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

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.

1. Introduction

Let 𝑀^𝑛_𝑖(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^𝑛₁(4𝑐) is Lorentzian. The curvature tensor 𝑅 of 𝑀^𝑛_𝑖(4𝑐) is given by 𝑅()⟩⟨⟩⟨⟩−⟨⟩⟨⟩}.𝑋,𝑌𝑍=𝑐{⟨𝑌,𝑍𝑋−𝑋,𝑍𝑌+𝐽𝑌,𝑍𝐽𝑋𝐽𝑋,𝑍𝐽𝑌+2𝑋,𝐽𝑌𝐽𝑍 (1)

Let ℂ^𝑛 denote the complex number 𝑛-space with complex coordinates 𝑧₁,…,𝑧_𝑛. The ℂ^𝑛 endowed with 𝑔_𝑠,𝑛, that is, the real part of the Hermitian form 𝑏_𝑠,𝑛(𝑧,𝜔)=−^𝑠_𝑘=1𝑧_𝑘𝜔_𝑘+^𝑛_𝑗=𝑠+1𝑧_𝑗𝜔_𝑗,𝑧,𝜔∈ℂ^𝑛, (2) defines a flat indefinite complex space form with complex index 𝑠. Denote the pair (ℂ^𝑛,𝑔_𝑠,𝑛) by ℂ^𝑛_𝑠 briefly, which is the flat Lorentzian complex 𝑛-space. In particular, ℂ²₁ is the flat Lorentzian complex plane.

Let us consider the differentiable manifold: 𝑆₂^2𝑛+1(𝑐)=𝑧∈ℂ₁^𝑛+1;𝑏_1,𝑛+1(𝑧,𝑧)=𝑐⁻¹>0, (3) which is an indefinite real space form of constant sectional curvature 𝑐. The Hopf fibration 𝜋∶𝑆₂^2𝑛+1(𝑐)⟶𝐶𝑃^𝑛₁(4𝑐)∶𝑧↦𝑧⋅ℂ^∗ (4) is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on 𝐶𝑃^𝑛₁(4𝑐) such that 𝜋 is a Riemannian submersion.

The pseudo-Riemannian manifold 𝐶𝑃^𝑛₁(4𝑐) is a Lorentzian complex space form of positive holomorphic sectional curvature 4𝑐.

Analogously, if 𝑐<0, consider 𝐻₂^2𝑛+1(𝑐)=𝑧∈ℂ₂^𝑛+1;𝑏_1,𝑛+1(𝑧,𝑧)=𝑐⁻¹<0, (5) which is an indefinite real space form of constant sectional curvature 𝑐. The Hopf fibration 𝜋∶𝐻₂^2𝑛+1(𝑐)⟶𝐶𝐻^𝑛₁(4𝑐)∶𝑧↦𝑧⋅ℂ^∗ (6) is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on 𝐶𝐻^𝑛₁(4𝑐) such that 𝜋 is a Riemannian submersion.

The pseudo-Riemannian manifold 𝐶𝐻^𝑛₁(4𝑐) is a Lorentzian complex space form of negative holomorphic sectional curvature 4𝑐.

It is well known that a complete simply connected complex space form 𝑀^𝑛_𝑖(4𝑐) is holomorphic isometric to ℂ^𝑛₁, 𝐶𝑃^𝑛₁(4𝑐), or 𝐶𝐻^𝑛₁(4𝑐), according to 𝑐=0, 𝑐>0, or 𝑐<0, respectively.

A real surface in a Kähler surface with almost complex structure 𝐽 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 Theorems 12 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

Let 𝑀 be a Lorentzian surface of a Lorentzian Kähler surface 𝑀²₁ equipped with an almost structure 𝐽 and metric 𝑔. Let ⟨,⟩ denote the inner product associated with 𝑔.

We denote the Levi-Civita connections of 𝑀 and 𝑀²₁ by ∇ and ∇, respectively. Gauss formula and Weingarten formula are given, respectively, by (see [1, 2]) ∇_𝑋𝑌=∇_𝑋(),∇𝑌+ℎ𝑋,𝑌_𝑋𝜉=−𝐴_𝜉𝑋+𝐷_𝑋𝜉, (7)(8) 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 ℎ(𝑋,𝑌),𝜉=𝐴_𝜉𝑋,𝑌, (9) for 𝑋,𝑌 tangent to 𝑀 and 𝜉 normal to 𝑀.

