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

Abstract

We study surfaces of revolution with a nonlightlike axis in 3-dimensional Minkowski space and classify such surfaces in terms of the Gauss map 𝐺 that satisfies the condition Δ^ℎ𝐺=Λ𝐺, with Λ being a 3×3 real matrix. Furthermore, this paper completes the classification problem of surfaces of revolution in Minkowski 3-space given by Jin et al. (2013).

1. 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 𝑀 be a connected surface in Euclidean 3-space ℝ³, and let 𝐺∶𝑀→𝕊²⊂ℝ³ be its Gauss map. It is well known [3] that 𝑀 has constant mean curvature if and only if Δ𝐺=‖Λ‖²𝐺, with Δ being the Laplace operator on 𝑀 corresponding to the induced metric on 𝑀 from ℝ³. As a special case, one can consider Euclidean surfaces whose Gauss map is an eigenfuction of the Laplacian; that is, Δ𝐺=𝜆𝐺,𝜆∈ℝ. (1)

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 ℝ³ such that their Gauss map 𝐺 satisfies the condition Δ𝐺=Λ𝐺,Λ∈Mat(3,ℝ) (2) 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 ℝ³₁ whose Gauss map 𝐺∶𝑀→𝑀²(𝜀) satisfies (2) are locally the following spaces:(1)ℝ²₁,𝑆¹₁×𝑅¹ and ℝ¹₁×𝑆¹ if 𝜀=1,(2)ℝ²,𝐻² and 𝐻¹×ℝ¹ if 𝜀=−1.

Recently, we studied [7] such surface and its Gauss map satisfy the following condition in Minkowski space: Δ^ℎ𝐺=Λ𝐺,Λ∈Mat(3,ℝ), (3) where Δ^ℎ is the Laplace operator with respect to the second fundamental form ℎ of the surface. This operator is formally defined by Δ^ℎ1=−|ℋ|²_𝑖,𝑗=1𝜕𝜕𝑥^𝑖|ℋ|ℎ^𝑖𝑗𝜕𝜕𝑥^𝑗 (4) for the components ℎ_𝑖𝑗 (𝑖,𝑗=1,2) of the second fundamental form ℎ on 𝑀, and we denote by (ℎ^𝑖𝑗) (resp., ℋ) the inverse matrix (resp., the determinant) of the matrix (ℎ_𝑖𝑗).

The main purpose of this note is to complete classification of surfaces of revolution in ℝ³₁ whose Gauss map satisfies the condition Δ^ℎ𝐺=Λ𝐺. Actually, we will show the de Sitter pseudosphere, the hyperbolic pseudosphere, and five kinds of catenoid satisfying the above condition.

2. Preliminaries

Let ℝ³₁ be a 3-dimensional Minkowski space with the scalar product and Lorentz cross product defined as ⟨𝐱,𝐲⟩=−𝑥₀𝑦₀+𝑥₁𝑦₁+𝑥₂𝑦₂,𝐱×𝐲=𝑥₂𝑦₁−𝑥₁𝑦₂,𝑥₂𝑦₀−𝑥₀𝑦₂,𝑥₀𝑦₁−𝑥₁𝑦₀ (5) for any vectors 𝐱=(𝑥₀,𝑥₁,𝑥₂) and 𝐲=(𝑦₀,𝑦₁,𝑦₂) in ℝ³₁.

A vector 𝐱 of ℝ³₁ 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 ℝ³₁ is said to be causal. Let 𝛾∶𝐼→ℝ³₁ be a smooth curve in ℝ³₁, where 𝐼 is an interval in ℝ. We call 𝛾 spacelike, timelike, or lightlike curve if the tangent vector 𝛾^′ at any point is spacelike, timelike, or lightlike, respectively.

Let 𝐼 be an open interval and 𝛾∶𝐼→Π a plane curve lying in a plane Π of ℝ³₁ and 𝑙 a straight line in Π which does not intersect with the curve 𝛾. A surface of revolution 𝑀 with axis 𝑙 in ℝ³₁ is defined to be invariant under the group of motions in ℝ³₁, which fixes each point of the line 𝑙 [8]. From this we obtain four kinds of surfaces of revolution in ℝ³₁. If the axis 𝑙 is timelike (resp., spacelike), then there is a Lorentz transformation by which the axis 𝑙 is transformed to the 𝑥₀-axis (resp., 𝑥₁-axis or 𝑥₂-axis). Hence, without loss of generality, we may consider as the axis of revolution with the 𝑥₀-axis or 𝑥₂-axis if 𝑙 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 𝑂𝑥₀𝑥₁.

We now introduce three different types of surfaces of revolution in ℝ³₁.

