Weighted minimal translation surfaces in the Galilean space with density

1 Introduction

Constant mean curvature and constant Gaussian curvature surfaces are one of main objects which have drawn geometers’ interest for a very long time. In particular, regarding the study of minimal surfaces, L. Euler found that the only minimal surfaces of revolution are the planes and the catenoids, and E. Catalan proved that the planes and the helicoids are the only minimal ruled surfaces in the Euclidean 3-space 𝔼³. Also, H. F. Scherk in 1835 studied translation surfaces in 𝔼³ defined as graph of the function z(x, y) = f(x) + g(y) and he proved that, besides the planes, the only minimal translation surfaces are the surfaces given by

where a is a non-zero constant.

Translation surfaces having constant mean curvature, in particular zero mean curvature, in the Euclidean space and the Minkowski space are described in [1]. Other results for minimal translation surfaces were obtained in [2, 3] when the ambient spaces are the affine space and the hyperbolic space, respectively.

As a new category in geometry, manifold with density (called also a weighted manifold) appears in many ways in mathematics, such as quotients of Riemannian manifolds or as Gauss spaces. It was instrumental in Perelman’s proof of the Poincare conjecture [4]. A manifold with density is a Riemannian manifold M with a positive density function e^ϕ used to weighted volume and area, that is, for a Riemannian volume dV₀ and a area dA₀ the new weighted volume dV and area dA are defined by

By using the first variation of the weighted area, the weighted mean curvature H_ϕ (also called ϕ-mean curvature) of a surface in the Euclidean 3-space 𝔼³ with density e^ϕ, is given by

where H is the mean curvature and N is the unit normal vector of the surface. The weighted mean curvature H_ϕ of a surface in 𝔼³ with density e^ϕ was introduced by Gromov [5] 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 𝔼³. For more details about manifolds with density and some relative topics we refer to [6–12]. In particular, Hieu and Hoang [8] studied ruled surfaces and translation surfaces in 𝔼³ with density e^Z and they classified the weighted minimal ruled surfaces and the weighted minimal translation surfaces. Lopez [10] considered a linear density e^{ax + by + cz} and he classified the weighted minimal translation surfaces and weighted minimal cyclic surfaces in the Euclidean 3-space 𝔼³. Also Belarbi and Belkhelfa [6] investigated properties of the weighted minimal graphs in 𝔼³ with a log-linear density.

In this article, we focus on a class of translation surfaces in the Galilean 3-space G₃. There are two types of translation surfaces according to a non-isotropic curve and an isotropic curve, called translation surfaces of type 1 and type 2, respectively. We classify the weighted minimal translation surfaces in G₃ with a log-linear density.

2 Preliminaries

In 1872, F. Klein in his Erlangen program proposed how to classify and characterize geometries on the basis of projective geometry and group theory. He showed that the Euclidean and non-Euclidean geometries could be considered as spaces that are invariant under a given group of transformations. The geometry motivated by this approach is called a Cayley-Klein geometry. Actually, the formal definition of Cayley-Klein geometry is pair (G, H), where G is a Lie group and H is a closed Lie subgroup of G such that the (left) coset G/H is connected. G/H is called the space of the geometry or simply Cayley-Klein geometry.

The Galilean geometry is the real Cayley-Klein geometry equipped with the projective metric of signature (0, 0, +, +). The absolute figure of the Galilean 3-space G₃ consists of an ordered triple {ω, f, I}, where ω is the ideal (absolute) plane, f the line (the absolute line) in ω and I the fixed elliptic involution of points of f.

Let x = (x₁, y₁, z₁) and y = (x₂, y₂, z₂) be vectors in G₃. A vector x is called isotropic if x₁ = 0, otherwise it is called non-isotropic. The Galilean scalar product 〈⋅,⋅〉 of x and y is defined by (cf. [13])

From this, the Galilean norm of a vector x in G₃ is given by ||x||=〈x,x〉 and all unit non-isotropic vectors are the form (1, y₁, z₁). For an isotropic vector x₁ = 0 holds. The Galilean cross product of x and y on G₃ is defined by

where e₂ = (0,1,0) and e₃ = (0,0,1).