A vector 𝑣 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 cone ℒ𝒞⊂ℂ²₁ by {𝑤∈ℂ²₁∣⟨𝑤,𝑤⟩=0}. A curve 𝑤(𝑡) is called null if 𝑤^′ is lightlike for any 𝑡.

For each normal vector 𝜉 of 𝑀 at 𝑥∈𝑀, the shape operator 𝐴_𝜉 is a symmetric endomorphism of the tangent space 𝑇_𝑥𝑀. The mean curvature vector is defined by 1𝐻=2traceℎ. (10) A Lorentzian surface 𝑀 in 𝑀²₁ is called minimal if its mean curvature vector 𝐻 vanishes at each point on 𝑀. And a Lorentzian surface 𝑀 in 𝑀²₁ is called marginally trapped (or quasiminimal) if its mean curvature vector is lightlike at each point on 𝑀.

For a Lorentzian surface 𝑀 in a Lorentzian complex space form 𝑀²₁, the Gauss and Codazzi and Ricci equations are given, respectively, by ⟨𝑅()⟩=𝑅()+⟨ℎ()(()()𝑅()𝑍𝑋,𝑌𝑍,𝑊𝑋,𝑌𝑍,𝑊𝑌,𝑍,ℎ𝑋,𝑊)⟩−⟨ℎ𝑋,𝑍,ℎ𝑌,𝑊⟩,𝑋,𝑌^⟂=∇ℎ(𝑋,𝑌,𝑍)−),𝑅∇ℎ(𝑌,𝑋,𝑍_𝐷(𝑋,𝑌)𝜉,𝜂=𝑅()+𝐴𝑋,𝑌𝜉,𝜂_𝜉,𝐴_𝜂𝑋,𝑌, (11) where 𝑋,𝑌,𝑍, and 𝑊 are vectors tangent to 𝑀 and ∇ℎ is defined by ∇ℎ(𝑋,𝑌,𝑍)=𝐷_𝑋ℎ(𝑌,𝑍)−ℎ∇_𝑋𝑌,𝑍−ℎ𝑌,∇_𝑋𝑍. (12)

2.2. Bitension Field

For smooth maps 𝜙∶(𝑀^𝑛,𝑔)→(𝑀^𝑚,⟨,⟩), the tension field 𝜏(𝜙) is a section of the vector bundle 𝜙^∗𝑇𝑀 defined by 𝜏𝜙=trace∇d𝜙=^𝑛_𝑖=1𝑒_𝑖,𝑒_𝑖∇^𝜙_𝑒𝑖𝑑𝜙𝑒_𝑖−𝑑𝜙∇_𝑒𝑖𝑒_𝑖, (13) 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 𝜏𝜙=𝑛𝐻. (14) 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 𝐸𝜙=12_𝑀𝑑𝜙²𝑣_𝑔, (15) for smooth maps 𝜙∶(𝑀^𝑛,𝑔)→(𝑀^𝑚,⟨,⟩).

The bitension field is defined by 𝜏₂𝜙=^𝑛_𝑖=1𝑒_𝑖,𝑒_𝑖∇^𝜙_𝑒𝑖∇^𝜙_𝑒𝑖−∇^𝜙_{∇𝑒𝑖𝑒𝑖}𝜏+𝑅^𝑁𝜏,𝑑𝜙𝑒_𝑖𝑑𝜙𝑒_𝑖, (16) where 𝑅𝑀 is the curvature tensor of 𝑀.

If 𝜙 is an isometric immersion and 𝑀 is the complex space form 𝑀^𝑛_𝑖(4𝑐), it follows from (1), (14), and (16) that 𝜏₂𝜙=−𝑛Δ𝐻+5𝑛𝑐𝐻, (17) where Δ=−∑^𝑛_𝑖=1⟨𝑒_𝑖,𝑒_𝑖⟩(∇_𝑒𝑖∇_𝑒𝑖−∇_{∇𝑒𝑖𝑒𝑖}).

A smooth map 𝜙 is called biharmonic if 𝜏₂(𝜙)=0 at each point on 𝑀. It is easy to see that harmonic maps are always biharmonic.