Type 1. The surface of revolution with timelike axis.

Without loss of generality, we choose 𝑥₀𝑂 as the axis. Meanwhile suppose that 𝛾 has a parameter as follows: 𝛾(𝑢)=𝑔(𝑢),𝑓(𝑢),0, (6) where 𝑓(𝑢) and 𝑔(𝑢) are smooth functions and 𝑓(𝑢)>0. Then the surface of revolution 𝑀 with 𝑥₀-axis may be given by 𝑥(𝑢,𝑣)=𝑔(𝑢),𝑓(𝑢)cos𝑣,𝑓(𝑢)sin𝑣. (7)

Type 2. The surface of revolution with spacelike axis.

Without loss of generality, we choose 𝑥₂𝑂 as the axis, and suppose that curve 𝛾 has a parameter as follows: 𝛾(𝑢)=0,𝑓(𝑢),𝑔(𝑢)or𝛾(𝑢)=𝑓(𝑢),0,𝑔(𝑢), (8) where 𝑓(𝑢) and 𝑔(𝑢) are smooth functions and 𝑓(𝑢)>0. Then the surface of revolution 𝑀 with 𝑥₂-axis may be given by 𝑥(𝑢,𝑣)=𝑓(𝑢)sinh𝑣,𝑓(𝑢)cosh𝑣,𝑔(𝑢) (9) or 𝑥(𝑢,𝑣)=𝑓(𝑢)cosh𝑣,𝑓(𝑢)sinh𝑣,𝑔(𝑢). (10)

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 𝛾 has a parameter as follows: 𝛾(𝑢)=𝑓(𝑢),𝑔(𝑢),0, (11) where 𝑓(𝑢) is smooth positive function and 𝑔(𝑢) is smooth function such that ℎ(𝑢)=𝑓(𝑢)−𝑔(𝑢)≠0. Then the surface of revolution 𝑀 with the line spanned by vector (1,1,0) as axis may be given by 𝑥()=𝑓(𝑢)+𝑣𝑢,𝑣²2ℎ(𝑢)(𝑢)+𝑣,𝑔²2ℎ(𝑢),ℎ(𝑢)𝑣. (12)

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 𝐺 on a surface 𝑀 in ℝ³₁. The map 𝐺∶𝑀→𝑄²(𝜀)⊂ℝ³₁ which sends each point of 𝑀 to the unit normal vector to 𝑀 at that point is called the Gauss map of surface 𝑀. Here 𝜀(=±1) denotes the sign of the vector field 𝐺 and 𝑄²(𝜀) is a 2-dimensional space form as follows: 𝑄²(𝜀)=𝑆²₁(1)inℝ³₁𝐻if𝜀=1,₂(−1)inℝ³₁if𝜀=−1. (13)

A surface 𝑀⊂ℝ³₁ is called minimal if and only if mean curvature 𝐻=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 𝑥(𝑢,𝑣)=(𝑢,sinh𝑢cos𝑣,sinh𝑢sin𝑣) (14) for sinh𝑢>0. Then the components of the first and the second fundamental forms are given by 𝑔₁₁=sinh²𝑢,𝑔₁₂=𝑔₂₁=0,𝑔₂₂=sinh²ℎ𝑢,₁₁=−1,ℎ₁₂=ℎ₂₁=0,ℎ₂₂=1. (15) So the mean curvature 𝐻 on the surface is ()𝐻=−1sinh²𝑢+1sinh²𝑢2sinh²𝑢sinh²𝑢=0. (16) Therefore, the surface of catenoid of the 1st kind is minimal.

Figure 1 

The catenoid of the 1st kind.

Example 3 (the catenoid of the 2nd kind is shown in Figure 2). A surface of catenoid of the 2nd kind is parameterized by 𝑥(𝑢,𝑣)=(cos𝑢cosh𝑣,cos𝑢sinh𝑣,𝑢) (17) for cos𝑢>0. Then the components of the first and the second fundamental forms are given by 𝑔₁₁=cos²𝑢,𝑔₁₂=𝑔₂₁=0,𝑔₂₂=cos²ℎ𝑢,₁₁=−1,ℎ₁₂=ℎ₂₁=0,ℎ₂₂=1. (18) So the mean curvature 𝐻 on the surface is ()𝐻=−1cos²𝑢+1cos²𝑢2cos²𝑢cos²𝑢=0. (19) Therefore, the surface of catenoid of the 2nd kind is minimal.

Figure 2 

The catenoid of the 2nd kind.

