Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group ${\mathbb{H}}_3$

Abstract

Purpose

In the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ₁(x)*γ₂(y), where γ₁ and γ₂ are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.

Design/methodology/approach

In this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔr_i = λ_ir_i. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.

Findings

Classification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.

Originality/value

The subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

Citation

Medjati, R., Zoubir, H. and Medjahdi, B. (2023), "Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group ${\mathbb{H}}_3$", Arab Journal of Mathematical Sciences, Vol. 29 No. 2, pp. 154-171. https://doi.org/10.1108/AJMS-03-2021-0071

1. Introduction

A Euclidean submanifold is said to be of finite Chen type if its coordinate functions are finite sum of eigenfunctions of its Laplacian [1]. Chen posed the problem of classifying the finite type surfaces in the three-dimensional Euclidean space E3. Further, the notion of finite type can be extended to any function on a submanifold of a Euclidean space or a pseudo-Euclidean space.

A well known result, due to Takahshi [2], states that the minimal surfaces and the spheres are the only surfaces in E3 satisfying the condition.

Δr=λr,   λ∈R

In [3] Ferrandez, Garay and Lucas proved that the surfaces of E3 satisfying

ΔH=AH,  A∈M3(R)

where H denotes the mean curvature vector field, are either minimal, or an open piece of a sphere or of a right circular cylinder.

In [4], F. Dillen, J. Pas and L. Verstraelen proved that the only surfaces in E3 satisfying

Δr=Ar+B,  A∈M3(R), B∈M3,1(R)

are the minimal surfaces, the spheres and the circular cylinders.

In [5], Baba-Hamed, Bekkar and Zoubir studied coordinate finite type translation surface in a three-dimensional Minkowski Space.

In [6], Yoon classified coordinate finite type translation surfaces in a three-dimensional Galilean space.

In [7], Bekkar and Senoussi studied the translation surfaces in the three-dimensional Euclidean and Lorentz-Minkowski spaces under the condition

ΔIIIri=μiri,   μi∈R

In [8–10] Bukcu, Karacan and Yoon classified translation surfaces of type 1 and type 2 that satisfy the condition

ΔI, II, IIIxi=λixi

in the three-dimensional simply isotropic space.

Recently, in [11] Cakmak, Karakan, Kiziltug and Yoon studied the translation surfaces in three-dimensional Galilean space under the condition.

ΔIIxi=λixi

In [12, 13] Rahmani and Rahmani has showed that modulo an automorphism of the Lie algebra, the three-dimensional Lorentz Heisenberg group H3 has the following classes of left-invariant Lorentz metrics:

g1=−dx2+dy2+(xdy+dz)2g2=dx2+dy2−(xdy+dz)2g3=dx2+(xdy+dz)2−1−xdy−dz2.

They proved that the metrics g₁, g₂, g₃ are non-isometrics and that g₃ is flat.

Let r: M2→H3 be an isometric immersion of surface in H3. Then r is said to be semi-Riemannian surface in H3 if the induced metric on M² is non-degenerate. In that case this metric is Riemannian or Lorentzian and the surface is said to be space-like or time-like, respectively.

A space-like surface is said to be maximal if H = 0.

A time-like surface is said to be extremal surface (or minimal surface) if H = 0.

In this paper, we classify space-like translation surfaces of H3 endowed with flat metric g₃ under the condition.

(1.1)Δri=λiri,   λi∈R

Indeed this study is closely linked to that of result, due to Dillen et al. [14] where they proved that a ruled surface M² in three-dimensional Minkowski space R13 is of finite type if and only if M² is minimal, or M² is a part of a circular cylinder, or M² is a part of a hyperbolic cylinder, or M² is an isoparametric surface with null rules.

2. Preliminaries

2.1 The Lorentz Heisenberg group H3

In this paragraph we shortly recall that the Heisenberg group H3 is a Lie group which is diffeomorphic to R3 [15] is defined as

(x,y,z)*(x¯,y¯,z¯)=(x+x¯,y+y¯,z+z¯−xy¯)

The identity of the group is 0,0,0 and the inverse of x,y,z is given by −x,−y,−xy−z. The left invariant Lorentz metric on H3 is

g3=dx2+(xdy+dz)2−1−xdy−dz2

The following set of left-invariant vector fields forms an pseudo-orthonormal basis for corresponding Lie-algebra

B=e1=∂∂x,   e2=∂∂y+1−x∂∂z,   e3=∂∂y−x∂∂z

The characterizing properties of this algebra are the following commutation relations:

e2,e3=0, e3,e1=e2−e3,e2,e1=e2−e3.

with

g3e1,e1=1, g3e2,e2=1,  g3e3,e3=−1.

