On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space

if and only if M is a plane, sphere, or cylinder. Baikoussis and Blair [3] proved that a ruled surface M in R satisfies condition (3) if and only if M is a plane, helicoidal surface, or spiral surface in R. Additionally, Choi and Aĺıas et al. [4–6] completely classified the surfaces of revolution and ruled surfaces in 3-dimensional Minkowski space that satisfy condition (3). Kim and Yoon [7] studied ruled surfaces inR 1 such that


Introduction
The notion of finite type immersions introduced by Chen in [1] has been widely used in studying submanifolds of Euclidean and pseudo-Euclidean spaces.Also, such a notion can be extended to smooth maps on submanifolds.Among them the Gauss map is a very useful and extensively used to deal with submanifolds [2].
Let  be a connected surface in Euclidean 3-space R 3 , and let  :  → S 2 ⊂ R 3 be its Gauss map.It is well known [3] that  has constant mean curvature if and only if Δ = ‖Λ‖ 2 , with Δ being the Laplace operator on  corresponding to the induced metric on  from R 3 .As a special case, one can consider Euclidean surfaces whose Gauss map is an eigenfuction of the Laplacian; that is, Δ = ,  ∈ R. ( On the other hand, Chen and Piccinni [2] proved that the only compact surface in a Euclidean 3-space satisfying (1) is a sphere.Jang [4] studied that an orientable, connected surface in a Euclidean 3-space satisfying (1) is a sphere or a circular cylinder.On the generalization of (1), Dillen et al. [5] studied surfaces of revolution in a Euclidean 3-space R 3 such that their Gauss map  satisfies the condition and proved that such surfaces are part of the planes, the spheres, and the circular cylinders.
As a Lorentz version of Dillen et al. 's result, the author proves the following [6].
Theorem 1.The only spacelike or timelike surfaces of revolution in R 3  1 whose Gauss map  :  →  2 () satisfies (2) are locally the following spaces: Recently, we studied [7] such surface and its Gauss map satisfy the following condition in Minkowski space: where Δ ℎ is the Laplace operator with respect to the second fundamental form ℎ of the surface.This operator is formally defined by for the components ℎ  (,  = 1, 2) of the second fundamental form ℎ on , and we denote by (ℎ  ) (resp., H) the inverse matrix (resp., the determinant) of the matrix (ℎ  ).

Abstract and Applied Analysis
The main purpose of this note is to complete classification of surfaces of revolution in R 3  1 whose Gauss map satisfies the condition Δ ℎ  = Λ.Actually, we will show the de Sitter pseudosphere, the hyperbolic pseudosphere, and five kinds of catenoid satisfying the above condition.

Preliminaries
Let R 3  1 be a 3-dimensional Minkowski space with the scalar product and Lorentz cross product defined as for any vectors x = ( 0 ,  1 ,  2 ) and y = ( 0 ,  1 ,  2 ) in R 3 1 .A vector x of R 3  1 is said to be spacelike if ⟨x, x⟩ > 0 or x = 0, timelike if ⟨x, x⟩ < 0, and lightlike or null if ⟨x, x⟩ = 0 and x ̸ = 0.A timelike or lightlike vector in R 3 1 is said to be causal.Let  :  → R 3  1 be a smooth curve in R 3 1 , where  is an interval in R. We call  spacelike, timelike, or lightlike curve if the tangent vector   at any point is spacelike, timelike, or lightlike, respectively.
Let  be an open interval and  :  → Π a plane curve lying in a plane Π of R 3  1 and  a straight line in Π which does not intersect with the curve .A surface of revolution  with axis  in R 3  1 is defined to be invariant under the group of motions in R 3  1 , which fixes each point of the line  [8].From this we obtain four kinds of surfaces of revolution in R 3  1 .If the axis  is timelike (resp., spacelike), then there is a Lorentz transformation by which the axis  is transformed to the  0 -axis (resp.,  1 -axis or  2 -axis).Hence, without loss of generality, we may consider as the axis of revolution with the  0 -axis or  2 -axis if  is not null.If the axis is null, then we may assume that this axis is the line spanned by vector (1, 1, 0) on the plane  0  1 .
We now introduce three different types of surfaces of revolution in R 3  1 .
Type 1.The surface of revolution with timelike axis.
Without loss of generality, we choose  0  as the axis.Meanwhile suppose that  has a parameter as follows: where () and () are smooth functions and () > 0.
Then the surface of revolution  with  0 -axis may be given by Type 2. The surface of revolution with spacelike axis.
Then the surface of revolution  with  2 -axis may be given by or Type 3. The surface of revolution with lightlike axis.
Without loss of generality, we choose a line spanned by the vector (1, 1, 0) as axis, and suppose that curve  has a parameter as follows: where () is smooth positive function and () is smooth function such that ℎ() = () − () ̸ = 0. Then the surface of revolution  with the line spanned by vector (1, 1, 0) as axis may be given by Here we only consider type 1 and type 2, as for type 3 we have already discussed in [7].Now, let us consider the Gauss map  on a surface  in R 3  1 .The map  :  →  2 () ⊂ R 3 1 which sends each point of  to the unit normal vector to  at that point is called the Gauss map of surface .Here (= ±1) denotes the sign of the vector field  and  2 () is a 2-dimensional space form as follows: A surface  ⊂ R 3 1 is called minimal if and only if mean curvature  = 0. Now we consider some examples of minimal surfaces which will be mentioned in theorems.
Example 2 (the catenoid of the 1st kind is shown in Figure 1).A surface of catenoid of the 1st kind is parameterized by for sinh  > 0. Then the components of the first and the second fundamental forms are given by So the mean curvature  on the surface is Therefore, the surface of catenoid of the 1st kind is minimal.Example 3 (the catenoid of the 2nd kind is shown in Figure 2).A surface of catenoid of the 2nd kind is parameterized by for cos  > 0. Then the components of the first and the second fundamental forms are given by So the mean curvature  on the surface is Therefore, the surface of catenoid of the 2nd kind is minimal.
Example 4 (the catenoid of the 3rd kind is shown in Figure 3).A surface of catenoid of the 3rd kind is parameterized by for cos  > 0. Then the components of the first and the second fundamental forms are given by  So the mean curvature  on the surface is Therefore, the surface of catenoid of the 3rd kind is minimal.
Example 5 (the catenoid of the 4th kind is shown in Figure 4).A surface of catenoid of the 4th kind is parameterized by for sinh  > 0. Then the components of the first and the second fundamental forms are given by So the mean curvature  on the surface is Therefore, the surface of catenoid of the 4th kind is minimal.Example 6 (the catenoid of the 5th kind is shown in Figure 5).A surface of catenoid of the 5th kind is parameterized by for cosh  > 0. Then the components of the first and the second fundamental forms are given by So the mean curvature  on the surface is Therefore, the surface of catenoid of the 5th kind is minimal.
Example 7 (the de Sitter pseudosphere is shown in Figure 6).The de Sitter pseudosphere centered at (0, 0, 0) with radius 1 is parameterized by Then its Gauss map  and Laplacian are given by By a straight computation, we get So we have that is, the de Sitter pseudosphere satisfies condition (3).