Example 4 (the catenoid of the 3rd kind is shown in Figure 3). A surface of catenoid of the 3rd kind is parameterized by 𝑥(𝑢,𝑣)=(𝑢,cos𝑢cos𝑣,cos𝑢sin𝑣) (20) for cos𝑢>0. Then the components of the first and the second fundamental forms are given by 𝑔₁₁=−cos²𝑢,𝑔₁₂=𝑔₂₁=0,𝑔₂₂=cos²ℎ𝑢,₁₁=cos²𝑢,ℎ₁₂=ℎ₂₁=0,ℎ₂₂=cos²𝑢. (21) So the mean curvature 𝐻 on the surface is 𝐻=cos²𝑢cos²𝑢+cos²𝑢−cos²𝑢2−cos²𝑢cos²𝑢=0. (22) Therefore, the surface of catenoid of the 3rd kind is minimal.

Figure 3 

The catenoid of the 3rd kind.

Example 5 (the catenoid of the 4th kind is shown in Figure 4). A surface of catenoid of the 4th kind is parameterized by 𝑥(𝑢,𝑣)=(sinh𝑢cosh𝑣,sinh𝑢sinh𝑣,𝑢) (23) for sinh𝑢>0. Then the components of the first and the second fundamental forms are given by 𝑔₁₁=−sinh²𝑢,𝑔₁₂=𝑔₂₁=0,𝑔₂₂=sinh²ℎ𝑢,₁₁=−1,ℎ₁₂=ℎ₂₁=0,ℎ₂₂=−1. (24) So the mean curvature 𝐻 on the surface is ()𝐻=−1sinh²𝑢+(−1)−sinh²𝑢2−sinh²𝑢sinh²𝑢=0. (25) Therefore, the surface of catenoid of the 4th kind is minimal.

Figure 4 

The catenoid of the 4th kind.

Example 6 (the catenoid of the 5th kind is shown in Figure 5). A surface of catenoid of the 5th kind is parameterized by 𝑥(𝑢,𝑣)=(cosh𝑢sinh𝑣,cosh𝑢cosh𝑣,𝑢) (26) for cosh𝑢>0. Then the components of the first and the second fundamental forms are given by 𝑔₁₁=cosh²𝑢,𝑔₁₂=𝑔₂₁=0,𝑔₂₂=−cosh²ℎ𝑢,₁₁=1,ℎ₁₂=ℎ₂₁=0,ℎ₂₂=1. (27) So the mean curvature 𝐻 on the surface is 1𝐻=−cosh²𝑢+1cosh²𝑢2cosh²𝑢−cosh²𝑢=0. (28) Therefore, the surface of catenoid of the 5th kind is minimal.

Figure 5 

The catenoid of the 5th kind.

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 𝑥(𝑢,𝑣)=(sinh𝑢,cosh𝑢cos𝑣,cosh𝑢sin𝑣). (29) Then its Gauss map 𝐺 and Laplacian are given by (),Δ𝐺=−sinh𝑢,−cosh𝑢cos𝑣,−cosh𝑢sin𝑣_ℎ=𝜕²𝜕𝑢²−1cosh²𝑢𝜕²𝜕𝑣²+sinh𝑢𝜕cosh𝑢.𝜕𝑢 (30) By a straight computation, we get Δ^ℎ𝐺=(−2sinh𝑢,−2cosh𝑢cos𝑣,−2cosh𝑢sin𝑣). (31) So we have Δ^ℎ𝐺=200020002𝐺; (32) that is, the de Sitter pseudosphere satisfies condition (3).

Figure 6 

The de Sitter pseudosphere.

3. The Surface of Revolution with Timelike Axis

In this section, we will classify the surfaces of revolution with timelike axis in ℝ³₁ that satisfy condition (3).

Theorem 8. The only surfaces of revolution with timelike axis in ℝ³₁ whose Gauss map 𝐺 satisfies Δ^ℎ𝐺=Λ𝐺,Λ∈Mat(3,ℝ) (33) are locally the catenoid of the 1st kind, the catenoid of the 3rd kind, the de Sitter pseudosphere, or the hyperbolic pseudosphere.

