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

Abstract

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.

1. 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 ${\mathbb{R}}^3$ [13]. Then, Beneki et al. [14] and Ji et al. [15] studied similar work in ${\mathbb{R}}_1^3$. Furthermore, Dae Won Yoon et al. studied the helicoidal surfaces with a prescribed weighted mean and Gaussian curvature in ${\mathbb{R}}^3$ with density [16] and Yıldız et al. have studied the helicoidal surfaces with prescribed weighted curvatures in ${\mathbb{R}}_1^3$ 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:

$$\begin{array}{}\mathrm{(1)}& \hfill H_{g_F}& =F{H}_{g_0}-〈\mathbf{N},gradF〉,\hfill \mathrm{(2)}& \hfill G_{g_F}& =F^2{G}_{g_0}-2H_{g_0}F〈\mathbf{N},\nabla F〉+{〈\mathbf{N},\nabla F〉}^2,\hfill \end{array}$$

where $\mathbf{N}$ is the unit normal vector of a surface and $\nabla F$ 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.

2. Preliminaries

Let M be a 3-dimensional complete manifold $\left({\mathbb{R}}^3,{〈,〉}_g\right)$ equipped with a metric ${〈,〉}_g$ that is conformal to the Euclidean metric $〈,〉$ such that:

$${〈,〉}_g=\frac{1}{F^2}〈,〉,$$

where $F:{\mathbb{R}}^3\to {\mathbb{R}}^{+}$ is a positive differentiable function.

A manifold with a positive density function $\phi$ is used to weight the volume and the hypersurface area. In terms of the underlying Riemannian volume $d{V}_0$ and area $d{A}_0$, the new, weighted volume and area are given by $dV=\phi d{V}_0$ and $dA=\phi d{A}_0,$ respectively. One of the most important examples of manifolds with density, with applications to probability and statistics, is a Gauss space with density $\phi =e^{a\left(-x^2-y^2-z^2\right)}$ for $a\in \mathbb{R}$ [22].

In Euclidean 3-space with density $e^{\phi },$ the weighted mean curvature is given by:

$$H_{{\phi }_{g_0}}=H_{g_0}-\frac{1}{2}〈\mathbf{N},▽\phi 〉,$$

(3)

where $H_{g_0}$ is the mean curvature of the surface, $\mathbf{N}$ is the unit normal vector of the surface, and $▽\phi$ is the gradient vector of $\phi$ [23]. If $H_{{\phi }_{g_0}}=0$, then the surface is called a weighted minimal surface. In Euclidean 3-space with density $e^{\phi },$ the weighted Gaussian curvature with density is:

$$G_{{\phi }_{g_0}}=G_{g_0}-△\phi ,$$

(4)

where $G_{g_0}$ is the Gaussian curvature of the surface and △ is the Laplacian operator [27].

Throughout this paper, for $x=\left(x_1,x_2,x_3\right)\in {\mathbb{R}}^3,$ we consider the positive density function and the conformal factor function as $e^{\phi }=e^{-x_1^2-x_2^2}$ and $F=\sqrt{x_1^2+x_2^2},$ respectively.

Let $\gamma$ be a $C^2–$curve on $x_1{x}_3–$plane, of type $\gamma \left(u\right)=\left(u,0,f\left(u\right)\right)$, where $u\in I$ for an open interval $I\subset {\mathbb{R}}^{+}.$ Using helicoidal motion on $\gamma ,$ we can obtain the helicoidal surface M as:

$$X\left(u,v\right)=\left[\begin{array}{ccc}cosv& -sinv& 0\\ sinv& cosv& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}u\\ 0\\ f\left(u\right)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ hv\end{array}\right],$$

with $x_3$-axis and a pitch $h\in \mathbb{R},$ so the parametric equation can be given in the form:

$$X\left(u,v\right)=\left(ucosv,usinv,f\left(u\right)+hv\right).$$

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:

$$\begin{array}{cc}\hfill H_{g_0}& =\frac{\left(u^2+h^2\right)u{f}^{″}\left(u\right)+u^2{f}^{{\prime }^3}\left(u\right)+\left(u^2+2h^2\right)f^{\prime }\left(u\right)}{2{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^{3/2}},\hfill \\ \hfill G_{g_0}& =\frac{u^3{f}^{\prime }\left(u\right)f^{″}\left(u\right)-h^2}{{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^2},\hfill \\ \hfill \mathbf{N}& =\frac{\left(hsinv-u{f}^{\prime }\left(u\right)cosv,-u{f}^{\prime }\left(u\right)sinv-hcosv,u\right)}{{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^{1/2}}.\hfill \end{array}$$