Consider a C^r-surface Σ, r ≥ 1, in G₃ parameterized by

Let Σ be a regular surface in G₃. Then the unit normal vector field N of the surface Σ is defined by

where the positive function ω is given by

Here the partial derivatives of the functions x, y and z with respect to u_i(i = 1,2) are denoted by x_ui, y_ui and z_ui, respectively. On the other hand, the matrix of the first fundamental form ds² of a surface Σ in G₃ is given by (cf. [14, 15])

where ds12=(g1du1+g2du2)2 and ds22=h11du12+2h12du1du2+h22du22. Here g_i = x_ui and h_ij = 〈x~ui,x~uj〉(i,j=1,2) stand for derivatives of the first coordinate function x(u₁, u₂) with respect to u₁, u₂, and for the Euclidean scalar product of the projections x~uk of vectors x_uk onto the yz-plane, respectively.

The Gaussian curvature K and the mean curvature H of a surface Σ are defined by means of the coefficients L_ij, i, j = 1, 2 of the second fundamental form, which are the normal components of x_uiui, i, j = 1, 2, that is,

Thus, the Gaussian curvature K of a regular surface is defined by

and the mean curvature H is given by

3 Translation surfaces in G₃

In this section, we define translation surfaces in G₃ that are obtained by translating two planar curves. According to the planar curves, we have two types as follows [16]:

Type 1. a non-isotropic curve (having its tangent non-isotropic) and an isotropic curve.

Type 2. non-isotropic curves.

There are no motions of the Galilean space that carry one type of a curve into another, so we will treat them separately.

First, we construct translation surfaces of type 1 in the Galilean 3-space G₃.

Let α(x) be a non-isotropic curve in the plane y = 0 and β(y) an isotropic curve in the plane x = 0. This means that

In this case, a translation surface of type 1 is parameterized by

where f and g are smooth functions. The unit normal vector field N of the surface is

By a straightforward computation, the mean curvature H is given by

Next, we construct translation surfaces of type 2 in the Galilean 3-space G₃.

Let α(x) be a non-isotropic curve in the plane y = 0 and β(y) an non-isotropic curve in the plane z = 0, that is,

Then the parametrization of the surface is given by

where f and g are smooth functions. The unit normal vector field N is

From (6) the mean curvature H of the surface is given by

where ω² = f^{′ 2}(x) + g^{′ 2}(y).

In [16], Šipuš and Divjak classified minimal translation surfaces in the Galilean 3-space G₃ and they proved the following theorems:

4 Weighted minimal translation surfaces of type 1

In this section, we classify translation surfaces of type 1 with zero weighted mean curvature in the Galilean 3-space.

Let Σ₁ be a translation surface of type 1 defined by

Suppose that Σ₁ is the surface in G₃ with a linear density e^ϕ, where ϕ = ax + by + cz, a, b, c not all zero. In this case, the weighted mean curvature H_ϕ of Σ₁ can be expressed as

where ∇ϕ is the gradient of ϕ.

If Σ₁ is the weighted minimal surface, then the weighted minimality condition H_ϕ = 0 with the help of (9) turns out to be

or equivalently,

Let us distinguish two cases according to the value of a.

Case 1. a ≠ 0.

In this case the vector (a, b, c) is non-isotropic and from (2) we get g″(y) = 0. Therefore Σ₁ is determined by

for some constants d₁, d₂.

In other words, the obtained surface is a ruled surface with rulings having the constant isotropic direction (0, 1, d₁) and it is a cylindrical surface.

Case 2. a = 0.

In this case the vector (0, b, c) is isotropic. From (14) we have the following ordinary differential equation:

Subcase 2.1. b = 0. The general solution of (15) is given by

where d₁, d₂ are constant.