Biharmonic maps 𝜙∶(𝑀^𝑛,𝑔)→(𝑀^𝑚,⟨,⟩) between Riemannian manifolds are critical points of the bienergy functional 𝐸₂𝜙=12_𝑀𝜏𝜙²𝑣_𝑔. (18) Sasahara proposed the notion of quasi-biharmonic submanifolds as follows.

Definition 1. A pseudo-Riemannian submanifold isometrically immersed in a pseudo-Riemannian manifold by 𝜙 is called quasi-biharmonic if 𝜏₂ is lightlike at each point on the submanifold.

3. Basic Results on Lorentzian Slant Surfaces

Let 𝑀 be a Lorentzian surface in a Lorentzian Kähler surface (𝑀²₁,𝑔,𝐽). For each tangent vector 𝑋 of 𝑀, we put 𝐽𝑋=𝑃𝑋+𝐹𝑋, (19) where 𝑃𝑋 and 𝐹𝑋 are the tangential and the normal components of 𝐽𝑋.

On the Lorentzian surface 𝑀 there exists a pseudoorthonormal local frame {𝑒₁,𝑒₂} such that 𝑒₁,𝑒₁=𝑒₂,𝑒₂=0,𝑒₁,𝑒₂=−1. (20) It follows from (19), (20), and ⟨𝐽𝑋,𝐽𝑌⟩=⟨𝑋,𝑌⟩ that 𝑃𝑒₁=(sinh𝜃)𝑒₁,𝑃𝑒₂=−(sinh𝜃)𝑒₂, (21) for some function 𝜃. This function 𝜃 is called the Wirtinger angle of 𝑀.

When the Wirtinger angle 𝜃 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.

A 𝜃-slant surface is called Lagrangian if 𝜃=0 and proper slant if 𝜃≠0.

If we put 𝑒₃=(sech𝜃)𝐹𝑒₁,𝑒₄=(sech𝜃)𝐹𝑒₂, (22) then we find from (19)–(22) that 𝐽𝑒₁=sinh𝜃𝑒₁+cosh𝜃𝑒₃,𝐽𝑒₂=−sinh𝜃𝑒₂+cosh𝜃𝑒₄,𝐽𝑒₃=−cosh𝜃𝑒₁−sinh𝜃𝑒₃,𝐽𝑒₄=−cosh𝜃𝑒₂+sinh𝜃𝑒₄,𝑒₃,𝑒₃=𝑒₄,𝑒₄=0,𝑒₃,𝑒₄=−1. (23)(24)(25) We call such a frame {𝑒₁,𝑒₂,𝑒₃,𝑒₄} an adapted pseudoorthonormal frame for the Lorentzian surface 𝑀 in 𝑀²₁.

Lemma 2. If 𝑀 is a slant surface in a Lorentzian Kähler surface 𝑀²₁, then with respect to an adapted pseudoorthonormal frame one has ∇_𝑋𝑒₁=𝜔(𝑋)𝑒₁,∇_𝑋𝑒₂=−𝜔(𝑋)𝑒₂,𝐷_𝑋𝑒₃=Φ(𝑋)𝑒₃,𝐷_𝑋𝑒₄=−Φ(𝑋)𝑒₄, (26)(27) for some 1-forms 𝜔,Φ on 𝑀.

For a Lorentzian surface 𝑀 in 𝑀²₁, we put ℎ𝑒_𝑖,𝑒_𝑗=ℎ³_𝑖𝑗𝑒₃+ℎ⁴_𝑖𝑗𝑒₄, (28) where {𝑒₁,𝑒₂,𝑒₃,𝑒₄} is an adapted pseudoorthonormal frame and ℎ is the second fundamental form of 𝑀.

Lemma 3 (see [3]). If 𝑀 is a 𝜃-slant surface in a Lorentzian Kähler surface 𝑀²₁, then with respect to an adapted pseudoorthonormal frame one has 𝜔_𝑗−Φ_𝑗=2ℎ³_1𝑗𝐴tanh𝜃,_𝐹𝑋𝑌=𝐴_𝐹𝑌𝐴𝑋,_𝑒3𝑒_𝑗=ℎ³_1𝑗𝑒₁+ℎ⁴_1𝑗𝑒₂,𝐴_𝑒4𝑒_𝑗=ℎ³_𝑗2𝑒₁+ℎ⁴_𝑗2𝑒₂, (29) for any 𝑋,𝑌∈𝑇𝑀 and 𝑗=1,2, where 𝜔_𝑗=𝜔(𝑒_𝑗) and Φ_𝑗=Φ(𝑒_𝑗).