The Surface of Revolution with Timelike Axis
In this section, we will classify the surfaces of revolution with timelike axis in R 3 1 that satisfy condition (3).
Theorem 8.The only surfaces of revolution with timelike axis in R 3 1 whose Gauss map  satisfies are locally the catenoid of the 1st kind, the catenoid of the 3rd kind, the de Sitter pseudosphere, or the hyperbolic pseudosphere.
Proof.Let  be a surface of revolution with timelike axis as (7).We may assume that the profile curve  is of unit speed; thus We will give detailed proof just for the case  = 1.Then  is a spacelike surface and we may put for the smooth function  = ().Using the natural frame {  ,  V } of  defined by we obtain the components of the first and the second fundamental forms of the surface  as follows: where Gauss map  is defined by So the matrix (ℎ  ) composed by the second fundamental form ℎ can be expressed as Since the surface has no parabolic points, so H = ℎ 11 ℎ 22 − ℎ 2 12 =    sinh  ̸ = 0 for every .Then the mean curvature  on  is given by By a straightforward computation, the Laplacian Δ ℎ of the second fundamental form ℎ on  with the help of ( 4), (35), and (37) turns out to be Accordingly, we get By the assumption (33) and the above equation, we get the following system of differential equations: where   (,  = 1, 2, 3) denote the components of the matrix Λ given by (33).In order to prove the theorem, we have to solve the above system of ordinary differential equations.From (42) we easily deduce that  12 =  21 =  13 =  23 =  31 =  32 = 0 and  22 =  33 ; that is, the matrix Λ is diagonal.We put  22 =  33 =  and  11 = .Then, the system (42) is reduced to the following equations: By the computation (43) × cosh  -(44) × sinh , we easily get We discuss five cases according to the constants  and .
Case 3 ( ̸ = 0,  = 0).In this case, (45) becomes (sinh /) +   = sinh 2 ; that is, and thus Substituting ( 54) and ( 55) into (43), we get Differentiating the above equation, we have If we take the differentiation of the equation once again, we get Since  is a positive function and  ̸ = 0, cosh  sinh 2  = 0 for every .Therefore,  = () is vanishing identically for every .Hence, we have It implies that  is a part of a Euclidean plane R 2 whose points are parabolic.Thus, there is no surface of revolution with timelike axis satisfying this case.
Case 4 ( = 0,  ̸ = 0).In this case, (45) becomes (sinh /) +   = −cosh 2 ; that is, and thus Furthermore, by ( 43), (60), and (61), we get where we put Differentiating (62) and using (60), we find where Combining ( 62) and (64), we can show where Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in sinh  satisfying where  1 = 540672,  2 = 6397952, . ..,  12 = 3072000 are coefficients as nonzero constant of the function sinh 8+2 .Since this polynomial is equal to zero for every , all its coefficients must be zero.Thus, we have  = 0.So we get a contradiction, and therefore, in this case, there are no surfaces of revolution with timelike axis.
Case 5 (let  ̸ = 0,  ̸ = 0, and  ̸ = ).In this case, (45) is unchanged; that is, and thus From which, (43) is written as where Differentiating (70) and using (68), we find where Combining ( 70) and (72), we have where Hence, by this procedure, (74) is reduced to a linear one with respect to the function .Therefore, if we repeat this method one more time, we can find the following polynomial: where  1 = 258048,  2 = 5046272, . ..,  12 = 3072000 are nonzero constants.Since this polynomial is equal to zero for every , all its coefficients must be zero.Therefore we conclude that  = 0; that is,  = , which is a contradiction.Consequently, there are no surfaces of revolution with timelike axis in this case.When  = −1,  is a timelike surface.In this case, we can assume that   () = sinh  and   () = cosh , and using the same algebraic techniques as for  = 1 easily prove that the 3rd kind of catenoid and the de Sitter pseudosphere satisfy condition (33).This completes the proof.

The Surface of Revolution with Spacelike Axis
In this section, we will classify the surfaces of revolution with spacelike axis in R 3 1 that satisfy condition (3).
Theorem 9.The only surfaces of revolution with spacelike axis in R 3 1 whose Gauss map  satisfies are locally the 2nd kind of catenoid, the 4th kind of catenoid, the 5th kind of catenoid, the hyperbolic pseudosphere, or the de Sitter pseudosphere.