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Weighted minimal translation surfaces in the Galilean space with density

  • Dae Won Yoon EMAIL logo
Published/Copyright: April 23, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

Keywords: Galilean space; Translation surface; Weighted manifold; Weighted mean curvature
MSC 2010: 53A35; 53C25

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

z=1alog⁡|cos⁡(ax)|−1alog⁡|cos⁡(ay)|,

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 eϕ 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

dV=eϕdV0,dA=eϕdA0.

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

Hϕ=H−12dϕdN, (1)

where H is the mean curvature and N is the unit normal vector of the surface. The weighted mean curvature Hϕ of a surface in 𝔼3 with density eϕ was introduced by Gromov [5] and it is a natural generalization of the mean curvature H of a surface.

A surface with Hϕ = 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 eZ and they classified the weighted minimal ruled surfaces and the weighted minimal translation surfaces. Lopez [10] considered a linear density eax + 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])

〈x,y〉=x1x2,ifx1≠0orx2≠0,y1y2+z1z2,ifx1=0andx2=0. (2)

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

x×y=0e2e3x1y1z1x2y2z2, (3)

where e2 = (0,1,0) and e3 = (0,0,1).

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

x(u1,u2)=(x(u1,u2),y(u1,u2),z(u1,u2)).

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

N=1ω(0,xu1zu2−xu2zu1,xu2yu1−xu1yu2),

where the positive function ω is given by

ω=(xu1zu2−xu2zu1)2+(xu2yu1−xu1yu2)2.

Here the partial derivatives of the functions x, y and z with respect to ui(i = 1,2) are denoted by xui, yui and zui, 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])

ds2=ds1200ds22, (4)

where ds12=(g1du1+g2du2)2 and ds22=h11du12+2h12du1du2+h22du22. Here gi = xui and hij = 〈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 xuk 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 Lij, i, j = 1, 2 of the second fundamental form, which are the normal components of xuiui, i, j = 1, 2, that is,

Lij=1g1〈g1x~uiuj−gi,jx~u1,N〉=1g2〈g2x~uiuj−gi,jx~u2,N〉.

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

K=L11L22−L122ω2 (5)

and the mean curvature H is given by

H=12ω2(g22L11−2g1g2L12+g12L22). (6)

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

α(x)=(x,0,f(x)),β(y)=(0,y,g(y)).

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

x(x,y)=(x,y,f(x)+g(y)), (7)

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

N=1g′2+1(0,−g′(y),1). (8)

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

H=g″(y)2(1+g′2(y))32. (9)

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,

α(x)=(x,0,f(x)),β(y)=(y,g(y),0).

Then the parametrization of the surface is given by

x(x,y)=(x+y,g(y),f(x)), (10)

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

N=1ω(0,f′(x),g′(y)). (11)

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

H=12ω3(f″(x)g′(y)+f′(x)g″(y)), (12)

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:

Theorem 3.1

A translation surface of type 1 of zero mean curvature in the Galilean 3-space G3 is congruent to a cylindrical surface with isotropic rulings.

Theorem 3.2

A translation surface of type 2 of zero mean curvature in the Galilean 3-space G3 is congruent to an isotropic plane or a non-cylindrical surface with isotropic rulings.

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

x(x,y)=(x,y,f(x)+g(y)).

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

Hϕ=H−12〈N,∇ϕ〉, (13)

where ∇ϕ is the gradient of ϕ.

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

g″(y)2(1+g′2(y))32=12(g′2(y)+1)12〈(0,−g′(y),1),(a,b,c)〉

or equivalently,

g″(y)=(g′2(y)+1)〈(0,−g′(y),1),(a,b,c)〉. (14)

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

z(x,y)=f(x)+d1y+d2

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:

g″(y)=(g′2(y)+1)(−bg′(y)+c). (15)

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

g(y)=−1cln⁡|cos⁡(cy+d1)|+d2,

where d1, d2 are constant.

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

g″(y)+b(g′3(y)+g′(y))=0. (16)

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

dpdy=−bp(p2+1). (17)

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

p(y)=±1e2(by+d1)−1.

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

g(y)=12btan−1⁡e2(by+d1)−1+d2, (18)

where d1, d2 are constant.

