Weighted minimal translation surfaces in the Galilean space with density

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.


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 E 3 . Also, H. F. Scherk in 1835 studied translation surfaces in E 3 defined as graph of the function z.x; y/ D f .x/ C g.y/ and he proved that, besides the planes, the only minimal translation surfaces are the surfaces given by z D 1 a log j cos.ax/j 1 a log j cos.ay/j; 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 used to weighted volume and area, that is, for a Riemannian volume d V 0 and a area dA 0 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 (also called -mean curvature) of a surface in the Euclidean 3-space E 3 with density e , is given by where H is the mean curvature and N is the unit normal vector of the surface. The weighted mean curvature H of a surface in E 3 with density e was introduced by Gromov [5] and it is a natural generalization of the mean curvature H of a surface. A surface with H D 0 is called a weighted minimal surface or a -minimal surface in E 3 . For more details about manifolds with density and some relative topics we refer to [6][7][8][9][10][11][12]. In particular, Hieu and Hoang [8] studied ruled surfaces and translation surfaces in E 3 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 axCbyCcz and he classified the weighted minimal translation surfaces and weighted minimal cyclic surfaces in the Euclidean 3-space E 3 . Also Belarbi and Belkhelfa [6] investigated properties of the weighted minimal graphs in E 3 with a log-linear density.
In this article, we focus on a class of translation surfaces in the Galilean 3-space G 3 . 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 G 3 with a log-linear density.

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; C; C/. The absolute figure of the Galilean 3-space G 3 consists of an ordered triple f!; f; I g, where ! is the ideal (absolute) plane, f the line (the absolute line) in ! and I the fixed elliptic involution of points of f . Let x D .x 1 ; y 1 ; z 1 / and y D .x 2 ; y 2 ; z 2 / be vectors in G 3 . A vector x is called isotropic if x 1 D 0, otherwise it is called non-isotropic. The Galilean scalar product h ; i of x and y is defined by (cf. [13]) From this, the Galilean norm of a vector x in G 3 is given by jjxjj D p hx; xi and all unit non-isotropic vectors are the form .1; y 1 ; z 1 /: For an isotropic vector x 1 D 0 holds. The Galilean cross product of x and y on G 3 is defined by where e 2 D .0; 1; 0/ and e 3 D .0; 0; 1/.
Consider a C r -surface †, r 1, in G 3 parameterized by Let † be a regular surface in G 3 . Then the unit normal vector field N of the surface † is defined by where the positive function ! is given by Here the partial derivatives of the functions x, y and z with respect to u i (i D 1; 2) are denoted by x u i , y u i and z u i , respectively. On the other hand, the matrix of the first fundamental form ds 2 of a surface † in G 3 is given by (cf. [14,15]) x u j i (i; j D 1; 2/ stand for derivatives of the first coordinate function x.u 1 ; u 2 / with respect to u 1 ; u 2 , and for the Euclidean scalar product of the projections Q x u k of vectors x u k onto the yz-plane, respectively. The Gaussian curvature K and the mean curvature H of a surface † are defined by means of the coefficients L ij ; i; j D 1; 2 of the second fundamental form, which are the normal components of x u i u j ; i; j D 1; 2, that is, Thus, the Gaussian curvature K of a regular surface is defined by and the mean curvature H is given by

Translation surfaces in G 3
In this section, we define translation surfaces in G 3 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 G 3 .
Let˛.x/ be a non-isotropic curve in the plane y D 0 andˇ.y/ an isotropic curve in the plane x D 0. This means that˛.
x/ D .x; 0; f .x//; .y/ D .0; y; g.y//: In this case, a translation surface of type 1 is parameterized by x.x; y/ D .x; y; f .x/ C g.y//; where f and g are smooth functions. The unit normal vector field N of the surface is N D 1 g 0 2 C 1 .0; g 0 .y/; 1/: By a straightforward computation, the mean curvature H is given by Next, we construct translation surfaces of type 2 in the Galilean 3-space G 3 .
Let˛.x/ be a non-isotropic curve in the plane y D 0 andˇ.y/ an non-isotropic curve in the plane z D 0, that is,˛.
x/ D .x; 0; f .x//; .y/ D .y; g.y/; 0/: Then the parametrization of the surface is given by x.x; y/ D .x C y; g.y/; f .x//; 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 ! 2 D f 0 2 .x/ C g 0 2 .y/.
In [16], Šipuš and Divjak classified minimal translation surfaces in the Galilean 3-space G 3 and they proved the following theorems: Theorem 3.1. A translation surface of type 1 of zero mean curvature in the Galilean 3-space G 3 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 G 3 is congruent to an isotropic plane or a non-cylindrical surface with isotropic rulings.