If ∇ is the Levi–Civita connection and R is the curvature tensor of ∇, we have

∇e1e1=∇e1e2=∇e1e3=0∇e2e1=∇e3e1=e2−e3∇e2e2=∇e2e3=∇e3e2=∇e3e3=−e1.

2.2 The Beltrami formula

We recall that a translation surfaces M2γ1,γ2 in the three-dimensional Lorentz Heisenberg group H3 is a surface parametrized by

r:M2→H3,   rx,y=γ1x* γ2y,

where γ₁ and γ₂ are any generating curves in R3.

M2,r is said to be of finite Chen-type k if the position vector r admits the following spectral decomposition

r=r0+∑i=1kri

where r_i are H3-valued eigenfunctions of the Laplacian of M2,r and Δr_i = λ_ir_i, λi∈R, i = 1, 2, …, k [1]. If λ_i are different, then M² is said to be of k-type.

The Laplacian Δ on M² is given by

(2.1)Δ=−1D ∑ij∂∂xiDgij ∂∂xj.

where G=gij is the matrix consisting of components of the induced metric on M², and G−1=gij is the inverse matrix of G and D = det G.

We set

Δr=Δr1,Δr2,Δr3

such that r=ru,v=r1u,v,r2u,v,r3u,v is a function of class C².

We recall that the Heisenberg space endowed with a flat Lorentzian metric has the same properties as the Minkowski space, however it is well known that the position vector of a surface in a pseudo-Euclidean space is twice the mean curvature vector [1]. So we have the following result:

Theorem 1.

A Beltrami formula in (H3,g3) is given by the following:

(2.2) Δr=2H

where Δ is the Laplacian of the surface and H is the mean curvature vector field of M².

3. Translation surfaces in (H3,g3)

In the Lorentz Heisenberg space H3, a translation surface is parametrized by rx,y=γ1x*γ2y, where γ₁ and γ₂ are two planar curves lying in planes, which are non-orthogonal and * denotes the group operation of H3.

Definition 1.

A translation surface M² in H3 is surface parametrized by γ1x*γ2y, where γ1:I⊂R→H3, γ2:J⊂R→H3 are curves in two coordinate planes of R3.

We distinguish six types of translation surfaces in H3.

3.1 Surfaces of type 1

Let us first consider a translation surface M² parametrized by

(3.1)r(x,y)=x,0,u(x)*(0,y,v(y))=(x,y,u(x)+v(y)−xy),

where u and v are two smooth functions.

Thus, the basis of the tangent space T_pM² is

rx=(1,0,ux−y)=∂x+ux−y∂zry=(0,1,vy−x)=∂y+vy−x∂z

That is

(3.2)rx=e1+ux−ye2−ux−ye3, ry=vye2+1−vye3

Therefore the coefficients of the first and second fundamental form are

E=g3rx,rx=1,   F=g3rx,ry=ux−y,   G=g3ry,ry=2vy−1

and

L=g3∇rxrx,N,   M=g3∇rxry,N,   N=g3∇ryry,N,

where N is a unit vector field normal on M², which satisfies the following system

g3rx,N=0,g3ry,N=0,g3N,N=−1.

and so,

(3.3)∇rxrx=uxxe2−uxxe3, ∇rxry=0,∇ryry=−e1+vyye2−vyye3.

The normal unit vector field N to M² is given by:

(3.4)N=−ux−yW e1+1−vyW e2+vyW e3

with

W=EG−F2=2vy−ux−y2−1.

Therefore

L=uxxW,M=0, N=vyy+ux−yW.

and the mean curvature vector field H is given by the formula:

H=HN,

with

H=EN+GL−2FM2EG−F2

where H is, the mean curvature of the surface M².