Proof. Let 𝑀 be a surface of revolution with timelike axis as (7). We may assume that the profile curve 𝛾 is of unit speed; thus 𝛾^′,𝛾^′=−𝑔^′2(𝑢)+𝑓^′2(𝑢)=𝜀(±1). (34) We will give detailed proof just for the case 𝜀=1. Then 𝑀 is a spacelike surface and we may put 𝑓^′(𝑢)=cosh𝑡,𝑔^′(𝑢)=sinh𝑡 (35) for the smooth function 𝑡=𝑡(𝑢). Using the natural frame {𝑥_𝑢,𝑥_𝑣} of 𝑀 defined by 𝑥_𝑢=𝑔^′,𝑓^′cos𝑣,𝑓^′,𝑥sin𝑣_𝑣=,𝑥0,−𝑓sin𝑣,𝑓cos𝑣_𝑢𝑢=𝑔^′′,𝑓^′′cos𝑣,𝑓^′′,𝑥sin𝑣_𝑢𝑣=0,−𝑓^′sin𝑣,𝑓^′,𝑥cos𝑣_𝑣𝑣=0,−𝑓cos𝑣,−𝑓sin𝑣, (36) we obtain the components of the first and the second fundamental forms of the surface 𝑀 as follows: 𝑔₁₁=𝑥_𝑢,𝑥_𝑢𝑔=1,₁₂=𝑔₂₁=⟨𝑥_𝑢,𝑥_𝑣𝑔⟩=0,₂₂=𝑥_𝑣,𝑥_𝑣=𝑓²,ℎ₁₁=𝑥_𝑢𝑢,𝐺=𝑓^′𝑔^′′−𝑓^′′𝑔^′=𝑡^′,ℎ₁₂=ℎ₂₁=𝑥_𝑢𝑣ℎ,𝐺=0,₂₂=𝑥_𝑣𝑣,𝐺=𝑓𝑔^′=𝑓sinh𝑡, (37) where Gauss map 𝐺 is defined by (𝑥_𝑢×𝑥_𝑣)/|𝑥_𝑢×𝑥_𝑣|=(−𝑓^′,−𝑔^′cos𝑣,−𝑔^′sin𝑣).
So the matrix (ℎ_𝑖𝑗) composed by the second fundamental form ℎ can be expressed as ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂=𝑡^′00𝑓sinh𝑡. (38) Since the surface has no parabolic points, so ℋ=ℎ₁₁ℎ₂₂−ℎ²₁₂=𝑡^′𝑓sinh𝑡≠0 for every 𝑡. Then the mean curvature 𝐻 on 𝑀 is given by 𝑓𝐻=^′𝑔^′′−𝑓^′′𝑔^′𝑓²+𝑓𝑔^′2𝑓²=12𝑡^′+sinh𝑡𝑓. (39) By a straightforward computation, the Laplacian Δ^ℎ of the second fundamental form ℎ on 𝑀 with the help of (4), (35), and (37) turns out to be Δ^ℎ1=−𝑡^′𝜕²𝜕𝑢²−1𝜕𝑓sinh𝑡𝜕𝑣²²+𝑡^′′2𝑡^′2−cosh𝑡2𝑓𝑡^′−cosh𝑡𝜕2sinh𝑡.𝜕𝑢 (40) Accordingly, we getΔ^ℎ3𝐺=2𝑡^′cosh𝑡+cosh𝑡+𝑡2𝑓2𝑡^′′′𝑡sinh𝑡_′𝑡sinh𝑡+^′′2𝑡^′1cosh𝑡−+2𝑓sinh²𝑡2𝑓+𝑡^′cosh²𝑡𝑡2sinh𝑡cos𝑣_′𝑡sinh𝑡+^′′2𝑡^′1cosh𝑡−+2𝑓sinh²𝑡2𝑓+𝑡^′cosh²𝑡2sinh𝑡sin𝑣. (41)By the assumption (33) and the above equation, we get the following system of differential equations: 32𝑡^′cosh𝑡+cosh𝑡+𝑡2𝑓2𝑡^′′′sinh𝑡+𝑎₁₁cosh𝑡+𝑎₁₂sinh𝑡cos𝑣+𝑎₁₃𝑡sinh𝑡sin𝑣=0,_′𝑡sinh𝑡+^′′2𝑡^′1cosh𝑡−+2𝑓sinh²𝑡2𝑓+𝑡^′cosh²𝑡2sinh𝑡+𝑎₂₂sinh𝑡cos𝑣+𝑎₂₁cosh𝑡+𝑎₂₃𝑡sinh𝑡sin𝑣=0_′𝑡sinh𝑡+^′′2𝑡^′1cosh𝑡−+2𝑓sinh²𝑡2𝑓+𝑡^′cosh²𝑡2sinh𝑡+𝑎₃₃sinh𝑡sin𝑣+𝑎₃₁cosh𝑡+𝑎₃₂sinh𝑡cos𝑣=0, (42) where 𝑎_𝑖𝑗 (𝑖,𝑗=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 𝑎₁₂=𝑎₂₁=𝑎₁₃=𝑎₂₃=𝑎₃₁=𝑎₃₂=0 and 𝑎₂₂=𝑎₃₃; that is, the matrix Λ is diagonal. We put 𝑎₂₂=𝑎₃₃=𝜆 and 𝑎₁₁=𝜇. Then, the system (42) is reduced to the following equations: 32𝑡^′cosh𝑡+cosh𝑡+𝑡2𝑓2𝑡^′′′𝑡sinh𝑡=−𝜇cosh𝑡,_′𝑡sinh𝑡+^′′2𝑡^′1cosh𝑡−+2𝑓sinh²𝑡2𝑓+𝑡^′cosh²𝑡2sinh𝑡=−𝜆sinh𝑡. (43)(44) By the computation (43) × cosh𝑡 − (44) × sinh𝑡, we easily get sinh𝑡𝑓+𝑡^′=−𝜇cosh²𝑡+𝜆sinh²𝑡. (45) We discuss five cases according to the constants 𝜆 and 𝜇.
