Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.


Introduction
It is well known that a helicoidal surface is a generalization of a rotation surface.There are many studies about these surfaces under some given certain conditions [1][2][3][4][5][6][7][8][9][10][11][12].Recently, the popular question has become whether a helicoidal surface can be constructed when its curvatures are prescribed.Several researchers have worked on this problem and obtained useful results.Firstly, Baikoussis et al. studied helicoidal surfaces with a prescribed mean and Gaussian curvature in R 3 [13].Then, Beneki et al. [14] and Ji et al. [15] studied similar work in R 3  1 .Furthermore, Dae Won Yoon et al. studied the helicoidal surfaces with a prescribed weighted mean and Gaussian curvature in R 3 with density [16] and Yıldız et al. have studied the helicoidal surfaces with prescribed weighted curvatures in R 3  1 with density [17].For more details on manifolds with density and surfaces in manifold with density, see References [18][19][20][21][22][23][24][25].
This problem is extended to complete manifolds.Lee et al. studied the helicoidal surfaces with a prescribed extrinsic curvature or mean curvature in a conformally flat 3-space [10].It is well known that a metric on a complete manifold is conformal to the Euclidean metric.For a given surface in a complete manifold with a conformal factor function F, the mean curvature and the extrinsic curvature are given by: where N is the unit normal vector of a surface and ∇F is the gradient of F, H g 0 is the mean curvature of the surface in Euclidean 3-space, and G g 0 is the Gaussian curvature of a surface in Euclidean 3-space [26].
In this paper, we study helicoidal surfaces in a 3-dimensional complete manifold with density.We construct a helicoidal surface with a prescribed weighted mean and weighted extrinsic curvature.Then, we give examples to illustrate our result.

Preliminaries
Let M be a 3-dimensional complete manifold R 3 , , g equipped with a metric , g that is conformal to the Euclidean metric , such that: where F : R 3 → R + is a positive differentiable function.
A manifold with a positive density function ϕ is used to weight the volume and the hypersurface area.In terms of the underlying Riemannian volume dV 0 and area dA 0 , the new, weighted volume and area are given by dV = ϕdV 0 and dA = ϕdA 0 , respectively.One of the most important examples of manifolds with density, with applications to probability and statistics, is a Gauss space with density ϕ = e a(−x 2 −y 2 −z 2 ) for a ∈ R [22].
In Euclidean 3-space with density e ϕ , the weighted mean curvature is given by: where H g 0 is the mean curvature of the surface, N is the unit normal vector of the surface, and ϕ is the gradient vector of ϕ [23].If H ϕ g 0 = 0, then the surface is called a weighted minimal surface.In Euclidean 3-space with density e ϕ , the weighted Gaussian curvature with density is: where G g 0 is the Gaussian curvature of the surface and is the Laplacian operator [27].Throughout this paper, for x = (x 1 , x 2 , x 3 ) ∈ R 3 , we consider the positive density function and the conformal factor function as e ϕ = e −x 2 1 −x 2 2 and F = x 2 1 + x 2 2 , respectively.Let γ be a C 2 −curve on x 1 x 3 −plane, of type γ (u) = (u, 0, f (u)), where u ∈ I for an open interval I ⊂ R + .Using helicoidal motion on γ, we can obtain the helicoidal surface M as: with x 3 -axis and a pitch h ∈ R, so the parametric equation can be given in the form: It is straightforward to see that the mean curvature H g 0 , the Gaussian curvature G g 0 , and the unit normal vector of helicoidal surface are: Using Equations ( 3) and ( 4), the weighted mean curvature H ϕ g 0 and the weighted Gaussian curvature G ϕ g 0 are obtained as: We assume that M is a surface in a 3-dimensional complete manifold with density.By considering Equations ( 1)-( 4), we can define the weighted mean curvature H ϕ g F and the weighted extrinsic curvature G ϕ g F as: We obtain H ϕ g F and G ϕ g F for M as:

Helicoidal Surfaces with Prescribed Weighted Mean or Weighted Extrinsic Curvature
In this section, we construct helicoidal surfaces with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with density e ϕ = e −x 2 1 −x 2   2 , where conformal factor Theorem 1.Let γ (u) = (u, 0, f (u)) be a profile curve of the helicoidal surface given by X (u, v) = (u cos v, u sin v, f (u) + hv) in the 3-dimensional complete manifold with density and H ϕ g F (u) be the weighted mean curvature at the point (u, 0, f (u)).Then, there exists a two-parameter family of the helicoidal surface given by the curves: where: Conversely, for a given smooth function H ϕ g F (u), one can obtain the two-parameter family of curves γ u, H ϕ g F (u), h, c 1 , c 2 being the two-parameter family of helicoidal surfaces, accepting H ϕ g F (u) as the weighted mean curvature h as a pitch.
Proof.Let us solve Equation (5), which is a second-order nonlinear ordinary differential equation.If we apply: into the equation, then we obtain the first-order linear ordinary differential equation: Then, the general solution of Equation ( 8) is: where c 1 ∈ R. Using Equations ( 7) and ( 9), we obtain: From the above equation, we obtain: By integrating Equation ( 11), we obtain: where c 2 ∈ R.
By contrast, for a given constant h ∈ R − {0}, a real-valued smooth function H ϕ g F (u) defined on an open interval I ⊂ R + and an arbitrary u 0 ∈ I, there exists an open subinterval u 0 ∈ I ⊂ I and an open interval J ⊂ R which contains: such that: for arbitrary (u, c 1 ).Since S (u 0 , c 1 ) = 1 > 0 and S is continuous, S is positive on I × J ⊂ R 2 .Thus, the two-parameter family of the curves can be given as: where: The following corollary is an immediate consequence of Theorem 1 and the definition of a minimal surface.

Corollary 1.
Let M be a minimal helicoidal surface in a complete manifold with density e ϕ.Then, M is an open part of either a helicoid or a surface parametrized by: where c 1 , c 2 ∈ R.
Example 1.Consider a helicoidal surface with the weighted mean curvature: and the pitch h = 1 in a complete manifold with density.Using Equation ( 12), we get γ (u) .Thus, we obtain the parametrization of the surface as follows: and the figure of the domain: is given in Figure 1.The difference between H ϕ g 0 and H ϕ g F of the helicoidal surface with density can be seen in Figure 2. Example 2. Consider a helicoidal surface with the weighted mean curvature: and the pitch h = 1 in a complete manifold with density.Using Equation ( 12), we get γ (u) .Thus, we obtain the parametrization of the surface as follows: and the figure of the domain: is given in Figure 3. Theorem 2. Let γ (u) = (u, 0, f (u)) be a profile curve of the helicoidal surface given by X (u, v) = (u cos v, u sin v, f (u) + hv) in a 3-dimensional complete manifold with density and G ϕ g F (u) be the weighted extrinsic curvature at the point (u, 0, f (u)).Then, there exists a two-parameter family of the helicoidal surface, which is given by the curves: , where: and c 1 and c 2 are constants.Conversely, for a given smooth function G ϕ g F , one can obtain the two-parameter family of curves γ u, G ϕ g F (u), h, c 1 , c 2 , being the two-parameter family of the helicoidal surfaces, accepting G ϕ g F as the weighted extrinsic curvature h as a pitch.
Proof.Let's solve the second-order nonlinear ordinary differantial Equation (6).We can rewrite Equation ( 6) as follows: where: The general solution of Equation ( 13) is: where c 1 ∈ R. Combining Equations ( 14) and ( 15),we get: If we set: then: It follows that: where c 2 ∈ R.
Conversely, for a given h ∈ R and a smooth function G ϕ g F (u), defined on an open interval I ⊂ R + and an arbitrary u 0 ∈ I, there exists an open subinterval I ⊂ I containing u 0 and an open interval J ⊂ R containing: which is defined on I × J. Thus, a two-parameter family of the curves can be given as: , where (u, c 1 ) ∈ I × J; c 2 , h ∈ R and G ϕ g F is a smooth function.
Example 3. Consider a helicoidal surface with the weighted extrinsic curvature: in a complete manifold with density.Using Equation ( 17), we obtain f (u) = ln u for h = 1, c 1 = 0, c 2 = 0 and the parametrization of the surface as follows: X (u, v) = (u cos v, u sin v, ln u + v) .
The figure of the surface of the domain: is given in Figure 4.The difference between G ϕ g 0 and G ϕ g F of the helicoidal surface with density can be seen in Figure 5.

Conclusions and Future Work
In this paper, using the conformal factor function F = x 2 1 + x 2 2 , we constructed a helicoidal surface with a prescribed weighted mean curvature and Gaussian curvature in a complete manifold with a positive density function.Different helicoidal surfaces can be obtained in a complete manifold with density using different conformal factor functions.In addition, if conformal factor function F is bounded, a manifold is called a conformally flat space.Thus, by considering a bounded function, one can study helicoidal surface in a conformally flat space with density.

Figure 1 .
Figure 1.The helicoidal surface with the weighted mean curvature.

Figure 3 .
Figure 3.The helicoidal surface with the weighted mean curvature.

Figure 4 .
Figure 4.The helicoidal surface with the weighted Gaussian curvature.