So the mean curvature of the surface M² parametrized by (x, y, u(x) + v(y) − xy) is given by

(3.5)H=12W3vyy+2vy−1uxx+ux−y

3.2 Surfaces of type 2

Now the translation surface M² is parametrized by

(3.6)r(x,y)=0,y,v(y)*(x,0,u(x))=(x,y,u(x)+v(y)),

where u and v are two smooth functions. Therefore

rx=(1,0,ux)=∂x+ux∂zry=(0,1,vy)=∂y+vy∂z

and so,

(3.7)rx=e1+uxe2−uxe3, ry=x+vye2+1−x−vye3

Therefore the coefficients of the first fundamental form are

E=1   F=ux  G=2vy+2x−1

We have

(3.8)∇rxrx=uxxe2−uxxe3, ∇rxry=e2−e3,∇ryry=−e1+vyye2−vyye3.

The normal unit vector field N to M² is given by:

(3.9)N=−uxW e1+1−x−vyW e2+x+vyW e3

with

W=EG−F2=2vy+x−1−ux2.

Then the coefficients of the second fundamental form of M² are

L=uxxW,M=1W,N=ux+vyyW.

We follow the same steps as the previous type to calculate the mean curvature of the translation surface M². We obtain

(3.10)H=12W3vyy+2vy+2x−1uxx−ux

3.3 Surfaces of type 3

The translation surface M² is parametrized by

(3.11)r(x,y)=x,0,u(x)*(v(y),y,0)=(x+v(y),y,u(x)−xy),

where u and v are two smooth functions. Therefore

rx=(1,0,ux−y)=∂x+ux−y∂zry=(vy,1,−x)=vy∂x+∂y−x∂z

and so,

(3.12)rx=e1+ux−ye2−ux−ye3, ry=vye1+e3,

whereby the coefficients of the first fundamental form are

E=g3rx,rx=1,   F=g3rx,ry=ux+vy−y,   G=g3ry,ry=vy2−1.

We have

(3.13)∇rxrx=uxxe2−uxxe3,∇rxry=0,∇ryry=vyy−1e1+vye2−vye3.

The normal unit vector field N to M² is given by:

(3.14)N=1Wux−ye1−vyux−y+1e2+vyux−ye3

with the space-like case W=−ux−y2−2vyux−y−1.

Then the coefficients of the second fundamental form of M² are

L=−uxxW  ,M=0  ,N=vyy−1ux−y−vyW.

We follow the same steps as the previous types to calculate the main curvature of the translation surface M², we obtain

(3.15)H=12W3ux−yvyy−1−vy−uxxvy2−1

3.4 Surfaces of type 4

The translation surface M² is parametrized by

(3.16)r(x,y)=v(y),y,0*(x,0,u(x))=(v(y)+x,y,u(x)),

where u and v are two smooth functions. Therefore

rx=(1,0,ux)=∂x+ux∂zry=(vy,1,0)=vy∂x+∂y

and so,

(3.17)rx=e1+uxe2−uxe3, ry=vye1+v+xe2+1−v−xe3

Thus, the coefficients of the first and second fundamental form are

E=g3rx,rx=1,   F=g3rx,ry=vy+ux,G=g3ry,ry=vy2+2v+2x−1.

We have

(3.18)∇rxrx=uxxe2−uxxe3,∇rxry=e2−e3,∇ryry=vyy−1e1+2vye2−2vye3.

The normal unit vector field N to M² is given by:

(3.19)N=1W−uxe1+uxvy+1−v−xe2+v+x−uxvye3

with the space-like case W=2v+x−uxvy−ux2−1.

Then the coefficients of the second fundamental form of M² are

L=uxxW,M=1W,N=2vy−vyy−1uxW.

We follow the same steps as the previous types to calculate the main curvature of the translation surface M², we obtain

(3.20)H=12W3vyy+1ux+vy2+2v+2x−1uxx

3.5 Surfaces of type 5

In this case, the translation surface M² is parametrized by

(3.21)r(x,y)=v(y),y,0*(x,0,u(x))=(y,v(y)+x,u(x)),

where u and v are two smooth functions. Therefore

rx=(0,1,ux)=∂y+ux∂zry=(1,vy,0)=∂x+vy∂y

and so,

(3.22)rx=y+uxe2−ux+y−1e3,ry=e1+yvye2+1−yvye3

Hence the coefficients of the first and second fundamental form are

E=g3rx,rx=2ux+y−1, F=g3rx,ry=ux+2y−1vy,G=g3ry,ry=2y−1vy2+1.

We have

(3.23)∇rxrx=−e1+uxxe2−uxxe3,∇rxry=−vye1+e2−e3,∇ryry=−vy2e1+2vy+yvyye2−2vy+y−1vyye3.