For Lorentzian slant surfaces in 𝑀²₁(4𝑐), the author with Hou has proved the following interesting result.

Theorem 4 (see [17]). Every slant surface in a nonflat Lorentzian complex space form 𝑀²₁(4𝑐) must be Lagrangian.

According to Theorem 4, we need only to consider the slant surfaces in Lorentzian complex plane ℂ²₁ because the case of Lagrangian marginally trapped surfaces has been considered in [16, 18].

4. The Bitension Field of Marginally Trapped Slant Surfaces

Let 𝑀 be a 𝜃-slant marginally trapped surface in a Lorentzian complex plane ℂ²₁. There is a pseudoorthonormal local frame field {𝑒₁̂,𝑒₂̂} such that 𝑒₁̂,𝑒₁̂=𝑒₂̂,𝑒₂̂=0,⟨𝑒₁̂,𝑒₂̂𝑒̂⟩=−1,𝐻=−ℎ₁,𝑒₂̂. (30) Since the mean curvature vector 𝐻 is lightlike at each point, we put ℎ(𝑒₁̂,𝑒₂̂)=𝛼𝐹𝑒₁̂ for some nonzero real-valued function 𝛼. By putting 𝑒₁=𝛼𝑒₁̂, 𝑒₂=𝛼⁻¹𝑒₂̂, we have 𝑒₁,𝑒₁=𝑒₂,𝑒₂=0,𝑒₁,𝑒₂ℎ𝑒=−1,₁,𝑒₂=𝐹𝑒₁. (31)(32) By applying (20) and the total symmetry of ⟨ℎ(𝑋,𝑌),𝐹𝑍⟩, we obtain ℎ𝑒₁,𝑒₁=𝜆𝐹𝑒₂,ℎ𝑒₁,𝑒₂=𝐹𝑒₁,ℎ𝑒₂,𝑒₂=𝛾𝐹𝑒₁+𝐹𝑒₂, (33) for two smooth functions 𝜆, 𝛾. It follows from (9), (22), (25), and (33) that 𝐴_𝐹𝑒1=cosh²𝜃01𝜆0,𝐴_𝐹𝑒2=cosh²𝜃1𝛾01. (34) By Lemma 2, (33), and differentiating the second fundamental form covariantly, we get ∇_𝑒2ℎ𝑒₁,𝑒₁=𝑒₂(𝜆)𝐹𝑒₂−𝜆Φ₂𝐹𝑒₂−2𝜔₂𝜆𝐹𝑒₂,∇_𝑒1ℎ𝑒₁,𝑒₂=Φ₁𝐹𝑒₁,∇_𝑒2ℎ𝑒₁,𝑒₂=Φ₂𝐹𝑒₁,∇_𝑒1ℎ𝑒₂,𝑒₂=𝑒₁𝛾𝐹𝑒₁+𝛾Φ₁𝐹𝑒₁−Φ₁𝐹𝑒₂+2𝜔₁𝛾𝐹𝑒₁+2𝜔₁𝐹𝑒₂. (35) By the Codazzi equation, comparing coefficients gives 𝜔₁=Φ₁=0,𝑒₂(𝜆)−𝜆Φ₂−2𝜔₂𝑒𝜆=0,₁𝛾−Φ₂=0. (36) On the other hand, from Lemma 3 and (33) we have 𝜔₂−Φ₂=2sinh𝜃. (37) Consequently, (36) becomes 𝜔₁=0,𝑒₂(𝜆)=3𝜔₂𝑒𝜆−2sinh𝜃𝜆,₁𝛾=𝜔₂−2sinh𝜃. (38) In order to express the bitension field of marginally trapped slant surfaces in ℂ²₁ with respect to a pseudoorthonormal frame (31), we need the following formula (cf. [1, 18]): Δ𝐻=Δ^𝐷𝐻−ℎ𝑒₁,𝐴_𝐻𝑒₂−ℎ𝑒₂,𝐴_𝐻𝑒₁−𝐴_{𝐷𝑒1𝐻}𝑒₂−𝐴_{𝐷𝑒2𝐻}𝑒₁−∇_𝑒1𝐴_𝐻𝑒₂−∇_𝑒2𝐴_𝐻𝑒₁, (39) where Δ and Δ^𝐷 are, respectively, given by Δ=∇_𝑒1∇_𝑒2+∇_𝑒2∇_𝑒1−∇_{∇𝑒1𝑒2}−∇_{∇𝑒2𝑒1},Δ^𝐷=𝐷_𝑒1𝐷_𝑒2+𝐷_𝑒2𝐷_𝑒1−𝐷_{∇𝑒1𝑒2}−𝐷_{∇𝑒2𝑒1}. (40)