Case 1  (𝜆=𝜇=0). In this case, we easily get 𝑡^′+(sinh𝑡/𝑓)=0, which implies that the mean curvature 𝐻 vanishes identically because of (39). Therefore, the surface is minimal; from theorem 1 in [9], it is the 1st kind of catenoid. Furthermore, the 1st kind of catenoid satisfies condition (33).
Case 2  (𝜆=𝜇≠0). By (45), we get 𝑡^′=−sinh𝑡𝑓−𝜆. (46) Differentiating (46) with respect to 𝑢, we have 𝑡^′′𝑡=−^′cosh𝑡𝑓+sinh𝑡cosh𝑡𝑓². (47) Combining (46), (47), and (43), we get 𝜆²𝑓²+4𝜆𝑓sinh𝑡+4sinh²𝑡=0, (48) from which 𝑓=−2sinh𝑡𝜆. (49) Furthermore, (49) together with (46) becomes 𝑡^′=−𝜆/2; that is, 𝑡(𝑢)𝜆=−2𝑢+𝑘,𝑘∈ℝ. (50) On the other hand, by (35) and (50), we get 𝑓(𝑢)2=−𝜆−𝜆sinh2,𝑔(𝑢)2𝑢+𝑘=−𝜆−𝜆cosh2𝑢+𝑘+𝑐,𝑐∈ℝ. (51) Then, the surface 𝑀 has the following expression: 𝑥()=−2𝑢,𝑣𝜆−𝜆cosh2−2𝑢+𝑘+𝑐,𝜆−𝜆sinh2−2𝑢+𝑘cos𝑣,𝜆−𝜆sinh2𝑢+𝑘sin𝑣,𝑐,𝑘∈ℝ. (52) Consequently, we have ⟨𝑥()()⟩2𝑢,𝑣−𝐂,𝑥𝑢,𝑣−𝐂=−𝜆²,𝐂=(𝑐,0,0), (53) which means that the surface 𝑀 is contained in the hyperbolic pseudosphere 𝐻²(−2/|𝜆|) centered at 𝐂 with radius 2/|𝜆|. Also, the hyperbolic pseudosphere satisfies condition (33).
Case 3  (𝜆≠0, 𝜇=0). In this case, (45) becomes (sinh𝑡/𝑓)+𝑡^′=𝜆sinh²𝑡; that is, 𝑡^′=−sinh𝑡𝑓+𝜆sinh²𝑡, (54) and thus 𝑡^′′𝑡=−^′cosh𝑡𝑓+sinh𝑡cosh𝑡𝑓²+2𝜆𝑡^′sinh𝑡cosh𝑡. (55) Substituting (54) and (55) into (43), we get 5𝜆²𝑓²sinh²𝑡−8𝜆𝑓sinh𝑡+4=0. (56) Differentiating the above equation, we have 5𝜆𝑓sinh𝑡−4=0. (57) If we take the differentiation of the equation once again, we get 𝜆²𝑓sinh²𝑡cosh𝑡=0. (58) Since 𝑓 is a positive function and 𝜆≠0, cosh𝑡sinh²𝑡=0 for every 𝑡. Therefore, 𝑡=𝑡(𝑢) is vanishing identically for every 𝑢. Hence, we have 𝑓(𝑢)𝑔(𝑢)=𝑢+𝑐,=𝑎,𝑎,𝑐∈ℝ. (59) It implies that 𝑀 is a part of a Euclidean plane ℝ² whose points are parabolic. Thus, there is no surface of revolution with timelike axis satisfying this case.