The normal unit vector field N to M² is given by:

(3.24)N=1Wuxvye1−ux+y−1e2+ux+ye3

with W=2y+ux−ux2vy2−1.

The coefficients of the second fundamental form of M² then are

L=uxx−uxvyW,M=1−uxvy2W,N=2vy−vyyux−uxvy3W.

As in the previous types, we obtain

(3.25)H=12W32y−1vy2+1uxx−2y+ux−1vyyux+1−2y−1+uxvy2uxvy

3.6 Surfaces of type 6

The translation surface M² is parametrized by

(3.26)r(x,y)=y,v(y),0*(0,x,u(x))=(y,v(y)+x,u(x)−xy),

where u and v are two smooth functions. Therefore

rx=(0,1,ux)=∂y+ux∂z, ry=(1,vy,0)=∂x+vy∂y

and so,

(3.27)rx=uxe2+1−uxe3,                   ry=e1+yvy−xe2+x−yvy+vye3

Therefore the coefficients of the first and second fundamental form are

E=g3rx,rx=2ux−1,F=g3rx,ry=vyux+yvy−vy−x,G=g3ry,ry=1−vy2+2vyyvy−x.

We have

(3.28)∇rxrx=−e1+uxxe2−uxxe3,∇rxry=−vye1,∇ryry=−vy2e1+2vy+yvyye2−2vy+yvyy−vyye3.

The normal unit vector field N to M² is given by:

(3.29)N=1Wuxvy−yvy+xe1+1−uxe2+uxe3

With W=2ux−uxvy−yvy+x2−1.

Then the coefficients of the second fundamental form of M² are

L=uxx−uxvy+yvy−xW, M=−uxvy−yvy+xvyW,N=−uxvy−yvy+xvy2+2vy+yvyy−vyyuxW

As above, we obtain

(3.30)H=12W32ux−1y−uxvyy+1−vy2+2vyyvy−xuxx+3ux+y−2vy−x

4. Translation surfaces in Lorentz Heisenberg space (H3,g3) satisfying Δr_i = λ_ir_i

This section is devoted to classify the translation surfaces of type 1, type 2, type 3, type 4, type 5 and type 6 that satisfy the condition Δr_i = λ_ir_i, λi∈R.

The result is:

Theorem 2.

Let M² be a translation surface of type 1, type 2, type 3, type 4, type 5 or type 6 in the three-dimensional Lorentz Heisenberg space (H3,g3). Then M² satisfies the condition Δr_i = λ_ir_i, i = 1, 2, 3, λi∈R, if and only if M² has zero mean curvature.

Proof of Theorem 2.

In the following we will prove Theorem 2.

4.1 Surfaces of type 1

By (2.2) and (3.4), the Laplacian operator Δr of r can be expressed as

(4.1)Δr=−2ux−yWHe1+21−vyWHe2+2vyWHe3

Next, suppose M² satisfies condition (1.1). Then, from (3.1) and (4.1), we obtain the following system of ordinary differential equations.

(4.2)−2ux−yWH=λ1x

(4.3)21−vyWH=λ2xy+λ3u+v−xy

(4.4)2vyWH=λ21−xy−λ3u+v−xy

Therefore, the problem of classifying the translation surfaces M² satisfying (1.1) is reduced to the integration of the system (4.2), (4.3) and (4.4) of ordinary differential equations.

Next we study it according to the constants λ₁, λ₂ and λ₃.

Combining Equations (4.3) and (4.4) yields

(4.5)−2ux−yWH=λ1x

(4.6)2WH=λ2xy

Case 1: λ₁λ₂ = 0. Then H = 0 and λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.5) into (4.6), we get

(4.7)−ux−y=λ1λ2xy

Taking the partial derivative of (4.7) with respect to x gives

(4.8)−uxx=λ1λ21y

The left-hand side in (4.8) is either a constant or a function of x, while other side is a function of y. That is not possible.

4.2 Surfaces of type 2

We follow the same steps as the previous type. By (2.2) and (3.9), the Laplacian Δr of r can be expressed as

(4.9)Δr=−2uxWHe1+21−vy−xWHe2+2vy+xWHe3

Suppose that M² satisfies the condition (1.1). Then, from (3.6) and (4.9), we obtain the following system of ordinary differential equations.

