion.As for helicoidal surfaces in R 3 , the surfaces with prescribed mean or Gaussian curvature have been studied by Baikoussis and Koufogiorgos [3].On the other hand, Beneki et al. [4] and Ji and Hou [5][6][7] extended it in a Minkowski space.Recently, in [8], Yoon et al. also constructed helicoidal surfaces in a Heisenberg group for such a case.
A density on a Riemannian manifold is a positive function Φ, weighting both volume and surface area.In terms of the underlying Riemannian volume dV 0 and area dA 0 , the weighted volume and area are given by dV = ΦdV 0 and dA = ΦdA 0 , respectively.Manifolds with densities (called also a weighted manifold) arise naturally in geometry as quotients of other Riemannian manifolds, in physics as spaces with different media, in probability as the famous Gauss space G 3 with Φ = ce a 2 r 2 for a, c ∈ R and r 2 = x 2 + y 2 + z 2 .Also, it was instrumental in Perelman's proof of the Poincare conjecture [9].
By using the first variation of the weighted area, the mean curvature H φ of a surface in the Euclidean 3-space R 3 with density Φ = e φ can be defined.It is given by where H and N are the mean curvature and the unit normal vector of a surface and φ is the gradient of φ, which is called the weighted mean curvature or the φ-mean curvature of a surface.The weighted mean curvature H φ of a surface in R 3 with density e φ was introduced by Gromov [10] and it is a natural generalization of the mean curvature H of a surface.A surface with H φ = 0 is called a weighted minimal surface or a φ-minimal surface in R 3 .
Another curvature for a surface in the Euclidean 3-space is the Gaussian curvature.In [11], authors introduced a generalized Gaussian curvature of a surface in a manifold with density e φ and it is defined by where G is the Riemannain Gaussian curvature of a surface and ∆ is the Laplacian operator, which is called the weighted Gaussian curvature or the φ-Gaussian curvature of a surface.Also, they obtained a generalization of the Gauss-Bonnet formula for a smooth disc in a smooth surface with density e φ .
For more details about manifolds with density and some relative topics, we refer readers to [12][13][14][15][16][17], etc.In particular, Hieu and Hoang [13] studied ruled surfaces and translation surfaces in R 3 with density e z and they classified the weighted minimal ruled surfaces and translation surfaces.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 R 3 .Also, Belarbi and Belkhelfa [18] investigated the properties of the weighted minimal graphs in R 3 with a linear density.
In this article, we focus on a class of helicoidal surfaces in the Euclidean 3-space R 3 with density e φ , where φ(p) = −x 2 − y 2 , p = (x, y, z) ∈ R 3 .In particular, we construct all helicoidal surfaces in the space, in terms of the weighted Gaussian curvature and mean curvature, as smooth functions.

Preliminaries
We consider a regular plane curve where h is a constant, is said to be the helicoidal surface with axis Oz, a pitch h and the profile curve γ.
That is, M can be parametrized by We assume, without loss of generality, γ(u) = (u, 0, f (u)) is the profile curve in the xz-plane defined on any open interval I of positive real numbers.Then, the helicoidal surface M in R 3 is given by where f is a smooth function defined on I.By a direct computation, the Gaussian curvature G and the mean curvature H of the surface are given by On the other hand, the unit normal vector N of the surface is Suppose that M is the surface in R 3 with density e φ , where φ = −x 2 − y 2 .Then, in this case, the weighted mean curvature H φ and the weighted Gaussian curvature G φ can be expressed as and respectively.

Main Theorems and Examples
In this section, we construct helicoidal surfaces with prescribed weighted Gaussian curvature and weighted mean curvature in the Euclidean 3-space R 3 with density e −x 2 −y 2 , where (x, y, z) ∈ R 3 .