Case 4  (𝜆=0, 𝜇≠0). In this case, (45) becomes (sinh𝑡/𝑓)+𝑡^′=−𝜇cosh²𝑡; that is, 𝑡^′=−sinh𝑡𝑓−𝜇cosh²𝑡, (60) and thus 𝑡^′′𝑡=−^′cosh𝑡𝑓+sinh𝑡cosh𝑡𝑓²−2𝜇𝑡^′sinh𝑡cosh𝑡. (61) Furthermore, by (43), (60), and (61), we get 𝛼₁𝑓²+𝛼₂𝑓+𝛼₃=0, (62) where we put 𝛼₁=𝜇²5sinh⁴𝑡+6sinh²,𝛼𝑡+1₂=𝜇8sinh³,𝛼𝑡+4sinh𝑡₃=4sinh²𝑡. (63) Differentiating (62) and using (60), we find 𝛽₁𝑓²+𝛽₂𝑓+𝛽₃=0, (64) where 𝛽₁=𝜇²90sinh⁸𝑡+172sinh⁶𝑡+104sinh⁴𝑡+20sinh²,𝛽𝑡−2₂=𝜇120sinh⁷𝑡+120sinh⁵𝑡+24sinh³,𝛽𝑡−8sinh𝑡₃=40sinh⁶𝑡−8sinh²𝑡. (65) Combining (62) and (64), we can show 𝛾₁𝑓+𝛾₂=0, (66) where 𝛾₁=𝛼₂𝛽₁−𝛼₁𝛽₂, 𝛾₂=𝛼₃𝛽₁−𝛼₁𝛽₃.
Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in sinh𝑡 satisfying 𝜇²¹²_𝑖=1𝑐_𝑖sinh^8+2𝑖𝑡=0, (67) where 𝑐₁=540672, 𝑐₂=6397952,…, 𝑐₁₂=3072000 are coefficients as nonzero constant of the function sinh^8+2𝑖𝑡. Since this polynomial is equal to zero for every 𝑡, all its coefficients must be zero. Thus, we have 𝜇=0. So we get a contradiction, and therefore, in this case, there are no surfaces of revolution with timelike axis.
Case 5  (let 𝜆≠0, 𝜇≠0, and 𝜆≠𝜇). In this case, (45) is unchanged; that is, 𝑡^′=−sinh𝑡𝑓−𝜇cosh²𝑡+𝜆sinh²𝑡, (68) and thus 𝑡^′′𝑡=−^′cosh𝑡𝑓+sinh𝑡cosh𝑡𝑓²−2𝜇𝑡^′sinh𝑡cosh𝑡+2𝜆𝑡^′sinh𝑡cosh𝑡. (69) From which, (43) is written as 𝜉₁𝑓²+𝜉₂𝑓+𝜉₃=0, (70) where 𝜉₁=5𝜃²sinh⁴𝑡−6𝜇𝜃sinh²𝑡+𝜇²,𝜉₂=−8𝜃sinh³𝜉𝑡+4𝜇sinh𝑡,₃=4sinh²𝑡,𝜃=𝜆−𝜇. (71) Differentiating (70) and using (68), we find 𝜂₁𝑓²+𝜂₂𝑓+𝜂₃=0, (72) where 𝜂₁=90𝜃⁴sinh⁸𝑡+172𝜇𝜃³sinh⁶𝑡+104𝜇²𝜃²sinh⁴𝑡+20𝜇³𝜃sinh²𝑡−2𝜇⁴,𝜂₂=−120𝜃³sinh⁷𝑡+120𝜇𝜃²sinh⁵𝑡−24𝜇²𝜃sinh³𝑡−8𝜇³𝜂sinh𝑡,₃=40𝜃²sinh⁶𝑡−8𝜇²sinh²𝑡. (73) Combining (70) and (72), we have 𝜁₁𝑓+𝜁₂=0, (74) where 𝜁₁=𝜉₂𝜂₁−𝜉₁𝜂₂,𝜁₂=𝜉₃𝜂₁−𝜉₁𝜂₃.
Hence, by this procedure, (74) is reduced to a linear one with respect to the function 𝑓. Therefore, if we repeat this method one more time, we can find the following polynomial: ¹²_𝑖=1𝑐_𝑖𝜃^2+𝑖𝜇^12−𝑖sinh^8+2𝑖𝑡, (75) where 𝑐₁=258048, 𝑐₂=5046272,…, 𝑐₁₂=3072000 are nonzero constants. Since this polynomial is equal to zero for every 𝑡, all its coefficients must be zero. Therefore we conclude that 𝜃=0; that is, 𝜆=𝜇, which is a contradiction. Consequently, there are no surfaces of revolution with timelike axis in this case.
When 𝜀=−1, 𝑀 is a timelike surface. In this case, we can assume that 𝑓^′(𝑢)=sinh𝑡 and 𝑔^′(𝑢)=cosh𝑡, and using the same algebraic techniques as for 𝜀=1 easily prove that the 3rd kind of catenoid and the de Sitter pseudosphere satisfy condition (33). This completes the proof.