Lemma 5. Let 𝑀 be a marginally trapped slant surface in ℂ²₁. Then, the Gauss curvatures 𝐾 and Δ^𝐷𝐻 are related by Δ^𝐷𝐻=−𝐾𝐻. (41)

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 𝐷_𝑒1𝐷_𝑒2𝐻=−𝑒₁Φ₂𝐹𝑒₁=−𝑒₁𝜔₂𝐹𝑒₁,𝐷_𝑒2𝐷_𝑒1𝐻=𝐷_{∇𝑒1𝑒2}𝐻=𝐷_{∇𝑒2𝑒1}𝐻=0. (42) Consequently, we obtain Δ^𝐷𝐻=−𝑒₁𝜔₂𝐹𝑒₁. (43) Recall the definition of the Gauss curvature 𝐾=−⟨𝑅(𝑒₁,𝑒₂)𝑒₂,𝑒₁⟩. It follows from Lemma 2 and 𝜔₁=0 that 𝐾=−𝑒₁𝜔₂, (44) 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 Δ^𝐷𝐻=−𝐾𝐻. Hence, we know from Lemma 5 that the formula Δ^𝐷𝐻=−𝐾𝐻 also holds for slant surfaces in Lorentzian complex space forms.

Lemma 7. Let 𝑀 be a marginally trapped slant surface in ℂ²₁. Then, the normal part of Δ𝐻 is expressed as (Δ𝐻)^⟂=2cosh²𝜃𝜆𝛾𝐹𝑒₁+𝐹𝑒₂. (45)

Proof. On one hand, it follows from (34) that 𝐴_𝐻𝑒₂=−cosh²𝜃𝑒₁,𝐴_𝐻𝑒₁=−cosh²𝜃𝜆𝑒₂. (46) Combining (46) with (33) gives ℎ𝑒₁,𝐴_𝐻𝑒₂+ℎ𝑒₂,𝐴_𝐻𝑒₁=−cosh²𝜃𝜆𝛾𝐹𝑒₁+2𝜆𝐹𝑒₂. (47) On the other hand, it follows from (33) and the Gauss equation that the Gauss curvature is given by 𝐾=cosh²𝜃𝜆𝛾. (48) By Lemma 5, we have Δ^𝐷𝐻=cosh²𝜃𝜆𝛾𝐹𝑒₁. Combining these with the first section of (39) completes the proof.

Lemma 8. Let 𝑀 be a marginally trapped slant surface in ℂ²₁. Then, the tangential part of Δ𝐻 is expressed as (Δ𝐻)^⊤=cosh²𝜃𝑒₂(𝜆)−𝜆𝜔₂−2𝜆sinh𝜃𝑒₂. (49)

Proof. By (27), (36), and (46), we obtain −𝐴_{𝐷𝑒1𝐻}𝑒₂−𝐴_{𝐷𝑒2𝐻}𝑒₁=−Φ₂𝐴_𝐻𝑒₁=cosh²𝜃Φ₂𝜆𝑒₂. (50) It follows from (26), (38), and (46) that ∇_𝑒1𝐴_𝐻𝑒₂=∇_𝑒1𝐴_𝐻𝑒₂−𝐴_𝐻∇_𝑒1𝑒₂=−2cosh²𝜃𝜔₁𝑒₁∇=0,_𝑒2𝐴_𝐻𝑒₁=∇_𝑒2𝐴_𝐻𝑒₁−𝐴_𝐻∇_𝑒2𝑒₁=−cosh²𝜃𝑒₂(𝜆)−2𝜆𝜔₂𝑒₂. (51) 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 ℂ²₁.

