# Biharmonic and Quasi-Biharmonic Slant Surfaces in Lorentzian Complex Space Forms

Authors: Yu Fu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In 1991, Chen and Ishikawa initially studied biharmonic marginally trapped surfaces in neutral pseudo-Euclidean 4-space. Recently, biharmonic and quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms were studied by Sasahara in 2007 and 2011, respectively. In
this paper we extend Sasahara's results to the case of slant surfaces in Lorentzian complex space forms. By results, we completely classify biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex space forms.


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## Body

## 1. Introduction

LetM~in(4c) be a simply connected Lorentzian complex space form of complex dimension n and complex index i(i≥0), where the complex index is defined as the complex dimension of the largest complex negative definite subspace of the tangent space. In particular, if i=1, we say that M~1n(4c) is Lorentzian. The curvature tensor R~ of M~in(4c) is given by
(1)R~(X,Y)Z=c{〈Y,Z〉X-〈X,Z〉Y+〈JY,Z〉JX-〈JX,Z〉JY+2〈X,JY〉JZ}.Letℂn denote the complex number n-space with complex coordinates z1,…,zn. The ℂn endowed with gs,n, that is, the real part of the Hermitian form
(2)bs,n(z,ω)=-∑k=1s‍z-kωk+∑j=s+1n‍z-jωj,z,ω∈ℂn,
defines a flat indefinite complex space form with complex index s. Denote the pair (ℂn,gs,n) by ℂsn briefly, which is the flat Lorentzian complex n-space. In particular, ℂ12 is the flat Lorentzian complex plane.Let us consider the differentiable manifold:(3)S22n+1(c)={z∈ℂ1n+1;b1,n+1(z,z)=c-1>0},
which is an indefinite real space form of constant sectional curvature c. The Hopf fibration
(4)π:S22n+1(c)⟶CP1n(4c):z↦z·ℂ*
is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on CP1n(4c) such that π is a Riemannian submersion.The pseudo-Riemannian manifoldCP1n(4c) is a Lorentzian complex space form of positive holomorphic sectional curvature 4c.Analogously, ifc<0, consider
(5)H22n+1(c)={z∈ℂ2n+1;b1,n+1(z,z)=c-1<0},
which is an indefinite real space form of constant sectional curvature c. The Hopf fibration
(6)π:H22n+1(c)⟶CH1n(4c):z↦z·ℂ*
is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on CH1n(4c) such that π is a Riemannian submersion.The pseudo-Riemannian manifoldCH1n(4c) is a Lorentzian complex space form of negative holomorphic sectional curvature 4c.It is well known that a complete simply connected complex space formM~in(4c) is holomorphic isometric to ℂ1n, CP1n(4c), or CH1n(4c), according to c=0, c>0, or c<0, respectively.A real surface in a Kähler surface with almost complex structureJ is called slant if its Wirtinger angle is constant (see [1–3]). From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a Lorentzian Kähler surface (M~,g~,J). It should be pointed out that slant surfaces arise naturally and play important roles in the studies of surfaces of Kähler surfaces in the complex space forms; see [4].In last years, the geometry of Lorentzian surfaces in Lorentzian complex space forms has been studied by a series of papers given by Chen and other geometers, for instance, [1, 3, 5–13]. Lorentzian geometry is a vivid field of mathematical research that represents the mathematical foundation of the general theory of relativity—which is probably one of the most successful and beautiful theories of physics. For Lorentzian surfaces immersed in Lorentzian complex space forms, Chen [7] proved that Ricci equation is a consequence of Gauss and Codazzi equations, which indicates that Lorentzian surfaces in Lorentzian complex space forms have many interesting properties.During the last decade, the theory of biharmonic submanifolds has advanced greatly. By definition, a submanifold is called biharmonic if the bitension field of the isometric immersion defining the submanifold vanishes identically. There are a lot classification results and nonexistence results, (see, e.g., [4, 14, 15]). Recently, Sasahara introduces the notion of quasi-biharmonic submanifold in [16], which is defined with the property that the bitension field of the isometric immersion defining the submanifold is lightlike at each point. It is shown in [16] that the class of quasi-biharmonic submanifolds is quite different from the class of biharmonic submanifolds.A surface of a pseudo-Riemannian manifold is called marginally trapped (or quasiminimal) if its mean curvature vector field is lightlike. In the theory of cosmic black holes, a marginally trapped surface in a space-time plays an extremely important role. From the viewpoint of differential geometry, some classification results on marginally trapped surfaces have been obtained by some geometers (see [1, 3, 9–12]). In particular, Chen and Dillen [9] gave a complete classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms.In this paper, we investigate the bitension field of marginally trapped slant surfaces in Lorentzian complex space forms. In particular, we completely classify biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex space forms, respectively (see Theorems12 and 13). Our classification results extend Sasahara’s results from Lagrangian case to the slant case in Lorentzian complex space forms.


## 2. Preliminaries

### 2.1. Basic Notation, Formulas, and Definitions

LetM be a Lorentzian surface of a Lorentzian Kähler surface M~12 equipped with an almost structure J and metric g~. Let 〈,〉 denote the inner product associated with g~.We denote the Levi-Civita connections ofM and M~12 by ∇ and ∇~, respectively. Gauss formula and Weingarten formula are given, respectively, by (see [1, 2])
(7)∇~XY=∇XY+h(X,Y),(8)∇~Xξ=-AξX+DXξ,
for vector fields X,Y tangent to M and ξ normal to M, where h, A, and D are the second fundamental form, the shape operator, and the normal connection. It is well known that the second fundamental form h and the shape operator A are related by
(9)〈h(X,Y),ξ〉=〈AξX,Y〉,
for X,Y tangent to M and ξ normal to M.A vectorv is called spacelike (timelike) if 〈v,v〉>0 or 〈v,v〉=0(〈v,v〉<0). A vector v is called lightlike if it is nonzero and it satisfies 〈v,v〉=0.We define the light coneℒ𝒞⊂ℂ12 by {w∈ℂ12∣〈w,w〉=0}. A curve w(t) is called null if w′ is lightlike for any t.For each normal vectorξ of M at x∈M, the shape operator Aξ is a symmetric endomorphism of the tangent space TxM. The mean curvature vector is defined by
(10)H=12traceh.
A Lorentzian surface M in M~12 is called minimal if its mean curvature vector H vanishes at each point on M. And a Lorentzian surface M in M~12 is called marginally trapped (or quasiminimal) if its mean curvature vector is lightlike at each point on M.For a Lorentzian surfaceM in a Lorentzian complex space form M~12, the Gauss and Codazzi and Ricci equations are given, respectively, by
(11)〈R(X,Y)Z,W〉=〈R~(X,Y)Z,W〉+〈h(Y,Z),h(X,W)〉-〈h(X,Z),h(Y,W)〉,(R~(X,Y)Z)⊥=(∇-h)(X,Y,Z)-(∇-h)(Y,X,Z),〈RD(X,Y)ξ,η〉=〈R-(X,Y)ξ,η〉+〈[Aξ,Aη]X,Y〉,
where X,Y,Z, and W are vectors tangent to M and ∇-h is defined by
(12)(∇-h)(X,Y,Z)=DXh(Y,Z)-h(∇XY,Z)-h(Y,∇XZ).


### 2.2. Bitension Field

For smooth mapsϕ:(Mn,g)→(M~m,〈,〉), the tension field τ(ϕ) is a section of the vector bundle ϕ*TM~ defined by
(13)τ(ϕ)=trace∇dϕ=∑i=1n‍〈ei,ei〉{∇eiϕdϕ(ei)-dϕ(∇eiei)},
where ∇ϕ is the induced connection by ϕ on the bundle ϕ*TM~, which is the pullback of ∇~. If ϕ is an isometric immersion, then τ(ϕ) and the mean curvature vector field H of M are related by
(14)τ(ϕ)=nH.
If τ(ϕ)=0 at each point on M, then ϕ is called a harmonic map. The harmonic maps between two Riemannian manifolds are critical points of the energy functional
(15)E(ϕ)=12∫M‍|dϕ|2vg,
for smooth maps ϕ:(Mn,g)→(M~m,〈,〉).The bitension field is defined by(16)τ2(ϕ)=∑i=1n‍〈ei,ei〉{(∇eiϕ∇eiϕ-∇∇eieiϕ)τ+RN(τ,dϕ(ei))dϕ(ei)(∇eiϕ∇eiϕ-∇∇eieiϕ)},
where RM~ is the curvature tensor of M~.Ifϕ is an isometric immersion and M~ is the complex space form M~in(4c), it follows from (1), (14), and (16) that
(17)τ2(ϕ)=-nΔH+5ncH,
where Δ=-∑i=1n‍〈ei,ei〉(∇~ei∇~ei-∇~∇eiei).A smooth mapϕ is called biharmonic if τ2(ϕ)=0 at each point on M. It is easy to see that harmonic maps are always biharmonic.Biharmonic mapsϕ:(Mn,g)→(M~m,〈,〉) between Riemannian manifolds are critical points of the bienergy functional
(18)E2(ϕ)=12∫M‍|τ(ϕ)|2vg.
Sasahara proposed the notion of quasi-biharmonic submanifolds as follows.Definition 1.
A pseudo-Riemannian submanifold isometrically immersed in a pseudo-Riemannian manifold byϕ is called quasi-biharmonic if τ2 is lightlike at each point on the submanifold.


## 2.1. Basic Notation, Formulas, and Definitions

LetM be a Lorentzian surface of a Lorentzian Kähler surface M~12 equipped with an almost structure J and metric g~. Let 〈,〉 denote the inner product associated with g~.We denote the Levi-Civita connections ofM and M~12 by ∇ and ∇~, respectively. Gauss formula and Weingarten formula are given, respectively, by (see [1, 2])
(7)∇~XY=∇XY+h(X,Y),(8)∇~Xξ=-AξX+DXξ,
for vector fields X,Y tangent to M and ξ normal to M, where h, A, and D are the second fundamental form, the shape operator, and the normal connection. It is well known that the second fundamental form h and the shape operator A are related by
(9)〈h(X,Y),ξ〉=〈AξX,Y〉,
for X,Y tangent to M and ξ normal to M.A vectorv is called spacelike (timelike) if 〈v,v〉>0 or 〈v,v〉=0(〈v,v〉<0). A vector v is called lightlike if it is nonzero and it satisfies 〈v,v〉=0.We define the light coneℒ𝒞⊂ℂ12 by {w∈ℂ12∣〈w,w〉=0}. A curve w(t) is called null if w′ is lightlike for any t.For each normal vectorξ of M at x∈M, the shape operator Aξ is a symmetric endomorphism of the tangent space TxM. The mean curvature vector is defined by
(10)H=12traceh.
A Lorentzian surface M in M~12 is called minimal if its mean curvature vector H vanishes at each point on M. And a Lorentzian surface M in M~12 is called marginally trapped (or quasiminimal) if its mean curvature vector is lightlike at each point on M.For a Lorentzian surfaceM in a Lorentzian complex space form M~12, the Gauss and Codazzi and Ricci equations are given, respectively, by
(11)〈R(X,Y)Z,W〉=〈R~(X,Y)Z,W〉+〈h(Y,Z),h(X,W)〉-〈h(X,Z),h(Y,W)〉,(R~(X,Y)Z)⊥=(∇-h)(X,Y,Z)-(∇-h)(Y,X,Z),〈RD(X,Y)ξ,η〉=〈R-(X,Y)ξ,η〉+〈[Aξ,Aη]X,Y〉,
where X,Y,Z, and W are vectors tangent to M and ∇-h is defined by
(12)(∇-h)(X,Y,Z)=DXh(Y,Z)-h(∇XY,Z)-h(Y,∇XZ).


## 2.2. Bitension Field

For smooth mapsϕ:(Mn,g)→(M~m,〈,〉), the tension field τ(ϕ) is a section of the vector bundle ϕ*TM~ defined by
(13)τ(ϕ)=trace∇dϕ=∑i=1n‍〈ei,ei〉{∇eiϕdϕ(ei)-dϕ(∇eiei)},
where ∇ϕ is the induced connection by ϕ on the bundle ϕ*TM~, which is the pullback of ∇~. If ϕ is an isometric immersion, then τ(ϕ) and the mean curvature vector field H of M are related by
(14)τ(ϕ)=nH.
If τ(ϕ)=0 at each point on M, then ϕ is called a harmonic map. The harmonic maps between two Riemannian manifolds are critical points of the energy functional
(15)E(ϕ)=12∫M‍|dϕ|2vg,
for smooth maps ϕ:(Mn,g)→(M~m,〈,〉).The bitension field is defined by(16)τ2(ϕ)=∑i=1n‍〈ei,ei〉{(∇eiϕ∇eiϕ-∇∇eieiϕ)τ+RN(τ,dϕ(ei))dϕ(ei)(∇eiϕ∇eiϕ-∇∇eieiϕ)},
where RM~ is the curvature tensor of M~.Ifϕ is an isometric immersion and M~ is the complex space form M~in(4c), it follows from (1), (14), and (16) that
(17)τ2(ϕ)=-nΔH+5ncH,
where Δ=-∑i=1n‍〈ei,ei〉(∇~ei∇~ei-∇~∇eiei).A smooth mapϕ is called biharmonic if τ2(ϕ)=0 at each point on M. It is easy to see that harmonic maps are always biharmonic.Biharmonic mapsϕ:(Mn,g)→(M~m,〈,〉) between Riemannian manifolds are critical points of the bienergy functional
(18)E2(ϕ)=12∫M‍|τ(ϕ)|2vg.
Sasahara proposed the notion of quasi-biharmonic submanifolds as follows.Definition 1.
A pseudo-Riemannian submanifold isometrically immersed in a pseudo-Riemannian manifold byϕ is called quasi-biharmonic if τ2 is lightlike at each point on the submanifold.


## 3. Basic Results on Lorentzian Slant Surfaces

LetM be a Lorentzian surface in a Lorentzian Kähler surface (M~12,g,J). For each tangent vector X of M, we put
(19)JX=PX+FX,
where PX and FX are the tangential and the normal components of JX.On the Lorentzian surfaceM there exists a pseudoorthonormal local frame {e1,e2} such that
(20)〈e1,e1〉=〈e2,e2〉=0,〈e1,e2〉=-1.
It follows from (19), (20), and 〈JX,JY〉=〈X,Y〉 that
(21)Pe1=(sinhθ)e1,Pe2=-(sinhθ)e2,
for some function θ. This function θ is called the Wirtinger angle of M.When the Wirtinger angleθ is constant on M, the Lorentzian surface M is called a slant surface (cf. [2, 3]). In this case, θ is called the slant angle; the slant surface is then called θ-slant.Aθ-slant surface is called Lagrangian if θ=0 and proper slant if θ≠0.If we put(22)e3=(sechθ)Fe1,e4=(sechθ)Fe2,
then we find from (19)–(22) that
(23)Je1=sinhθe1+coshθe3,Je2=-sinhθe2+coshθe4,(24)Je3=-coshθe1-sinhθe3,Je4=-coshθe2+sinhθe4,(25)〈e3,e3〉=〈e4,e4〉=0,〈e3,e4〉=-1.
We call such a frame {e1,e2,e3,e4} an adapted pseudoorthonormal frame for the Lorentzian surface M in M~12.Lemma 2.
IfM is a slant surface in a Lorentzian Kähler surface M~12, then with respect to an adapted pseudoorthonormal frame one has
(26)∇Xe1=ω(X)e1,∇Xe2=-ω(X)e2,(27)DXe3=Φ(X)e3,DXe4=-Φ(X)e4,
for some 1-forms ω,Φ on M.For a Lorentzian surfaceM in M~12, we put
(28)h(ei,ej)=hij3e3+hij4e4,
where {e1,e2,e3,e4} is an adapted pseudoorthonormal frame and h is the second fundamental form of M.Lemma 3 (see [3]).
IfM is a θ-slant surface in a Lorentzian Kähler surface M~12, then with respect to an adapted pseudoorthonormal frame one has
(29)ωj-Φj=2h1j3tanhθ,AFXY=AFYX,Ae3ej=h1j3e1+h1j4e2,Ae4ej=hj23e1+hj24e2,
for any X,Y∈TM and j=1,2, where ωj=ω(ej) and Φj=Φ(ej).For Lorentzian slant surfaces inM~12(4c), the author with Hou has proved the following interesting result.Theorem 4 (see [17]).
Every slant surface in a nonflat Lorentzian complex space formM~12(4c) must be Lagrangian.According to Theorem4, we need only to consider the slant surfaces in Lorentzian complex plane ℂ12 because the case of Lagrangian marginally trapped surfaces has been considered in [16, 18].


## 4. The Bitension Field of Marginally Trapped Slant Surfaces

LetM be a θ-slant marginally trapped surface in a Lorentzian complex plane ℂ12. There is a pseudoorthonormal local frame field {e^1,e^2} such that
(30)〈e^1,e^1〉=〈e^2,e^2〉=0,〈e^1,e^2〉=-1,H=-h(e^1,e^2).
Since the mean curvature vector H is lightlike at each point, we put h(e^1,e^2)=αFe^1 for some nonzero real-valued function α. By putting e1=αe^1, e2=α-1e^2, we have
(31)〈e1,e1〉=〈e2,e2〉=0,〈e1,e2〉=-1,(32)h(e1,e2)=Fe1.
By applying (20) and the total symmetry of 〈h(X,Y),FZ〉, we obtain
(33)h(e1,e1)=λFe2,h(e1,e2)=Fe1,h(e2,e2)=γFe1+Fe2,
for two smooth functions λ, γ. It follows from (9), (22), (25), and (33) that
(34)AFe1=cosh2θ(01λ0),AFe2=cosh2θ(1γ01).
By Lemma 2, (33), and differentiating the second fundamental form covariantly, we get
(35)(∇-e2h)(e1,e1)=e2(λ)Fe2-λΦ2Fe2-2ω2λFe2,(∇-e1h)(e1,e2)=Φ1Fe1,(∇-e2h)(e1,e2)=Φ2Fe1,(∇-e1h)(e2,e2)=e1(γ)Fe1+γΦ1Fe1-Φ1Fe2+2ω1γFe1+2ω1Fe2.
By the Codazzi equation, comparing coefficients gives
(36)ω1=Φ1=0,e2(λ)-λΦ2-2ω2λ=0,e1(γ)-Φ2=0.
On the other hand, from Lemma 3 and (33) we have
(37)ω2-Φ2=2sinhθ.
Consequently, (36) becomes
(38)ω1=0,e2(λ)=3ω2λ-2sinhθλ,e1(γ)=ω2-2sinhθ.
In order to express the bitension field of marginally trapped slant surfaces in ℂ12 with respect to a pseudoorthonormal frame (31), we need the following formula (cf. [1, 18]):
(39)ΔH=ΔDH-h(e1,AHe2)-h(e2,AHe1)-ADe1He2-ADe2He1-(∇e1AH)e2-(∇e2AH)e1,
where Δ and ΔD are, respectively, given by
(40)Δ=∇~e1∇~e2+∇~e2∇~e1-∇~∇e1e2-∇~∇e2e1,ΔD=De1De2+De2De1-D∇e1e2-D∇e2e1.Lemma 5.
LetM be a marginally trapped slant surface in ℂ12. Then, the Gauss curvatures K and ΔDH are related by
(41)ΔDH=-KH.Proof.
It follows from (33) that the mean curvature vector H is given by H=-Fe1. By (22), (26), (27), (36), and (37), we have
(42)De1De2H=-e1(Φ2)Fe1=-e1(ω2)Fe1,De2De1H=D∇e1e2H=D∇e2e1H=0.
Consequently, we obtain
(43)ΔDH=-e1(ω2)Fe1.
Recall the definition of the Gauss curvature K=-〈R(e1,e2)e2,e1〉. It follows from Lemma 2 and ω1=0 that
(44)K=-e1(ω2),
which completes the proof of Lemma 5.Remark 6.
For marginally trapped Lagrangian surfaces immersed into Lorentzian complex space forms, Sasahara [15] has proved the formula ΔDH=-KH. Hence, we know from Lemma 5 that the formula ΔDH=-KH also holds for slant surfaces in Lorentzian complex space forms.Lemma 7.
LetM be a marginally trapped slant surface in ℂ12. Then, the normal part of ΔH is expressed as
(45)(ΔH)⊥=2cosh2θλ(γFe1+Fe2).Proof.
On one hand, it follows from (34) that
(46)AHe2=-cosh2θe1,AHe1=-cosh2θλe2.
Combining (46) with (33) gives
(47)h(e1,AHe2)+h(e2,AHe1)=-cosh2θ(λγFe1+2λFe2).
On the other hand, it follows from (33) and the Gauss equation that the Gauss curvature is given by
(48)K=cosh2θλγ.
By Lemma 5, we have ΔDH=cosh2θλγFe1. Combining these with the first section of (39) completes the proof.Lemma 8.
LetM be a marginally trapped slant surface in ℂ12. Then, the tangential part of ΔH is expressed as
(49)(ΔH)⊤=cosh2θ(e2(λ)-λω2-2λsinhθ)e2.Proof.
By (27), (36), and (46), we obtain
(50)-ADe1He2-ADe2He1=-Φ2AHe1=cosh2θΦ2λe2.
It follows from (26), (38), and (46) that
(51)(∇e1AH)e2=∇e1(AHe2)-AH(∇e1e2)=-2cosh2θω1e1=0,(∇e2AH)e1=∇e2(AHe1)-AH(∇e2e1)=-cosh2θ(e2(λ)-2λω2)e2.
Combing (50)-(51) with (37) gives the conclusion.Hence, by (17) and Lemmas 7 and 8, we get the expression of the bitension field of marginally trapped slant surfaces in Lorentzian complex plane ℂ12.Lemma 9.
Letϕ:M→ℂ12 be a marginally trapped slant immersed in ℂ12. Then the bitension field is given by
(52)τ2(ϕ)=-2cosh2θ(e2(λ)-λω2-2λsinhθ)e2-4cosh2θλ(γFe1+Fe2).


## 5. Classification Results

From now on, let us consider the biharmonic and quasi-biharmonic Marginally trapped slant surfaces in Lorentzian complex planeℂ12.By the definition of biharmonic surfaces, we can conclude the following from Lemma9.Lemma 10.
Letϕ:M→ℂ12 be a marginally trapped slant immersed in ℂ12. Then the immersion is biharmonic if and only if the function λ satisfies λ=0.Similarly, by the definition of quasi-biharmonic submanifolds, the bitension fieldτ2(ϕ) is lightlike. Therefore, we also have the following.Lemma 11.
Letϕ:M→ℂ12 be a marginally trapped slant immersed in ℂ12. Then the immersion is quasi-biharmonic if and only if the functions λ and γ satisfy λ≠0 and γ=0.Since the Gauss curvatureK is given by (48), we deduce from Lemmas 10 and 11 that K=0 in both cases. Moreover, (44) implies that e1(ω2)=0.We deduce from (26) and (38) that [e1,e2]=-ω2e1. There is a nonzero smooth function β satisfying
(53)e2(β)+βω2=0,
such that [βe1,e2]=0. Thus, there exist local coordinates (x,y) on M such that ∂/∂x=βe1 and ∂/∂y=e2. Then the metric tensor of M is given by
(54)g=-β(dx⊗dy+dy⊗dx),
and the Levi-Civita connection of g satisfies
(55)∇∂/∂x∂∂x=βxβ∂∂x,∇∂/∂x∂∂y=0,∇∂/∂y∂∂y=βyβ∂∂y.
Moreover, it follows from (33) and (54) that
(56)h(∂∂x,∂∂x)=λβ2F∂∂y,h(∂∂x,∂∂y)=F∂∂x,h(∂∂y,∂∂y)=γβ-1F∂∂x+F∂∂y.
Since e1(ω2)=0, it follows that ω2 is a function depending only on y; that is, ω2=ω2(y). Using local coordinates, (53) becomes
(57)βy+βω2(y)=0.
Solving (57) gives
(58)β=e-∫‍ω2(y)dy.
Without loss of generality, we may assume that β is only depending on variable y.By applying (23), (55), (56), (58), and Gauss formula (7), we have the following PDE system:
(59)Lxx=λβ2(i+sinhθ)Ly,(60)Lxy=(i-sinhθ)Lx,(61)Lyy=β-1γ(i-sinhθ)Lx+(βyβ+i+sinhθ)Ly.
By solving (60), we obtain that
(62)L(x,y)=f(x)e(i-sinhθ)y+z(y),
for some vector-valued functions f(x) and z(y) in ℂ12.Theorem 12.
Up to rigid motions ofℂ12, every biharmonic marginally trapped θ-slant surface in ℂ12 is given by a flat slant surface defined by
(63)L(x,y)=c1xe(i-sinhθ)y+w(y),
where c1 is lightlike vector and w(y) is a null curve in ℂ12 satisfying
(64)〈c1(i-sinhθ)e(i-sinhθ)y,w′(y)〉=0,〈c1e(i-sinhθ)y,w′(y)〉=-β(y)
for some nonzero real-valued function β(y).Proof.
LetM be a biharmonic marginally trapped θ-slant surface in ℂ12. According to Lemma 10, we have λ=0. Substituting (62) into (59), we have
(65)f′′(x)=0,
which yields f(x)=c1x+c2 for two constant vectors c1 and c2 in ℂ12. Hence, the immersion becomes
(66)L(x,y)=c1xe(i-sinhθ)y+w(y).
Note that w(y)=c2e(i-sinhθ)y+z(y) here. Moreover, (66) yields
(67)Lx=c1e(i-sinhθ)y,Ly=c1(i-sinhθ)xe(i-sinhθ)y+w′(y).
It follows from (54) and (67) that
(68)〈c1,c1〉=〈w′(y),w′(y)〉=〈c1(i-sinhθ)e(i-sinhθ)y,w′(y)〉=0,〈c1e(i-sinhθ)y,w′(y)〉=-β(y).
This completes the proof of Theorem 12.Theorem 13.
Up to rigid motions ofℂ12, every quasi-biharmonic marginally trapped θ-slant surface in ℂ12 is given by a flat slant surface defined by
(69)L(x,y)=p(x)e(i-sinhθ)y,
where p(x) is a null curve lying in the light cone ℒ𝒞 satisfying
(70)〈p(x),p(x)〉=〈p′(x),p′(x)〉=0,〈p(x),ip′(x)〉=1.Proof.
LetM be a quasi-biharmonic marginally trapped θ-slant surface in ℂ12. According to Lemma 11, we have γ=0 and λ≠0. In this case, the second and third equations of (38) become
(71)ω2=2sinhθ,λy=4sinhθλ.
Solving λ, we get
(72)λ=c1(x)e4sinhθy.
By applying (71), (58) yields
(73)β=ae-2sinhθy
for some nonzero constant number a. Consequently, with the previous information, the PDE system (59)–(61) becomes
(74)Lxx=ac1(x)Ly,(75)Lxy=(i-sinhθ)Lx,(76)Lyy=(i-sinhθ)Ly.
Substituting (62) into (76), we have
(77)z′′(y)=(i-sinhθ)z′(y),
which yields
(78)z(y)=c2e(i-sinhθ)y+c3
for two constant vectors c2 and c3 in ℂ12. We denote p(x) by p(x)=f(x)+c2. Hence, up to rigid motions of ℂ12 the immersion becomes
(79)L(x,y)=p(x)e(i-sinhθ)y.
Substituting (79) into (74), we obtain(80)p′′(x)=-acosh2θc1(x)p(x).
Moreover, (79) gives
(81)Lx=p′(x)e(i-sinhθ)y,Ly=(i-sinhθ)p(x)e(i-sinhθ)y.
It follows from (54), (73), and (81) that
(82)〈p(x),p(x)〉=〈p′(x),p′(x)〉=0,〈p′(x),(i-sinhθ)p(x)〉=-a.
In view of (73), we can assume a=1. Hence, we get the conclusion.Remark 14.
According to Theorem4, combining Theorems 12 and 13 with Sasahara’s results in [16, 18], we finish the complete classifications of biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex forms, respectively.



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*Source: 412709-2013-04-03.xml*