4. The Surface of Revolution with Spacelike Axis

In this section, we will classify the surfaces of revolution with spacelike axis in ℝ³₁ that satisfy condition (3).

Theorem 9. The only surfaces of revolution with spacelike axis in ℝ³₁ whose Gauss map 𝐺 satisfies Δ^ℎ𝐺=Λ𝐺,Λ∈Mat(3,ℝ) (76) 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.

Proof. Let 𝑀 be a surface of revolution with spacelike axis as (9). We may assume that the profile curve 𝛾 is of unit speed; thus ⟨𝛾^′,𝛾^′⟩=𝑓^′2(𝑢)+𝑔^′2(𝑢)=1. (77) Then we may put 𝑓^′(𝑢)=cos𝑡,𝑔^′(𝑢)=sin𝑡 (78) for the smooth function 𝑡=𝑡(𝑢). Using the natural frame {𝑥_𝑢,𝑥_𝑣} of 𝑀 defined by 𝑥_𝑢=𝑓^′sinh𝑣,𝑓^′cosh𝑣,𝑔^′,𝑥_𝑣=,𝑥𝑓cosh𝑣,𝑓sinh𝑣,0_𝑢𝑢=𝑓^′′sinh𝑣,𝑓^′′cosh𝑣,𝑔^′′,𝑥_𝑢𝑣=𝑓^′cosh𝑣,𝑓^′,𝑥sinh𝑣,0_𝑣𝑣=𝑓sinh𝑣,𝑓cosh𝑣,0, (79) we obtain the components of the first and the second fundamental forms of the surface as follows: 𝑔₁₁=𝑥_𝑢,𝑥_𝑢=1,𝑔₁₂=𝑔₂₁=𝑥_𝑢,𝑥_𝑣𝑔=0,₂₂=𝑥_𝑣,𝑥_𝑣=−𝑓²,ℎ₁₁=⟨𝑥_𝑢𝑢,𝐺⟩=𝑓^′′𝑔^′−𝑓^′𝑔^′′=−𝑡^′,ℎ₁₂=ℎ₂₁=⟨𝑥_𝑢𝑣ℎ,𝐺⟩=0,₂₂=𝑥_𝑣𝑣,𝐺=𝑓𝑔^′=𝑓sin𝑡, (80) where Gauss map 𝐺 is (𝑥_𝑢×𝑥_𝑣)/|𝑥_𝑢×𝑥_𝑣|=(𝑔^′sinh𝑣,𝑔^′cosh𝑣,−𝑓^′).
So the matrix (ℎ_𝑖𝑗) composed by the second fundamental form ℎ can be expressed as ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂=−𝑡^′00𝑓sin𝑡. (81) Since the surface has no parabolic points, so ℋ=ℎ₁₁ℎ₂₂−ℎ²₁₂=𝑡^′𝑓sin𝑡≠0 for every 𝑡. Then the mean curvature 𝐻 on 𝑀 is given by 𝑓𝐻=^′′𝑔^′−𝑓^′𝑔^′′−𝑓²+𝑓𝑔^′2−𝑓²1=−2𝑡^′+sin𝑡𝑓. (82) By a straightforward computation, the Laplacian Δ^ℎ of the second fundamental form ℎ on 𝑀 with the help of (4), (78), and (80) turns out to be Δ^ℎ=1𝑡^′𝜕²𝜕𝑢²−1𝜕𝑓sin𝑡𝜕𝑣²²+−𝑡^′′2𝑡^′2+cos𝑡2𝑓𝑡^′+cos𝑡𝜕2sin𝑡.𝜕𝑢 (83) Accordingly, we getΔ^ℎ𝐺=−𝑡^′𝑡sin𝑡+^′′2𝑡^′1cos𝑡−−2𝑓sin²𝑡2𝑓+𝑡^′cos²𝑡2sin𝑡sinh𝑣−𝑡^′𝑡sin𝑡+^′′2𝑡^′1cos𝑡−−2𝑓sin²𝑡2𝑓+𝑡^′cos²𝑡32sin𝑡cosh𝑣2𝑡^′cos𝑡+cos𝑡+𝑡2𝑓2𝑡^′′′sin𝑡. (84)By the assumption (76) and the above equation, we get the following system of differential equations: −𝑡^′𝑡sin𝑡+^′′2𝑡^′1cos𝑡−−2𝑓sin²𝑡2𝑓+𝑡^′cos²𝑡2sin𝑡−𝑎₁₁sin𝑡sinh𝑣−𝑎₁₂sin𝑡cosh𝑣+𝑎₁₃cos𝑡=0,−𝑡^′𝑡sin𝑡+^′′2𝑡^′1cos𝑡−−2𝑓sin²𝑡2𝑓+𝑡^′cos²𝑡2sin𝑡−𝑎₂₂sin𝑡cosh𝑣−𝑎₂₁sin𝑡sinh𝑣+𝑎₂₃3cos𝑡=0,2𝑡^′cos𝑡+cos𝑡+𝑡2𝑓2𝑡^′′′sin𝑡−𝑎₃₁sin𝑡sinh𝑣−𝑎₃₂sin𝑡cosh𝑣+𝑎₃₃cos𝑡=0, (85) where 𝑎_𝑖𝑗 (𝑖,𝑗=1,2,3) denote the components of the matrix Λ given by (76). In order to prove the