(4.10)−2uxWH=λ1x

(4.11)21−vy−xWH=λ2xy+λ3u+v

(4.12)2vy+xWH=λ21−xy−λ3u+v

Therefore, the problem of classifying the translation surfaces M² satisfying (1.1) is reduced to the integration of the system (4.10), (4.11) and (4.12) of ordinary differential equations. Applying similar algebraic methods, used in the case of surface of type 1, we will study this system according to the values of the constants λ₁, λ₂, λ₃.

By combining Equations (4.11) and (4.12) we obtain

(4.13)−2uxWH=λ1x

(4.14)2WH=λ2y

Case 1: λ₁λ₂ = 0.

1. If λ₂ = 0, then we have H = 0. Thus, the surface M² is minimal.

2. If λ₂ ≠ 0, from (4.13), we obtain u_x = 0. Substituting (3.10) into (4.14), with W=2vy+x−1−ux2, we get

(4.15)vyy=λ2y2vy+x−12

Taking the partial derivative of (4.15) with respect to x gives

(4.16)vy=1−2x2

The left-hand side in (4.16) is either a constant or a function of y, whereas other side is a function of x. That is not possible.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.13) into (4.14), we get

(4.17)−ux=λ1λ2xy

Taking the partial derivative of (4.17) with respect to x gives

(4.18)−uxx=λ1λ21y

The left-hand side in (4.18) is a constant or a function of x, whilst other side is a function of y. That is not possible.

4.3 Surfaces of type 3

By following the same steps as the previous types, by (2.2) and (3.14), we obtain

(4.19)Δr=2ux−yWHe1−2vyux−y+1WHe2+2vyux−yWHe3

Suppose that M² satisfies condition (1.1). Then, from (3.11) and (4.19), we obtain the following system of ordinary differential equations.

(4.20)2ux−yWH=λ1x+v

(4.21)−2vyux−y+1WH=λ2x+vy+λ3u−xy

(4.22)2vyux−yWH=1−x+vλ2y−λ3u−xy

Combining Equations (4.21) and (4.22) we have

(4.23)2ux−yWH=λ1x+v

(4.24)−2WH=λ2y

Case 1: λ₁λ₂ = 0. Then H = 0 and λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.23) into (4.24), we get

(4.25)−ux−y=λ1λ2x+vy

Taking the partial derivative of (4.25) with respect to x gives

(4.26)−uxx=λ1λ21y

We have an identity of two functions, one depending only on x and the other one depending only on y. That is not possible.

4.4 Surfaces of type 4

By (2.2) and (3.19), we obtain

(4.27)Δr=−2uxWHe1+2uxvy+1−v−xWHe2+2v+x−uxvyWHe3

Suppose that M² satisfies the condition (1.1). Then, from (3.16) and (4.27), we obtain the following system of ordinary differential equations.

(4.28)−2uxWH=λ1v+x

(4.29)2uxvy+1−v−xWH=λ2v+xy+λ3u

(4.30)2v+x−uxvyWH=λ21−v+xy−λ3u

Combining Equations (4.29) and (4.30) we have

(4.31)−2uxWH=λ1v+x

(4.32)2WH=λ2y

Case 1: λ₁λ₂ = 0.

1. If λ₂ = 0, then we have H = 0. Thus, the surface M² is minimal.

2. If λ₂ ≠ 0, from (4.31), we obtain u_x = 0. Then from (3.20) we have H = 0 and λ₂ = 0, which is contradiction.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.31) into (4.32), we get

(4.33)−ux=λ1λ2v+xy

Taking the partial derivative of (4.33) with respect to x gives

(4.34)−uxx=λ1λ21y

We have an identity of two functions, one depending only on x and the other one depending only on y. That is impossible.

4.5 Surfaces of type 5

By (2.2) and (3.24), the Laplacian Δ of r can be expressed as

(4.35)Δr=2uxvyWHe1−2ux+y−1WHe2+2ux+yWHe3

Suppose that M² satisfies condition (1.1). Then, from (3.21) and (4.35), we obtain the following system of ordinary differential equations.

(4.36)2uxvyWH=λ1y

(4.37)−2ux+y−1WH=λ2yv+x+λ3u

(4.38)2ux+yWH=λ21−yv+x−λ3u

Combining the Equations (4.37) and (4.38) we have

(4.39)2uxvyWH=λ1y

(4.40)2WH=λ2v+x

(4.41)2ux+2y−1WH=−λ3u

Case 1: λ₃ = 0. Since u depends only on x, then from (4.41) H = 0 and λ₁ = λ₂ = λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₃ ≠ 0.