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 † 1 be a translation surface of type 1 defined by x.x; y/ D .x; y; f .x/ C g.y//: Suppose that † 1 is the surface in G 3 with a linear density e , where D ax C by C cz, a; b; c not all zero. In this case, the weighted mean curvature H of † 1 can be expressed as where r is the gradient of . If † 1 is the weighted minimal surface, then the weighted minimality condition H D 0 with the help of (9) turns out to be g 00 .y/ 2.1 C g 0 2 .y// Let us distinguish two cases according to the value of a.
In this case the vector .a; b; c/ is non-isotropic and from (2) we get g 00 .y/ D 0. Therefore † 1 is determined by for some constants d 1 ; d 2 .
In other words, the obtained surface is a ruled surface with rulings having the constant isotropic direction .0; 1; d 1 / and it is a cylindrical surface. Case 2. a D 0.
In this case the vector .0; b; c/ is isotropic. From (14) we have the following ordinary differential equation: g 00 .y/ D .g 0 2 .y/ C 1/. bg 0 .y/ C c/: The general solution of (15) is given by where d 1 ; d 2 are constant. Subcase 2.2. c D 0. In the case, equation (15) writes as g 00 .y/ C b.g 0 3 .y/ C g 0 .y// D 0: In order to solve the equation, we put g 0 .y/ D p.y/. Then equation (16) can be rewritten as the form dp dy D bp.p 2 C 1/: A function g.y/ D d 1 , d 1 2 R is a solution of (16), and in that case the surface † 1 is given by z.x; y/ D f .x/Cd 1 . Suppose that p D g 0 .y/ ¤ 0. A direct integration of (17) yields p.y/ D˙1 p e 2.byCd 1 / 1 : Thus, the general solution of (16) appears in the form where d 1 ; d 2 are constant. Subcase 2.3. bc ¤ 0. We put p D g 0 .y/ in (15), then we have dp dy D .p 2 C 1/.bp c/: From this, the function p.y/ satisfies the following equation: Thus the weighted minimal translation surface † 1 in G 3 with a linear density e byCcz is given by z.x; y/ D f .x/ C g.y/, where f .x/ is any smooth function and a function g.y/ satisfies b ln.g 0 2 .y/ C 1/ C 2c tan 1 g 0 .y/ 2b ln jbg 0 .y/ cj C 2.b 2 C c 2 /y C d D 0; where d is constant.
Remark. In general, the normal vector of a surface in G 3 is always isotropic, therefore weighted minimal translation surfaces in G 3 with a log-linear density e axCbyCcz for a ¤ 0 is just the one in G 3 with density 1. Thus, such a surface is minimal in G 3 and classified by Šipuš and Divjak [16].
Suppose that m ¤ 0. The solution of the ODE g 000 .y/ C mg 00 .y/ D 0 is g.y/ D m 2 e myCd 1 C d 2 y C d 3 : Substituting the function g.y/ into (22), we get a polynomial on e myCd 1 with functions of x as coefficients. In the polynomial we can obtain the coefficient of e 3. myCd 1 / and it is m 2 c, a contradiction. Subcase 2.2. c D 0. This subcase is similar to the previous one. Thus, there are no weighted minimal translation surfaces of type 2 in G 3 with a linear density e by . Subcase 2.3. bc ¤ 0. Dividing (23) by f 00 .x/g 00 .y/, we get f 000 .x/ f 00 .x/ 2cf 0 .x/ D 2bg 0 .y/ g 000 .y/ g 00 .y/ : Hence, we deduce the existence of a real number m 2 R such that If m D 0, the general solution of (24) is given by