theorem, we have to solve the above system of ordinary differential equations. From (85) we easily deduce that 𝑎₁₂=𝑎₂₁=𝑎₁₃=𝑎₂₃=𝑎₃₁=𝑎₃₂=0 and 𝑎₁₁=𝑎₂₂; that is, the matrix Λ is diagonal. We put 𝑎₁₁=𝑎₂₂=𝜆 and 𝑎₃₃=𝜇. Then, the system (85) is reduced to the following equations: 32𝑡^′cos𝑡+cos𝑡+𝑡2𝑓2𝑡^′′′sin𝑡=−𝜇cos𝑡,−𝑡^′𝑡sin𝑡+^′′2𝑡^′1cos𝑡−−2𝑓sin²𝑡2𝑓+𝑡^′cos²𝑡2sin𝑡=𝜆sin𝑡. (86)(87) By the computation (87) × sin𝑡 − (86) × cos𝑡, we easily get −sin𝑡𝑓−𝑡^′=𝜆sin²𝑡+𝜇cos²𝑡. (88) We discuss five cases according to the constants 𝜆 and 𝜇.
Case 1  (𝜆=𝜇=0). In this case, we easily get −𝑡^′−(sin𝑡/𝑓)=0, which implies that the mean curvature 𝐻 vanishes identically because of (82). Therefore, the surface is minimal; from theorem 2 in [9], it is the 5th kind catenoid. Furthermore, the 5th kind catenoid satisfies the condition (76).
Case 2  (𝜆=𝜇≠0). By (88), we get 𝑡^′=−sin𝑡𝑓−𝜆. (89) Differentiating (89) with respect to 𝑢, we have 𝑡^′′𝑡=−^′cos𝑡𝑓+sin𝑡cos𝑡𝑓². (90) Combining (89), (90), and (86), we get 𝜆²𝑓²+4𝜆𝑓sin𝑡+4sin²𝑡=0 (91) from which 𝑓=−2sin𝑡𝜆. (92) Furthermore, (92) together with (89) becomes 𝑡^′=−𝜆/2; that is, 𝑡(𝑢)𝜆=−2𝑢+𝑘,𝑘∈ℝ. (93) On the other hand, by (78) and (93), we have 𝑓(𝑢)2=−𝜆−𝜆sin2,𝑔(𝑢)=2𝑢+𝑘𝜆−𝜆cos2𝑢+𝑘+𝑐,𝑐∈ℝ. (94)
Then, the surface 𝑀 has the following expression: 𝑥()=−2𝑢,𝑣𝜆−𝜆sin2−2𝑢+𝑘sinh𝑣,𝜆−𝜆sin22𝑢+𝑘cosh𝑣,𝜆−𝜆cos2𝑢+𝑘+𝑐,𝑐,𝑘∈ℝ. (95) Consequently, we have ⟨𝑥()()⟩=2𝑢,𝑣−𝐂,𝑥𝑢,𝑣−𝐂𝜆²,𝐂=(0,0,𝑐), (96) which means that the surface 𝑀 is contained in the de Sitter pseudosphere 𝑆²₁(2/|𝜆|) centered at 𝐂 with radius 2/|𝜆|. Also, the de Sitter pseudosphere satisfies condition (76). In the cases of 3, 4, and 5, we will use the same method of Section 3 and easily get that there are no surfaces of revolution with spacelike axis satisfies condition (76).
When the surface of revolution has the expression given by (10), we can similarly prove that the 2nd kind of catenoid, 4th kind of catenoid, the de Sitter pseudosphere, and the hyperbolic pseudosphere satisfy condition (76). This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests in this work.

Acknowledgments

The authors were supported by NSF of China (no. 11271063) and Youth Science Foundation of Heilongjiang Institute of Technology (no. 2012QJ19).