Using Equations (3) and (4), the weighted mean curvature $H_{{\phi }_{g_0}}$ and the weighted Gaussian curvature $G_{{\phi }_{g_0}}$ are obtained as:

$$\begin{array}{cc}\hfill H_{{\phi }_{g_0}}& =\frac{\left(u^2+h^2\right)u{f}^{″}\left(u\right)+\left(u^2-2u^4\right)f^{{\prime }^3}\left(u\right)+\left(u^2+2h^2-2u^4-2h^2{u}^2\right)f^{\prime }\left(u\right)}{2{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^{3/2}},\hfill \\ \hfill G_{{\phi }_{g_0}}& =\frac{u^3{f}^{\prime }\left(u\right)f^{″}\left(u\right)-h^2}{{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^2}+4.\hfill \end{array}$$

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 $H_{{\phi }_{g_F}}$ and the weighted extrinsic curvature $G_{{\phi }_{g_F}}$ as:

$$\begin{array}{cc}\hfill H_{{\phi }_{g_F}}& =F{H}_{g_0}-\frac{1}{2}F〈\mathbf{N},▽\phi 〉-〈\mathbf{N},▽F〉\hfill \\ \hfill G_{{\phi }_{g_F}}& =F^2{G}_{g_0}-2F{H}_{g_0}〈\mathbf{N},▽F〉-F^2△\phi +〈\mathbf{N},▽\phi 〉〈\mathbf{N},▽F〉F+{〈\mathbf{N},▽F〉}^2.\hfill \end{array}$$

We obtain $H_{{\phi }_{g_F}}$ and $G_{{\phi }_{g_F}}$ for M as:

$$\begin{array}{cc}\hfill H_{{\phi }_{g_F}}& =\frac{u\left[\left(3u^2-2u^4-2h^2\left(u^2-2\right)\right)f^{\prime }\left(u\right)+\left(3u^2-2u^4\right)f^{{\prime }^3}\left(u\right)+u\left(u^2+h^2\right)f^{″}\left(u\right)\right]}{2{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^{3/2}},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill G_{{\phi }_{g_F}}& =\frac{\begin{array}{c}{u}^2\left[4h^4+4u^4+h^2\left(8u^2-1\right)+\left(h^2\left(6u^2+3\right)+2\left(u^2+3u^4\right)\right)f^{{\prime }^2}\left(u\right)\right.\\ \left.+2\left(u^2+u^4\right)f^{{\prime }^4}\left(u\right)+u\left(2u^2+h^2\right)f^{\prime }\left(u\right)f^{″}\left(u\right)\right]\end{array}}{{\left(u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2\right)}^2}.\hfill \end{array}$$

(6)

3. 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 $e^{\phi }=e^{-x_1^2-x_2^2}$, where conformal factor $F=\sqrt{x_1^2+x_2^2}$ and $x=\left(x_1,x_2,x_3\right)\in {\mathbb{R}}^3$.

Theorem 1.

Let $\gamma \left(u\right)=\left(u,0,f\left(u\right)\right)$ be a profile curve of the helicoidal surface given by $X\left(u,v\right)=\left(ucosv,usinv,f\left(u\right)+hv\right)$ in the 3-dimensional complete manifold with density and $H_{{\phi }_{g_F}}\left(u\right)$ be the weighted mean curvature at the point $\left(u,0,f\left(u\right)\right)$. Then, there exists a two-parameter family of the helicoidal surface given by the curves:

$$\gamma \left(u,H_{{\phi }_{g_F}}\left(u\right),h,c_1,c_2\right)=\left(u,0,f\left(u\right)\right),$$

where:

$$f=\pm \int \frac{\sqrt{u^2+h^2}\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}{\sqrt{1-u^2{\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}^2}}du+c_2.$$

Conversely, for a given smooth function $H_{{\phi }_{g_F}}\left(u\right),$ one can obtain the two-parameter family of curves $\gamma \left(u,H_{{\phi }_{g_F}}\left(u\right),h,c_1,c_2\right)$ being the two-parameter family of helicoidal surfaces, accepting $H_{{\phi }_{g_F}}\left(u\right)$ 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:

$$\mathsf{\Psi }=\frac{f^{\prime }\left(u\right)}{\sqrt{u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2}},$$

(7)

into the equation, then we obtain the first-order linear ordinary differential equation:

$$H_{{\phi }_{g_F}}=\left(2u-u^3\right)\mathsf{\Psi }+\frac{u^2}{2}{\mathsf{\Psi }}^{\prime }.$$

(8)