1. If λ₂ = 0, from (4.40) H = 0, and λ₃ = 0, which is contradiction.

2. If λ₁ = 0 and λ₂ ≠ 0 from (4.39) we obtain u_xv_y = 0. Then

   • If u_x = 0. From (3.25) we obtain H = 0 and λ₃ = 0, which is a contradiction.

   • If v_y = 0, then v is constant v=a. By combining Equations (3.25) and (4.40) and the fact that W=2y+ux−ux2vy2−1, we conclude

(4.42)uxx=λ2a+x2y+2ux−12

Taking the partial derivative of (4.42) with respect to y gives

(4.43)ux=1−2y2

The right-hand side in (4.43) is either a function of y while other side is a constant or a function of x. That is impossible.

1. If λ₁λ₂ ≠ 0. Substituting (4.39) into (4.40) and (4.41) into (4.40) we get

(4.44)uxvy=λ1λ2yv+x

(4.45)ux+2y−1=−λ3λ2uv+x

Combining Equations (4.44) and (4.45) we have

(4.46)λ22y−1v+xvy=λ3uvy+λ1y

Taking the partial derivative of (4.46) with respect to x and the fact that v_y ≠ 0, we obtain

(4.47)ux=λ2λ32y−1

That is impossible because (4.47) is an identity of two functions, one depending only on x and the other one depending only on y.

4.6 Surfaces of type 6

By (2.2) and (3.29), we have

(4.48)Δr=2uxvy−yvy+xWHe1+21−uxWHe2+2uxWHe3

Suppose that M² satisfies condition (1.1). Then, from (3.26) and (4.48), we obtain the following system of ordinary differential equations.

(4.49)2uxvy−yvy+xWH=λ1y

(4.50)21−uxWH=λ2yv+x+λ3u−xy

(4.51)2x+ux−yWH=λ21−yv+x−λ3u−xy

Combining the Equations (4.50) and (4.51) we have

(4.52)2uxvy−yvy+xWH=λ1y

(4.53)2x−y+1WH=λ2v+x

(4.54)21−ux−x−y+1yWH=λ3u−xy

Case 1: λ₃ = 0 and H ≠ 0. Then from (4.54) we obtain

(4.55)1−ux−x−y+1y=0

Taking the partial derivative of (4.55) with respect to x, we obtain

(4.56)uxx=y

That is impossible because u_xx is function of x. Then H = 0 and λ₁ = λ₂ = λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₃ ≠ 0.

1. If λ₁λ₂ = 0. from (4.52), we have

(4.57)uxvy−yvy+x=0

Taking the partial derivative of (4.57) with respect to x, we obtain

(4.58)uxx=−1vy

Then both sides have to equal a nonzero constant, namely

uxx=a=−1vy

Which implies that vy=−1a. Considering it into (4.57) gives

(4.59)−uxa+x=−ya

The left-hand side in (4.59) is either constant or a function of x while other side is a function of y. That is impossible.

1. If λ₁λ₂ ≠ 0. Substituting (4.53) into (4.54), we get

(4.60)1−ux−x−y+1yv+x=λ3λ2u−xyx−y+1

The partial derivative of (4.60) with respect to x and y yields

(4.61)2x−2y+1+v−uxx−yvy=λ3λ22x−2y+1+ux

Taking partial derivative of (4.61) with respect to x and y leads to

uxxxvyy=0

We have two situations:

1. u_xxx = 0 i.e.

ux=b1x+b2.

Then (4.61) reduces to

(4.62)2x+1−λ3λ22x+1+b1x+b2=b1−yvy−v+2y1−λ3λ2

We have an identity of two functions, one depending only on x and the other one depending only on y. That is impossible.

1. v_yy = 0, i.e.

v=c1y+c2.

Then (4.61) reduces to

(4.63)2x−uxxc1−λ3λ22x+1+ux+c2+1=21−c1−λ3λ2y

The left-hand side in (4.63) is either constant or a function of x while other side is a function of y. That is impossible.

5. Conclusion

In this work, we give another characterization of translation surfaces in the Lorentz Heisenberg space H3 equipped with a flat metric. This is done by showing that these translation surfaces satisfy Condition Δr_i = λ_ir_i, i = 1, 2, 3, λi∈R, if and only if they have zero mean curvature.