Subcase 2.2. c = 0. In the case, equation (15) writes as

In order to solve the equation, we put g′(y) = p(y). Then equation (16) can be rewritten as the form

A function g(y) = d₁, d₁ ∈ ℝ is a solution of (16), and in that case the surface Σ₁ is given by z(x, y) = f(x) + d₁. Suppose that p = g′(y) ≠ 0. A direct integration of (17) yields

Thus, the general solution of (16) appears in the form

where d₁, d₂ are constant.

Subcase 2.3. bc ≠ 0. We put p = g′(y) in (15), then we have

From this, the function p(y) satisfies the following equation:

Thus the weighted minimal translation surface Σ₁ in G₃ with a linear density e^{by + cz} is given by z(x, y) = f(x) + g(y), where f(x) is any smooth function and a function g(y) satisfies

where d is constant.

5 Weighted minimal translation surfaces of type 2

Let Σ₂ be a translation surface of type 2 defined by

where f and g are smooth functions.

If Σ₂ is a surface in G₃ with a linear density e^ϕ, where ϕ = ax + by + cz, a, b, c not all zero, then the weighted minimality condition H_ϕ = 0 becomes

Let us distinguish two cases according to the value of a.

Case 1. a ≠ 0.

In this case, H = 0 and from (12) we have

If f or g is constant, then Σ₂ is a plane. Now, we assume that f′g′ ≠ 0. Then, the above equation writes as

which implies there exists a real number m ∈ ℝ such that

If m = 0, then the functions f and g are linear functions, which generates an isotropic plane.

If m ≠ 0, then the general solutions of (21) are given by

where d₁, d₂ are constant.

Case 2. a = 0.

From (12) and (20) the weighted minimality condition H_ϕ = 0 becomes

Taking the derivative with respect to x and next with respect to y, we have the following ordinary differential equation:

If f″(x) = 0 or g″(y) = 0, then f or g is a linear function. Now we assume that f″(x)g″(y) ≠ 0.

Subcase 2.1. b = 0. Dividing (23) by f″(x)g″(y), we have

Then there exists a real number m ∈ ℝ such that

If m = 0, g(y)=12d1y2+d2y+d3 with d₁, d₂, d₃ ∈ ℝ. Substituting the function g(y) into (22), equation (22) is a polynomial in y with functions of x as coefficients. Thus, all the coefficients must be zero. The coefficient of the highest degree y³ in (22) is cd13, thus it follows d₁ = 0 and g″(y) = 0, a contradiction.

Suppose that m ≠ 0. The solution of the ODE g‴(y) + mg″(y) = 0 is

Substituting the function g(y) into (22), we get a polynomial on e^{−my + d₁} with functions of x as coefficients. In the polynomial we can obtain the coefficient of e^{3(−my + d₁)} and it is −m²c, a contradiction.

Subcase 2.2. c = 0. This subcase is similar to the previous one. Thus, there are no weighted minimal translation surfaces of type 2 in G₃ with a linear density e^by.

Subcase 2.3. bc ≠ 0. Dividing (23) by f″(x)g″(y), we get

Hence, we deduce the existence of a real number m ∈ ℝ such that

If m = 0, the general solution of (24) is given by

where d_i(i = 1, ⋯, 4) are constant. Substituting the function (25) into (22) we can obtain the equation:

which is impossible.

Suppose m ≠ 0. Equation (24) can be written as

First, in order to solve the first equation of (26) we put p = f′(x). Then we have

and its solution is

Thus a first integration implies

By using the similar method, the function g satisfying the second equation of (26) is given by

When substituting (27) and (28) into (22), equation (22) appears a polynomial in e^{−m(x + d₁)} and e^m(y+d₁). The coefficient of e^{− 3m(x + d₁)} in the polynomial is −cm³, which is impossible because m and c are non-zero constant.

Thus we have: