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

Authors: Minghao Jin; Donghe Pei; Shu Xu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

By studying the Gauss mapG and Laplace operator Δh of the second fundamental form h, we will classify surfaces of revolution with a lightlike axis in 3-dimensional Minkowski space and also obtain the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere that satisfy condition ΔhG=ΛG,Λ being a 3×3 real matrix.


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## Body

## 1. Introduction

The Gauss map is a useful tool for studying surfaces in Euclidean space and pseudo-Euclidean space.Suppose thatM is a connected surface in ℝ3 and G is the Gauss map on M. According to a theorem proved by Ruh and Vilms [1], M has constant mean curvature if and only if
(1)ΔG=∥dG∥2G,
where Δ is the Laplace operator on M that corresponds to the metric induced on M from ℝ3. A special case of (1) is given by
(2)ΔG=λG,
where the Gauss map G is an eigenfunction of the Laplacian Δ on M. As a more general form of (1), Dillen et al. [2] proved that a surface of revolution M in ℝ3 satisfies the condition
(3)ΔG=ΛG,Λ∈Mat(3,ℝ),
if and only if M is a plane, sphere, or cylinder. Baikoussis and Blair [3] proved that a ruled surface M in ℝ3 satisfies condition (3) if and only if M is a plane, helicoidal surface, or spiral surface in ℝ3. Additionally, Choi and Alías 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 in ℝ1m such that
(4)ΔG=ΛG,Λ∈Mat(N,ℝ),N=(m2).Recently, an interesting question was raised: what surfaces of revolution without parabolic points in Euclidean or pseudo-Euclidean space satisfy the following condition?(5)ΔhG=ΛG,Λ∈Mat(3,ℝ),
where Δh is the Laplace operator with respect to the second fundamental form h of the surface. This operator is formally defined by (6)Δh=-1|ℋ|∑i,j=12‍∂∂xi(|ℋ|hij∂∂xj)
for the components hij(i,j=1,2) of the second fundamental form h on M, and we denote by (hij) (resp., ℋ) the inverse matrix (resp., the determinant) of the matrix (hij).In [8], the authors studied surfaces of revolution without parabolic points in Euclidean 3-space ℝ3 and presented some classification theorems. In this paper, we will consider surfaces of revolution with lightlike axis in ℝ13 and present some classification results.


## 2. Preliminaries

Letℝ13 be a 3-dimensional Minkowski space with the scalar product and Lorentz cross-product defined as
(7)〈x,y〉=-x02+x12+x22,x×y=(x2y1-x1y2,x2y0-x0y2,x0y1-x1y0)
for every vector x=(x0,x1,x2) and y=(y0,y1,y2) in ℝ13.A vectorx of ℝ13 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 ℝ13 is said to be causal. Let γ:I→ℝ13 be a smooth curve in ℝ13, 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.LetI be an open interval and γ:I→Π a plane curve lying in a plane Π of ℝ13 and l a straight line in Π which does not intersect with the curve γ. A surface of revolution M with axis l in ℝ13 is defined to be invariant under the group of motions in ℝ13, which fixes each point of the line l [9]. Because the present paper discusses the case of lightlike axis, without loss of generality, we may assume that the axis is the line spanned by vector (1,1,0) in the plane Ox0x1.So, we choose the line spanned by the vector(1,1,0) as axis and express the suppose curve γ as follows:
(8)γ(u)=(f(u),g(u),0),
where f(u) is a smooth positive function and g(u) is a smooth function such that h(u)=f(u)-g(u)≠0. Then, the surface of revolution M with such axis may be given by (9)x(u,v)=(f(u)+v22h(u),g(u)+v22h(u),h(u)v).Now, let us consider the Gauss mapG on a surface M in ℝ13. The map G:M→Q2(ɛ)⊂ℝ13, 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, ɛ(=±1) denotes the sign of the vector field G and Q2(ɛ) is a 2-dimensional space form as follows:
(10)Q2(ɛ)={S12(1)inℝ13ifɛ=1,H2(-1)inℝ13ifɛ=-1.A surfaceM⊂ℝ13 is called minimal if and only if its mean curvature H is zero. As de Woestijne ([10]) proved, we have the following theorems.Theorem 1 (see [10]).
Every minimal, spacelike surface of revolutionM⊂ℝ13 is congruent to a part of one of the following surfaces:(1)
a spacelike plane;(2)
the catenoid of the 1st kind;(3)
the catenoid of the 2nd kind;(4)
the surface of Enneper of the 2nd kind.Theorem 2 (see [10]).
Every minimal, timelike surface of revolutionM⊂ℝ13 is congruent to a part of one of the following surfaces:(1)
a Lorentzian plane;(2)
the catenoid of the 3rd kind;(3)
the catenoid of the 4th kind;(4)
the catenoid of the 5th kind;(5)
the surface of Enneper of the 3rd kind.Now, we consider some examples of surfaces of revolution which are mentioned in our theorems.Example 1 (The surface of Enneper of the 2nd kind is shown in Figure1).
The surface of Enneper of the 2nd kind is parameterized by(11)x(u,v)=(u3-u-v2u,u3+u-v2u,-2uv)
for u<0. Then, the components of the first and the second fundamental forms are given by
(12)g11=12u2,g12=g21=0,g22=4u2,h11=-24u2|xu×xv|,h12=h21=0,h22=8u2|xu×xv|.
So, the mean curvature H on the surface is
(13)H=(-24u2)(4u2)+(8u2)(12u2)2(12u2)(4u2)|xu×xv|=0.
Therefore, the surface of Enneper of the 2nd kind is minimal.Figure 1
The surface of Enneper of the 2nd kind.Example 2 (The surface of Enneper of the 3rd kind is shown in Figure2).
The surface of Enneper of the 3rd kind is parameterized by(14)x(u,v)=(-u3-u-v2u,-u3+u-v2u,-2uv)
for u<0. Then, the components of the first and the second fundamental forms are given by
(15)g11=-12u2,g12=g21=0,g22=4u2,h11=24u2|xu×xv|,h12=h21=0,h22=8u2|xu×xv|.
So, the mean curvature H on the surface is
(16)H=(24u2)(4u2)+(8u2)(-12u2)2(-12u2)(4u2)|xu×xv|=0.
Therefore, the surface of Enneper of the 3rd kind is minimal.Figure 2
The surface of Enneper of the 3rd kind.Example 3 (The de Sitter pseudosphere is shown in Figure3).
The de Sitter pseudosphere with radius1 can be expressed as
(17)x(u,v)=(sinhu,coshucosv,coshusinv).
Then, its Gauss map G and Laplacian are given by
(18)G=(-sinhu,-coshucosv,-coshusinv),Δh=∂2∂u2-1cosh2u∂2∂v2+sinhucoshu∂∂u.
By a straight computation, we get
(19)ΔhG=(-2sinhu,-2coshucosv,-2coshusinv),
which means
(20)ΔhG=(200020002)G;
that is, the de Sitter pseudosphere satisfies condition (1).Figure 3
The de Sitter pseudosphere.Example 4 (The hyperbolic pseudosphere is shown in Figure4).
The hyperbolic pseudosphere with radius1 is parameterized by
(21)x(u,v)=(coshu,sinhucosv,sinhusinv).
Then, its Gauss map G and Laplacian are given by
(22)G=(-coshu,-sinhucosv,-sinhusinv),Δh=-∂2∂u2-1sinh2u∂2∂v2-coshusinhu∂∂u.
By a straight computation, we get
(23)ΔhG=(2coshu,2sinhucosv,2sinhusinv).
So, we have
(24)ΔhG=(-2000-2000-2)G;
that is, the hyperbolic pseudosphere satisfies condition (1).Figure 4
The future hyperbolic pseudosphere.


## 3. The Surface of Revolution with Lightlike Axis

In this section, we will classify the surfaces of revolution with lightlike axis inℝ13 that satisfy condition (5).Theorem 3.
The only surfaces of revolution with lightlike axis inℝ13, whose Gauss map G satisfies
(25)ΔhG=ΛG,Λ∈Mat(3,ℝ),
are locally the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere.Proof.
LetM be a surface of revolution with lightlike axis as (9); then we may assume that the profile curve γ is of unit speed; thus
(26)〈γ′,γ′〉=-f′2(u)+g′2(u)=ɛ(±1).
Without lost of generality, we assume that h=f(u)-g(u)>0 and give a detailed proof just for the case ɛ=1.
Then, we may put(27)f′(u)=sinht,g′(u)=cosht
for the smooth function t=t(u). Using the natural frame {xu,xv} of M defined by
(28)xu=(f′+v22h′,g′+v22h′,h′v),xv=(vh,vh,h),xuu=(f′′+v22h′′,g′′+v22h′′,h′′v),xuv=(vh′,vh′,0),xvv=(h,h,0),
we obtain the components of the first and the second fundamental forms of the surface as follows:
(29)g11=〈xu,xu〉=1,g12=g21=〈xu,xv〉=0,g22=〈xv,xv〉=h2,h11=〈xuu,G〉=f′′g′-f′g′′=t′,h12=h21=〈xuv,G〉=0,h22=〈xvv,G〉=-hh′,
where Gauss map G is defined by (xu×xv)/|xu×xv|=(-g′+(v2/2)h′, -f′+(v2/2)h′, vh′).
So, the matrix(hij) is composed by second fundamental form h as follows:
(30)(h11h12h21h22)=(t′00-hh′).
Since ℋ=h11h22-h122=0 makes Laplacian Δh degenerate, so we can assume that ℋ≠0 for every t. Then, the mean curvature H on M is given by
(31)H=h2t′-hh′2h2=12(t′-h′h).
By a straightforward computation, the Laplacian Δh of the second fundamental form h on M with the help of (2), (27), and (29) turns out to be
(32)Δh=-1t′∂2∂u2+1hh′∂2∂v2+(t′′2t′2-h′2ht′-h′′2h′t′)∂∂u.
Accordingly, we get(33)ΔhG=((-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′)v2+g′′′t′-t′′g′′2t′2+h′g′′2t′h+h′′g′′2t′h′+1h(-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′)v2+f′′′t′-t′′f′′2t′2+h′f′′2t′h+h′′f′′2t′h′+1h(-h′′′t′+t′′h′′2t′2-h′h′′2t′h-h′′22t′h′)v).
By the assumption (25) and the above equation, we get the following system of differential equations:
(34)(-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′-a11+a122h′)v2-a13h′v+g′′′t′-t′′g′′2t′2+h′g′′2t′h+h′′g′′2t′h′+1h+a11g′+a12f′=0,(-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′-a21+a222h′)v2-a23h′v+f′′′t′-t′′f′′2t′2+h′f′′2t′h+h′′f′′2t′h′+1h+a21g′+a22f′=0,-a31+a322h′v2+(-h′′′t′+t′′h′′2t′2-h′h′′2t′h-h′′22t′h′-a33h′)v+a31g′+a32f′=0,
where aij(i,j=1,2,3) denote the components of the matrix Λ given by (25).
In order to prove the theorem, we have to solve the above system of ordinary differential equations. So, we get three systems of ODE, equivalently:(35)-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′-a11+a122h′=0,-a13h′=0,g′′′t′-t′′g′′2t′2+h′g′′2t′h+h′′g′′2t′h′+1h+a11g′+a12f′=0,-h′′′2t′+t′′h′′4t′2-h′h′′4t′h-h′′24t′h′-a21+a222h′=0-a23h′=0,f′′′t′-t′′f′′2t′2+h′f′′2t′h+h′′f′′2t′h′+1h+a21g′+a22f′=0,-a31+a322h′=0,-h′′′t′+t′′h′′2t′2-h′h′′2t′h-h′′22t′h′-a33h′=0,a31g′+a32f′=0.
From (35), we easily deduce that a13=a23=a31=a32=0 and a33=(a11+a22)/2=a11+a12=a21+a22. We put a11=λ and a22=μ. Therefore, the matrix Λ satisfies
(36)Λ=(λ12(μ-λ)012(λ-μ)μ00012(λ+μ)).
Then, three systems (35) now reduce to the following equations:
(37)g′′′t′-t′′g′′2t′2+h′g′′2t′h′+h′′g′′2t′h′+1h=-λg′-μ-λ2f′,(38)f′′′t′-t′′f′′2t′2+h′f′′2t′h′+h′′f′′2t′h+1h=-μf′-λ-μ2g′,(39)-h′′′t′+t′′h′′2t′2-h′h′′2t′h-h′′22t′h′=μ+λ2h′.
By the computation (37) ×cosht- (38) ×sinht and using f′=sinht, f′′=t′cosht, f′′′=t′2sinht+t′′cosht, g′=cosht, g′′=t′sinht, and g′′′=t′2cosht+t′′sinht, we easily get
(40)t′-h′h=-λcosh2t+μsinh2t+(λ-μ)sinhtcosht.
On the other hand, substituting h′′=-h′t′ and h′′′=h′(t′2-t′′) into (39) equivalently, we get the following equation:
(41)t′′-3t′2+h′ht′=(λ+μ)t′.
Now, we discuss five cases according to the constants λ and μ.
Case
1  (λ=μ=0). In this case, we easily get t′-(h′/h)=0, which implies that the mean curvature H vanishes identically because of (31). Therefore, the surface is minimal; from Theorem 1 it is the surface of Enneper of the 2nd kind. Furthermore, a surface of Enneper of the 2nd kind satisfies the condition (25).
Case
2 (λ=μ≠0). By (40), we get
(42)t′=h′h-λ.
Differentiating (42) with respect to u, we have
(43)t′′=-h′ht′-(h′h)2.
Substituting (42) and (43) into (41), we get
(44)4(h′h)2+4λh′h+λ2=0
from which
(45)h′h=λ2.
Furthermore, (45) together with (42) becomes t′=-(λ/2); that is,
(46)t(u)=-λ2u+k,k∈ℝ.
On the other hand, by (27), (45), and (46), we have
(47)f(u)=-2λcosh(-λ2u+k)+c,g(u)=-2λsinh(-λ2u+k)+c,c∈ℝ.
Then, the surface M has the following expression:
(48)x(u,v)=(-2λcosh(-λ2u+k)+v22h+c,-2λsinh(-λ2+k)+v22h+c,hv),
where h=f-g=-(2/λ)e(λ/2)-k, c, k∈ℝ. From this, we easily get
(49)〈x(u,v)-C,x(u,v)-C〉=-(2λ)2,C=(c,c,0).
This equation means that the surfaceM is contained in the hyperbolic pseudosphere H2(-(2/|λ|)) centered at C with radius 2/|λ|. Also, the hyperbolic pseudosphere satisfies condition (25).
Case
3  (λ≠0,μ=0). In this case, (40) becomes t′-(h′/h)=-λcosh2t+λsinhtcosht; that is,
(50)t′=h′h-λcosh2t+λsinhtcosht
and thus
(51)t′′=h′′h-(h′h)2-2λt′sinhtcosht+λt′sinh2t+λt′cosh2t.
Substituting (50) and (51) into (41), we get
(52)Φ1h2+Φ2h+Φ3=0,
where we put
(53)Φ1=λ2(-3cosh4t+8sinhtcosh3tΦ1=λ2-7sinh2tcosh2t+2sinh3tcosht),Φ2=λ(-6cosh3t+14sinhtcosh2tΦ2=λ-10sinh2tcosht+2sinh3t),Φ3=-4sinh2t+8sinhtcosht-4cosh2t.
Differentiating (52) and using (50), we find
(54)Ψ1f2+Ψ2f+Ψ3=0,
where
(55)Ψ1=λ2(-4sinh8tcosht+40sinh7tcosh2t-182sinh6tcosh3t+474sinh5tcosh4t-760sinh4tcosh5t+764sinh3tcosh6t-470sinh2tcosh7t+162sinhtcosh8t-24cosh9t),Ψ2=λ(-4sinh8t+44sinh7tcosht-220sinh6tcosh2t+612sinh5tcosh3t-1020sinh4tcosh4t+1044sinh3tcosh5t-644sinh2tcosh6t+220sinhtcosh7t-32cosh8t),Ψ3=8sinh7t-64sinh6tcosht+208sinh5tcosh2t-360sinh4tcosh3t+360sinh3tcosh4t-208sinh2tcosh5t+64sinhtcosh6t-8cosh7t.
Combining (52) and (54), we show that
(56)χ1f+χ2=0,
where χ1=Φ2Ψ1-Φ1Ψ2, χ2=Φ3Ψ1-Φ1Ψ3.
Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial insinht and cosht satisfying
(57)μ2(∑i=131‍cisinh31-itcosh5+it)=0,
where c1=1024, c2=-24064,…, and c31=170496 are nonzero coefficients of the function sinh31-itcosh5+it. Since this polynomial is equal to zero for every t, all its coefficients must be zero. Thus, we have μ=0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case.
Case
4  (λ=0,μ≠0). In this case, (40) becomes t′-(h′/h)=μsinh2t-μsinhtcosht; that is,
(58)t′=h′h+μsinh2t-μsinhtcosht
and thus
(59)t′′=h′′h-(h′h)2+2μt′sinhtcosht-μt′sinh2t-μt′cosh2t.
Substituting (58) and (59) into (41), we get
(60)ι1h2+ι2h+ι3=0,
where we put
(61)ι1=μ2(-3sinh4t+8sinh3tcoshtι1=-7sinh2tcosh2t+2sinhtcosh3t),ι2=μ(-6sinh3t+14sinh2tcoshtι2=-10sinhtcosh2t+2cosh3t),ι3=-4sinh2t+8sinhtcosht-4cosh2t.
Differentiating (60) and using (58), we find
(62)κ1f2+κ2f+κ3=0,
where
(63)κ1=μ2(108sinh9t-246sinh8tcosht+106sinh7tcosh2t-64sinh6tcosh3t+424sinh5tcosh4t-530sinh4tcosh5t+238sinh3tcosh6t-40sinh2tcosh7t+4sinhtcosh8t),κ2=μ(116sinh8t-220sinh7tcosht-28sinh6tcosh2t+76sinh5tcosh3t+432sinh4tcosh4t-612sinh3tcosh5t+276sinh2tcosh6t-44sinhtcosh7t+4cosh8t),κ3=8sinh7t-64sinh6tcosht+208sinh5tcosh2t-360sinh4tcosh3t+360sinh3tcosh4t-208sinh2tcosh5t+64sinhtcosh6t-8cosh7t.
Combining (60) and (62), we show that
(64)ω1f+ω2=0,
where ω1=ι2κ1-ι1κ2, ω2=ι3κ1-ι1κ3.
Differentiating once again this equation and using the same method above, we find the following trigonometric polynomial insinht and cosht satisfying
(65)μ2(∑i=131‍cisinh37-itcoshi-1t)=0,
where c1=86420736, c2=-4471635456,…, and c31=-8192 are nonzero coefficients of the function sinh37-itcoshi-1t. Since this polynomial is equal to zero for every t, all its coefficients must be zero. Thus, we have μ=0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis.
Case
5   (λ≠0,μ≠0,λ≠μ). In this case, (40) is unchanged; that is,
(66)t′=h′h-λcosh2t+μsinh2t+(λ-μ)sinhtcosht
and thus
(67)t′′=h′′h-(h′h)2-2(λ-μ)t′sinhtcosht+(λ-μ)t′sinh2t+(λ-μ)t′cosh2t.
Substituting (66) and (67) into (41), we get
(68)P1h2+P2h+P3=0,
where we put
(69)P1=(2λμ-5μ2)sinh4tP1=+(2λ2-12λμ+10μ2)sinh3tcoshtP1=+(-7λ2+18λμ-5μ2)sinh2tcosh2tP1=+(8λ2-8λμ)sinhtcosh3t-3λ2cosh4t,P2=(-4λ-2μ)sinh3tP2=+(-4λ+10μ)sinh2tcoshtP2=+(14λ-8μ)sinhtcosh2t-6λcosh3tP3=-4sinh2t+8sinhtcosht-4cosh2t.
Differentiating (68) and using (66), we find
(70)Q1f2+Q2f+Q3=0,
where
(71)Q1=(8λ3μ-44λ2μ2+16λμ3+20μ4)sinh9t+(8λ4-100λ3μ+144λ2μ2+118λμ3-170μ4)sinh8tcosht+⋯+(-48λ4+48λ3μ+42λ2μ2+…+-24λμ3-18μ4)cosh9t,Q2=(8λ3-52λ2μ+4λμ2-20μ3)sinh8t+(-64λ3+180λ2μ-28λμ2-40μ3)×sinh7tcosht+⋯+(-80λ3+80λ2μ+48λμ2-24μ3)×cosh8t,Q3=(8λ2-32λμ)sinh7t+(-64λ2+176λμ-40μ2)sinh6tcosht+⋯+(-32λ2+32λμ+24μ2)cosh7t.
Combining (68) and (70), we show that
(72)R1f+R2=0,
where R1=P2Q1-P1Q2, R2=P3Q1-P1Q3.
Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial insinht and cosht satisfying
(73)∑i=137‍ci(λ,μ)sinh37-itcoshi-1t=0,
where
(74)c1(λ,μ)=-331776λ11μ3+6819840λ10μ4+⋯-4352000μ14,⋮c37(λ,μ)=-18874368λ14+54263808λ13μ1+⋯+2985984μ14,
where ci(λ,μ)(i=1,…,37) are the known polynomials in λ and μ. Since this polynomial is equal to zero for every t, all its coefficients must be zero. Therefore, λ=μ=0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case.
Whenɛ=-1, we can assume that f′(u)=cosht and g′(u)=sinht. Using the same algebraic techniques as for ɛ=1, we easily prove from theorem (9) that the surfaces of Enneper of the 3rd kind and the de Sitter pseudosphere satisfy condition (25). This completes the proof.



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