Then, the general solution of Equation (8) is:

$$\mathsf{\Psi }=\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right),$$

(9)

where $c_1\in \mathbb{R}$. Using Equations (7) and (9), we obtain:

$$\begin{array}{c}\left(1-u^2{\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}^2\right)f^{{\prime }^2}\hfill \\ =\left(u^2+h^2\right){\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}^2.\hfill \end{array}$$

(10)

From the above equation, we obtain:

$$f^{\prime }=\pm \frac{\sqrt{u^2+h^2}\mathsf{\Psi }}{\sqrt{1-u^2{\mathsf{\Psi }}^2}}.$$

(11)

By integrating Equation (11), we obtain:

$$f=\pm \int \frac{\sqrt{u^2+h^2}\mathsf{\Psi }}{\sqrt{1-u^2{\mathsf{\Psi }}^2}}du+c_2,$$

(12)

where $c_2\in \mathbb{R}$.

By contrast, for a given constant $h\in \mathbb{R}-\left\{0\right\}$, a real-valued smooth function $H_{{\phi }_{g_F}}\left(u\right)$ defined on an open interval $I\subset {\mathbb{R}}^{+}$ and an arbitrary $u_0\in I$, there exists an open subinterval $u_0\in I^{\prime }\subset I$ and an open interval $J\subset \mathbb{R}$ which contains:

$${\tilde{c}}_1=-\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du\right)\left(u_0\right),$$

such that:

$$S\left(u,c_1\right)=1-u^2{\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}^2>0,$$

for arbitrary $\left(u,c_1\right)$. Since $S\left(u_0,{\tilde{c}}_1\right)=1>0$ and S is continuous, S is positive on $I^{\prime }\times J\subset {\mathbb{R}}^2.$ Thus, the two-parameter family of the curves can be given as:

$$\gamma \left(u,H_{{\phi }_{g_F}}\left(u\right),h,c_1,c_2\right)=\left(u,0,f\left(u\right)\right),$$

where:

$$f=\pm \int \frac{\sqrt{u^2+h^2}\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}{\sqrt{1-u^2{\left(\frac{1}{e^{-u^2}{u}^4}\left(\int \left(e^{-u^2}{u}^4\right)H_{{\phi }_{g_F}}du+c_1\right)\right)}^2}}du+c_2.$$

☐

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 $e^{\phi .}$ Then, M is an open part of either a helicoid or a surface parametrized by:

$$X\left(u,v\right)=\left(ucosv,usinv,\pm \int \frac{c_1\sqrt{u^2+h^2}}{\sqrt{e^{-2u^2}{u}^8-u^2{c}_1^2}}du+c_2+hv\right),$$

where $c_1,c_2\in \mathbb{R}.$

Example 1.

Consider a helicoidal surface with the weighted mean curvature:

$$H_{{\phi }_{g_F}}=\frac{5e^{-u^2}{u}^4-2e^{-u^2}{u}^6}{e^{-u^2}{u}^4},$$

and the pitch $h=1$ in a complete manifold with density. Using Equation (12), we get $\gamma \left(u\right).$ Thus, we obtain the parametrization of the surface as follows:

$$X\left(u,v\right)=\left(ucosv,usinv,-\sqrt{1-u^2}+v\right),$$

and the figure of the domain:

$$\left\{\begin{array}{c}0<u<1\\ -4<v<4\end{array},\right.$$

is given in Figure 1.

The difference between $H_{{\phi }_{g_0}}$ and $H_{{\phi }_{g_F}}$ of the helicoidal surface with density can be seen in Figure 2.

Example 2.

Consider a helicoidal surface with the weighted mean curvature:

$$H_{{\phi }_{g_F}}=\frac{1-2u^2}{u^4},$$

and the pitch $h=1$ in a complete manifold with density. Using Equation (12), we get $\gamma \left(u\right).$ Thus, we obtain the parametrization of the surface as follows:

$$X\left(u,v\right)=\left(ucosv,usinv,-\frac{\sqrt{-1+u^4}arctan\left(\frac{\sqrt{1+u^2}}{\sqrt{-1+u^4}}\right)}{\sqrt{1-\frac{1}{u^4}}{u}^2}+v\right),$$

and the figure of the domain:

$$\left\{\begin{array}{c}2<u<10\\ -5<v<5\end{array},\right.$$

is given in Figure 3.

Theorem 2.