Lemma 9. Let 𝜙∶𝑀→ℂ²₁ be a marginally trapped slant immersed in ℂ²₁. Then the bitension field is given by 𝜏₂𝜙=−2cosh²𝜃𝑒₂(𝜆)−𝜆𝜔₂−2𝜆sinh𝜃𝑒₂−4cosh²𝜃𝜆𝛾𝐹𝑒₁+𝐹𝑒₂. (52)

5. Classification Results

From now on, let us consider the biharmonic and quasi-biharmonic Marginally trapped slant surfaces in Lorentzian complex plane ℂ²₁.

By the definition of biharmonic surfaces, we can conclude the following from Lemma 9.

Lemma 10. Let 𝜙∶𝑀→ℂ²₁ be a marginally trapped slant immersed in ℂ²₁. 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 field 𝜏₂(𝜙) is lightlike. Therefore, we also have the following.

Lemma 11. Let 𝜙∶𝑀→ℂ²₁ be a marginally trapped slant immersed in ℂ²₁. Then the immersion is quasi-biharmonic if and only if the functions 𝜆 and 𝛾 satisfy 𝜆≠0 and 𝛾=0.

Since the Gauss curvature 𝐾 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 𝑒₂𝛽+𝛽𝜔₂=0, (53) 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 𝑔=−𝛽𝑑𝑥⊗𝑑𝑦+𝑑𝑦⊗𝑑𝑥, (54) and the Levi-Civita connection of 𝑔 satisfies ∇_{𝜕/𝜕𝑥}𝜕=𝛽𝜕𝑥𝛽^𝑥𝜕𝜕𝑥,∇_{𝜕/𝜕𝑥}𝜕𝜕𝑦=0,∇_{𝜕/𝜕𝑦}𝜕=𝛽𝜕𝑦𝛽^𝑦𝜕.𝜕𝑦 (55) Moreover, it follows from (33) and (54) that ℎ𝜕,𝜕𝜕𝑥𝜕𝑥=𝜆𝛽²𝐹𝜕𝜕𝜕𝑦,ℎ,𝜕𝜕𝑥𝜕𝜕𝑦=𝐹,ℎ𝜕𝜕𝑥,𝜕𝜕𝑦𝜕𝑦=𝛾𝛽⁻¹𝐹𝜕𝜕𝜕𝑥+𝐹.𝜕𝑦 (56) 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 𝛽_𝑦+𝛽𝜔₂𝑦=0. (57) Solving (57) gives 𝛽=𝑒^{−∫𝜔2(𝑦)𝑑𝑦}. (58) 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: 𝐿_𝑥𝑥=𝜆𝛽²(𝑖+sinh𝜃)𝐿_𝑦,𝐿_𝑥𝑦=(𝑖−sinh𝜃)𝐿_𝑥,𝐿_𝑦𝑦=𝛽⁻¹𝛾(𝑖−sinh𝜃)𝐿_𝑥+𝛽_𝑦𝛽+𝑖+sinh𝜃𝐿_𝑦. (59)(60)(61) By solving (60), we obtain that 𝐿𝑥,𝑦=𝑓(𝑥)𝑒^{(𝑖−sinh𝜃)𝑦}+𝑧𝑦, (62) for some vector-valued functions 𝑓(𝑥) and 𝑧(𝑦) in ℂ²₁.

Theorem 12. Up to rigid motions of ℂ²₁, every biharmonic marginally trapped 𝜃-slant surface in ℂ²₁ is given by a flat slant surface defined by 𝐿𝑥,𝑦=𝑐₁𝑥𝑒^{(𝑖−sinh𝜃)𝑦}+𝑤𝑦, (63) where 𝑐₁ is lightlike vector and 𝑤(𝑦) is a null curve in ℂ²₁ satisfying 𝑐₁(𝑖−sinh𝜃)𝑒^{(𝑖−sinh𝜃)𝑦},𝑤^′𝑦𝑐=0,₁𝑒^{(𝑖−sinh𝜃)𝑦},𝑤^′𝑦=−𝛽𝑦 (64) for some nonzero real-valued function 𝛽(𝑦).