The Solution of Equation (5)
Equation ( 5) is a second-order nonlinear ordinary differential equation.To solve it, we put Then, Equation ( 5) can be expressed in the form: It is a first-order linear ordinary differential equation with respect to A and its general solution is given by where c 1 ∈ R. On the other hand, Equations ( 7) and ( 9) imply Since thus the general solution of Equation (10) becomes where c 2 is constant.
Conversely, let h be a given non-zero real constant and H φ (u) be a real-valued smooth function defined on an open interval I ⊂ (0, +∞).Then, for any u 0 ∈ I, there exist an open subinterval I of u 0 (I ⊂ I) and an open interval J of R containing such that the function for any (u, c 1 ) ∈ I × J.In fact, because F(u 0 , c 1 ) = u 2 0 > 0, by the continuity of F, it is positive in a subset of I × J ⊂ R 2 .Therefore, for any (u, c 1 ) ∈ I × J, c 2 ∈ R, h ∈ R and any given smooth function H φ , we can define the two-parameter family of curves Applying the one-parameter subgroup Φ h t on these curves, we can obtain a two-parameter family of helicoidal surfaces with the weighted mean curvature H φ .Theorem 1.Let γ(u) = (u, 0, f (u)) be a profile curve of the helicoidal surface Equation (4) in the Euclidean 3-space with density e −x 2 −y 2 of which the weighted mean curvature at the point (u, 0, f (u)) is given by H φ (u).Then, for some constants c 1 , c 2 and h, there exists the two-parameter family of helicoidal surfaces generated by plane curves Conversely, let H φ (u) be a smooth function.Then, we can construct the two-parameter family of curves γ(u, H φ (u), c 1 , c 2 ) and so it is the two-parameter family of helicoidal surfaces with the weighted mean curvature H φ (u) and a pitch h.

Corollary 1.
Let M be a weighted minimal helicoidal surface in the Euclidean 3-space with density e −x 2 −y 2 .Then, M is an open part of either a helicoid or a surface parameterized by for some constants c 1 and c 2 .
Proof.If f is a constant function, it is a trivial solution for H φ = 0.It follows that a helicoidal surface is a helicoid.Otherwise, we can easily compute Equation ( 11), in such case f is given by (12).
Example 1.We consider a helicoidal surface with the constant weighted mean curvature and the pitch h = 1 in the Euclidean 3-space with density e −x 2 −y 2 .Then, by Equation (11), we can calculate the profile curve γ(u); from this, the parametrization of the surface is given by (see Figure 1) Example 2. Consider a helicoidal surface with the weighted mean curvature and the pitch h = 1 in the Euclidean 3-space with density e −x 2 −y 2 .By a direct computation with the help of Equation (11), we can compute the profile curve γ(u), which implies that the parametrization of the surface is expressed in the form (see Figure 2):  (1 − 2u 2 ).

The Solution of Equation (6)
To solve the second-order nonlinear ordinary differential Equation ( 6), we put Then, the weighted Gaussian curvature G φ can be rewritten as: The solution of the last equation is for some constant c 1 .Combining Equations ( 13) and ( 15), one gets the general solution of Equation ( 16) is given by where c 2 ∈ R.
Conversely, let h be a given real number and G φ be a smooth function defined on an open interval I ⊂ (0, +∞).Let Thus, we can find an open subinterval I ⊂ I containing u 0 and an open interval J of R containing c 1 such that the function F(u, c 1 ) is positive for any (u, c 1 ) ∈ I × J.In fact, because F(u 0 , c 1 ) = 1, by the continuity of F, it is positive in a subset of I × J ⊂ R 2 .Theeqrefore, for any (u, c 1 ) ∈ I × J, h ∈ R, c 2 ∈ R and given the smooth function G φ , we can define the two-parameter family of curves Consequently, we get a two-parameter family of helicoidal surfaces with the weighted Gaussian curvature G φ (u), u ∈ I and we have the following theorem.Theorem 2. Let γ(u) = (u, 0, f (u)) be a profile curve of the helicoidal surface (4) in the Euclidean 3-space with density e −x 2 −y 2 of which the weighted Gaussian curvature at the point (u, 0, f (u)) is given by G φ (u).Then, for some constants c 1 , c 2 and h, there exists the two-parameter family of the helicoidal surface generated by plane curves Conversely, let G φ (u) be a smooth function.Then, for any u 0 ∈ I, we can construct the two-parameter family of curves γ(u, G φ (u), h, c 1 , c 2 ), u ∈ I ⊂ I and so it is the two-parameter family of helicoidal surfaces with the weighted Gaussian curvature G φ (u), u ∈ I .
Example 3. We consider a helicoidal surface in the Euclidean 3-space with density e −x 2 −y 2 with a negative weighted Gaussian curvature In such a case, an integration of Equation (17) implies f (u) = u for h = 1, c 1 = 0 and c 2 = 0.It follows that the helicoidal surface is parametrized by (see Figure 3) X(u, v) = (u cos v, u sin v, u + v).Then, from Equation (17), we have f (u) = sin −1 u for h = 1, c 1 = 0 and c 2 = 0 and, in this case, a parametrization of the helicoidal surface is given by (see Figure 4) X(u, v) = (u cos v, u sin v, sin −1 u + v).