Let $\gamma \left(u\right)=\left(u,0,f\left(u\right)\right)$ be a profile curve of the helicoidal surface given by $X\left(u,v\right)=\left(ucosv,usinv,f\left(u\right)+hv\right)$ in a 3-dimensional complete manifold with density and $G_{{\phi }_{g_F}}\left(u\right)$ be the weighted extrinsic curvature at the point $\left(u,0,f\left(u\right)\right)$. Then, there exists a two-parameter family of the helicoidal surface, which is given by the curves:

$$\gamma \left(u,G_{{\phi }_{g_F}},h,c_1,c_2\right)=\left(u,0,\pm \int {\left(\frac{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u+\left(u^2+h^2\right)B}{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u-u^{2}B}\right)}^{\frac{1}{2}}du+c_2\right),$$

where:

$$B=\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}}\right)du+c_1,$$

and $c_1$ and $c_2$ are constants. Conversely, for a given smooth function $G_{{\phi }_{g_F}},$ one can obtain the two-parameter family of curves $\gamma \left(u,G_{{\phi }_{g_F}}\left(u\right),h,c_1,c_2\right)$, being the two-parameter family of the helicoidal surfaces, accepting $G_{{\phi }_{g_F}}$ 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:

$${\mathsf{\Phi }}^{\prime }+\left(\frac{5h^2+6u^2-4h^2{u}^2-4u^4}{2u^3+u{h}^2}\right)\mathsf{\Phi }=\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}},$$

(13)

where:

$$\mathsf{\Phi }=\frac{u{f}^{{\prime }^2}\left(u\right)-u}{u^2{f}^{{\prime }^2}\left(u\right)+u^2+h^2}.$$

(14)

The general solution of Equation (13) is:

$$\mathsf{\Phi }=\frac{\left(\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}}\right)du+c_1\right)}{e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}},$$

(15)

where $c_1\in \mathbb{R}.$ Combining Equations (14) and (15), we get:

$$\begin{array}{c}\left(\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u\right.\hfill \\ \left.-u^2\left(\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}}\right)du+c_1\right)\right)f^{{\prime }^2}\left(u\right)\hfill \\ =u\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\hfill \\ +\left(u^2+h^2\right)\left(\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}}\right)du+c_1\right).\hfill \end{array}$$

If we set:

$$B=\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{g_F}}\right)du+c_1,$$

then:

$$f^{{\prime }^2}=\frac{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u+\left(u^2+h^2\right)B}{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u-u^{2}B}.$$

(16)

It follows that:

$$f\left(u\right)=\pm \int {\left(\frac{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u+\left(u^2+h^2\right)B}{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u-u^{2}B}\right)}^{\frac{1}{2}}du+c_2$$

(17)

where $c_2\in \mathbb{R}.$

Conversely, for a given $h\in \mathbb{R}$ and a smooth function $G_{{\phi }_{g_F}}\left(u\right)$, defined on an open interval $I\subset {\mathbb{R}}^{+}$ and an arbitrary $u_0\in I$, there exists an open subinterval $I^{\prime }\subset I$ containing $u_0$ and an open interval $J\subset \mathbb{R}$ containing:

$${\tilde{c}}_1=-\left(\int \left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)\left(\left(\frac{-4-8h^2-12u^2}{2u^2+h^2}\right)+\frac{2}{u^2}{G}_{{\phi }_{gF}}\right)du\right)\left(u_0\right),$$

such that:

$$S\left(u,c_1\right)=\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)-uB>0,$$

which is defined on $I^{\prime }\times J.$ Thus, a two-parameter family of the curves can be given as:

$$\gamma \left(u,G_{{\phi }_{g_F}},h,c_1,c_2\right)=\left(u,0,\pm \int {\left(\frac{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u+\left(u^2+h^2\right)B}{\left(e^{-u^2}{u}^5{\left(h^2+2u^2\right)}^{-1-\frac{h^2}{2}}\right)u-u^{2}B}\right)}^{\frac{1}{2}}du+c_2\right),$$

where $\left(u,c_1\right)\in I^{\prime }\times J;$ $c_2,h\in \mathbb{R}$ and $G_{{\phi }_{g_F}}$ is a smooth function. ☐

Example 3.

Consider a helicoidal surface with the weighted extrinsic curvature:

$$G_{{\phi }_{g_F}}\left(u\right)=\frac{4+11u^2+14u^4+4u^6}{{\left(2+u^2\right)}^2},$$

in a complete manifold with density. Using Equation (17), we obtain $f\left(u\right)=lnu$ for $h=1,c_1=0,c_2=0$ and the parametrization of the surface as follows:

$$X\left(u,v\right)=\left(ucosv,usinv,lnu+v\right).$$

The figure of the surface of the domain:

$$\left\{\begin{array}{c}0<u<3\\ -4<v<4\end{array},\right.$$

is given in Figure 4.

The difference between $G_{{\phi }_{g_0}}$ and $G_{{\phi }_{g_F}}$ of the helicoidal surface with density can be seen in Figure 5.

4. Conclusions and Future Work

In this paper, using the conformal factor function $F=\sqrt{x_1^2+x_2^2}$, 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.