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

dpdy=−(p2+1)(bp−c).

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

1 b 2 + c 2 b 2 ln ⁡ ( p 2 + 1 ) + c tan − 1 ⁡ p − b ln ⁡ | b p − c | + y + d 1 = 0.

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

bln⁡(g′2(y)+1)+2ctan−1⁡g′(y)−2bln⁡|bg′(y)−c|+2(b2+c2)y+d=0, (19)

where d is constant.

Theorem 4.1

Let Σ1 be a translation surface of type 1 in the Galilean 3-space G3 with a log-linear density eax + by + cz. Suppose Σ1 is weighted minimal. Then Σ1 is parameterized as

x(x,y)=(x,y,f(x)+g(y)),

where either

  1. g is constant, or

  2. g(y) = d1 y + d2, or

  3. g(y)=−1cln⁡(cos⁡(cy+d1))+d2, or

  4. g(y)=12btan−1⁡e2(by+d1)−1+d2, or

  5. g satisfies (19).

Remark

In general, the normal vector of a surface in G3 is always isotropic, therefore weighted minimal translation surfaces in G3 with a log-linear density eax + by + cz for a ≠ 0 is just the one in G3 with density 1. Thus, such a surface is minimal in G3 and classified by Šipuš and Divjak [16].

5 Weighted minimal translation surfaces of type 2

Let Σ2 be a translation surface of type 2 defined by

x(x,y)=(x+y,g(y),f(x)),

where f and g are smooth functions.

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

H−12ω〈(0,f′(x),g′(y)),(a,b,c)〉=0. (20)

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

f″(x)g′(y)+f′(x)g″(y)=0.

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

f″(x)f′(x)=−g″(y)g′(y)

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

f″(x)f′(x)=−g″(y)g′(y)=m. (21)

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

f(x)=1memx+d1,g(y)=−1me−my+d2,

where d1, d2 are constant.

Case 2. a = 0.

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

f″(x)g′(y)+f′(x)g″(y)=(f′2(x)+g′2(y))(bf′(x)+cg′(y)). (22)

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

f‴(x)g″(y)+f″(x)g‴(y)=2bf″(x)g′(y)g″(y)+2cf′(x)f″(x)g″(y). (23)

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

f‴(x)f″(x)−2cf′(x)=−g‴(y)g″(y).

Then there exists a real number m ∈ ℝ such that

f‴(x)f″(x)−2cf′(x)=−g‴(y)g″(y)=m.

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

g(y)=m2e−my+d1+d2y+d3.

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 e3(−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 eby.

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

f‴(x)f″(x)−2cf′(x)=2bg′(y)−g‴(y)g″(y).

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

f‴(x)f″(x)−2cf′(x)=2bg′(y)−g‴(y)g″(y)=m. (24)

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

f(x)=−1cln⁡|−c(x+d2)|−d1c,g(y)=−1bln⁡|−b(y+d4)|−d3b, (25)

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

b41(x+d2)3+c41(y+d4)3=0,

which is impossible.

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

f‴(x)−c(f′2(x))′=mf″(x),g‴(y)−b(g′2(y))′=−mg″(y). (26)

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

dpdx=p(cp+m)

and its solution is

p=me−m(x+d1)−c.

Thus a first integration implies

f(x)=m2∫1e−m(x+d1)−c. (27)

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

g(y)=−m2∫1em(y+d1)−b. (28)

When substituting (27) and (28) into (22), equation (22) appears a polynomial in e−m(x + d1) and em(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:

Theorem 5.1

Let Σ2 be a translation surface of type 2 in the Galilean 3-space G3 with a log-linear density eax + by + cz. If Σ2 is weighted minimal, then Σ2 is an isotropic plane or parameterized as x(x, y) = (x + y, −1me−my+d1,1memx+d2) with m ≠ 0, d1, d2 ∈ ℝ.

Acknowledgement

The author wishes to express his sincere thanks to the referee for making several useful comments.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2015R1D1A1A01060046).

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Received: 2016-4-25
Accepted: 2017-3-1
Published Online: 2017-4-23

© 2017 Yoon

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0043
Keywords for this article
Galilean space; Translation surface; Weighted manifold; Weighted mean curvature
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BY-NC-ND 3.0

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