Proof. Let 𝑀 be a biharmonic marginally trapped 𝜃-slant surface in ℂ²₁. According to Lemma 10, we have 𝜆=0. Substituting (62) into (59), we have 𝑓^′′(𝑥)=0, (65) which yields 𝑓(𝑥)=𝑐₁𝑥+𝑐₂ for two constant vectors 𝑐₁ and 𝑐₂ in ℂ²₁. Hence, the immersion becomes 𝐿𝑥,𝑦=𝑐₁𝑥𝑒^{(𝑖−sinh𝜃)𝑦}+𝑤𝑦. (66) Note that 𝑤(𝑦)=𝑐₂𝑒^{(𝑖−sinh𝜃)𝑦}+𝑧(𝑦) here. Moreover, (66) yields 𝐿_𝑥=𝑐₁𝑒^{(𝑖−sinh𝜃)𝑦},𝐿_𝑦=𝑐₁(𝑖−sinh𝜃)𝑥𝑒^{(𝑖−sinh𝜃)𝑦}+𝑤^′𝑦. (67) It follows from (54) and (67) that 𝑐₁,𝑐₁=𝑤^′𝑦,𝑤^′𝑦=𝑐₁(𝑖−sinh𝜃)𝑒^{(𝑖−sinh𝜃)𝑦},𝑤^′𝑦𝑐=0,₁𝑒^{(𝑖−sinh𝜃)𝑦},𝑤^′𝑦=−𝛽𝑦. (68) This completes the proof of Theorem 12.

Theorem 13. Up to rigid motions of ℂ²₁, every quasi-biharmonic marginally trapped 𝜃-slant surface in ℂ²₁ is given by a flat slant surface defined by 𝐿𝑥,𝑦=𝑝(𝑥)𝑒^{(𝑖−sinh𝜃)𝑦}, (69) where 𝑝(𝑥) is a null curve lying in the light cone ℒ𝒞 satisfying 𝑝(𝑥),𝑝(𝑥)=𝑝^′(𝑥),𝑝^′(𝑥𝑝(𝑥))=0,,𝑖𝑝^′(𝑥)=1. (70)

Proof. Let 𝑀 be a quasi-biharmonic marginally trapped 𝜃-slant surface in ℂ²₁. According to Lemma 11, we have 𝛾=0 and 𝜆≠0. In this case, the second and third equations of (38) become 𝜔₂=2sinh𝜃,𝜆_𝑦=4sinh𝜃𝜆. (71) Solving 𝜆, we get 𝜆=𝑐₁(𝑥)𝑒^{4sinh𝜃𝑦}. (72) By applying (71), (58) yields 𝛽=𝑎𝑒^{−2sinh𝜃𝑦} (73) for some nonzero constant number 𝑎. Consequently, with the previous information, the PDE system (59)–(61) becomes 𝐿_𝑥𝑥=𝑎𝑐₁(𝑥)𝐿_𝑦,𝐿_𝑥𝑦=(𝑖−sinh𝜃)𝐿_𝑥,𝐿_𝑦𝑦=(𝑖−sinh𝜃)𝐿_𝑦. (74)(75)(76) Substituting (62) into (76), we have 𝑧^′′𝑦=(𝑖−sinh𝜃)𝑧^′𝑦, (77) which yields 𝑧𝑦=𝑐₂𝑒^{(𝑖−sinh𝜃)𝑦}+𝑐₃ (78) for two constant vectors 𝑐₂ and 𝑐₃ in ℂ²₁. We denote 𝑝(𝑥) by 𝑝(𝑥)=𝑓(𝑥)+𝑐₂. Hence, up to rigid motions of ℂ²₁ the immersion becomes 𝐿𝑥,𝑦=𝑝(𝑥)𝑒^{(𝑖−sinh𝜃)𝑦}. (79) Substituting (79) into (74), we obtain𝑝^′′(𝑥)=−𝑎cosh²𝜃𝑐₁(𝑥)𝑝(𝑥). (80) Moreover, (79) gives 𝐿_𝑥=𝑝^′(𝑥)𝑒^{(𝑖−sinh𝜃)𝑦},𝐿_𝑦=(𝑖−sinh𝜃)𝑝(𝑥)𝑒^{(𝑖−sinh𝜃)𝑦}. (81) It follows from (54), (73), and (81) that 𝑝(𝑥),𝑝(𝑥)=𝑝^′(𝑥),𝑝^′(𝑥𝑝)=0,_′(𝑥),(𝑖−sinh𝜃)𝑝(𝑥)=−𝑎. (82) In view of (73), we can assume 𝑎=1. Hence, we get the conclusion.

Remark 14. According to Theorem 4, 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.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 71271045), Program for Liaoning Excellent Talents in University (no. LJQ2012099), and General Project for Scientific Research of Liaoning Educational Committee (no. W2012186).