Classification of Marginally Trapped Surfaces

Symposium Valenciennes 51

Classiﬁcation of marginally trapped surfaces

BANG-YEN CHEN

Department of Mathematics, Michigan State University, [email protected]

Abstract The concept of trapped surfaces introduced by R. Penrose in [Phys. Rev. Lett. 14, 57–59, 1965] plays an extremely important role in cosmology and general relativity. It is considered as a cornerstone for the achievement of the singularity theorems, the analysis of gravitational collapse, the cosmic censorship hypothesis, etc. In term of mean curvature vector, a surface in a space-time (or more generally, in a semi-Riemannian manifold) is marginally trapped if its mean curvature vector is light-like at each point. In this article we present some very recent results concerning the classification of marginally trapped surfaces in Lorentzian space forms and in Robertson-Walker space-times. We also present the classification of Lagrangian (or slant) marginally trapped surfaces in complex Lorentzian space forms; 4 and the classification of flat (or biharmonic) marginally trapped surfaces in E2 . Keywords : Marginally trapped surface, Lagrangian surface, slant surface, Lorentzian space form, Robertson-Walker space-time, biharmonic surface.

1 Introduction

For physical reasons, a space-time is mathematically defined as a four-dimensional, smooth, connected semi-Riemannian manifold together with a smooth Lorentz metric of signature (−,+,+,+). A tangent vector v of a semi-Riemannian manifold is called space-like (respectively, time-like) if hv,vi > 0 (respectively, hv,vi < 0). A vector v is called light-like (or null) if it is nonzero and it satisfies hv,vi = 0. The light-cone LC is defined by LC = {v : hv,vi = 0}.

The concept of trapped surfaces introduced by Roger Penrose in [24] plays an extremely important role in cosmology and general relativity. It is considered as a cornerstone for the achievement of the singularity theorems, the analysis of gravitational collapse, the cosmic censorship hypothesis, etc (see, for instance, [20, 21, 22, 25]). In the theory of cosmic black holes, if there is a massive source inside the surface, then close enough to a massive enough source, the outgoing light rays may also be converging; a trapped surface. Everything inside is trapped. Nothing can escape, not even light. It is believed that there will be a marginally trapped surface, separating the trapped surfaces from the untrapped ones, where the outgoing light rays are instantaneously parallel. The surface of a black hole is located by the marginally trapped surface. As times develops, the marginally trapped surface generates a hypersurface in spacetime, a trapping horizon. Very little are known on the geometry of marginally trapped surfaces. In particular, no classification results are previously known for marginally trapped surfaces in space-times. Moreover. the issue of differentiability of the boundary of the trapped region is still wide open. However, for strictly stable outer marginally trapped surfaces, the following two results were shown in [2]: (i) Local existence of a trapping horizon; and (ii) Outgoing light rights are converging just inside and diverging just outside such a surface. In term of the mean curvature vector, a codimension-two surface is future trapped if its mean curvature vector is timelike and future-pointing at each point (similarly, for passed trapped); and marginally 52 B.Y. Chen trapped if its mean curvature vector is light-like at each point on the surface. (A surface with light-like mean curvature vector field is also known as a quasi-minimal surface by R. Rosca). In this article we present some very recent results concerning the classification of marginally trapped surfaces in Lorentzian space forms and in Robertson-Walker space-times. We also present the classification of Lagrangian (or slant) marginally trapped surfaces in complex Lorentzian space forms; and the 4 classification of flat (or biharmonic) marginally trapped surfaces in E2 .

2 Preliminaries

2.1 Basic notation and formulas

4 Let M be a surface in a space-time ST1 . Denote by ∇ and R the Levi Civita connections and the curvature tensor on M, respectively. Similarly, we denote by ∇˜ and R˜ the Levi Civita connections and 4 the curvature tensor on ST1 , respectively. Let X and Y be vector fields tangent to M and let ξ be a normal vector field of M. The formulas of Gauss and Weingarten give a decomposition of the vector fields ∇˜ XY and ∇˜ X ξ into a tangent and a normal component (cf. [23]):

∇˜ XY = ∇XY + h(X,Y), (1) ˜ ∇X ξ = −AξX + DX ξ. (2) These formulae deﬁne h,A and D, which are called the second fundamental form, the shape operator and the normal connection respectively. ⊥ For each ξ ∈ Tx M, the shape operator Aξ is a symmetric endomorphism of the tangent space TxM at x ∈ M. The shape operator and the second fundamental form are related by

hh(X,Y),ξi = AξX,Y (3)

−→ 1 for X,Y tangent to M and ξ normal to M. The mean curvature vector is deﬁned by H = 2 traceh. The equations of Gauss and Codazzi are given respectively by

hR(X,Y)Z,Wi = Ah(Y,Z)X,W − Ah(X,Z)Y,W + R˜(X,Y)Z,W , (4) (R˜(X,Y)Z)⊥ = (∇h)(X,Y,Z) − (∇h)(Y,X,Z), (5) where X,Y,Z,W are vector tangent to M, (R˜(X,Y)Z)⊥ the normal component of R˜(X,Y)Z, and ∇h is deﬁned by (∇h)(X,Y,Z) = DX h(Y,Z) − h(∇XY,Z) − h(Y,∇X Z). (6)

The relative null space at a point p ∈ M in ST is deﬁned by

Np(M) = {X ∈ TpM |h(X,Y) = 0 for all Y ∈ TpM}. (7)

4 The dimension of Np(M), denoted by νp(M), is called the relative nullity at p. The surface M in ST1 is said to have positive relative nullity if νp(M) > 0 for each p ∈ M. 4 4 If ST1 is of constant sectional curvature c, then the curvature tensor of ST1 satisﬁes R˜(X,Y)Z = c{hY,ZiX − hX,ZiY}. (8)

2.2 Lorentzian space forms

n Let Es denote the semi-Euclidean n-space with metric tensor given by s n 2 2 g0 = − ∑ dxi + ∑ dx j , (9) i=1 j=s+1 Symposium Valenciennes 53

n n where {x1,...,xn} is the rectangular coordinate system of Es . Then (Es ,g0) is a ﬂat semi-Riemannian manifold with index s. We put n n+1 −2 Ss (c) = {x ∈ Es |hx,xi = c > 0}, (10) n n+1 −2 Hs (c) = {x ∈ Es+1 |hx,xi = c < 0}, (11) n where h , i is the indeﬁnite inner product on the pseudo-Euclidean space. It is well-known that Ss (c) n and Hs (c) are complete semi-Riemannian manifolds with index s of constant sectional curvature c > 0 and c < 0, respectively. 4 4 4 The semi-Riemannian manifolds E1 ,S1(c) and H1 (c) are known as the Minkowski space-time, the de Sitter space-time and the anti-de Sitter space-time, respectively. These space-times are called Lorentzian space forms.

2.3 Lorentzian complex space forms

˜ n Let Mi (4ε) be an indefinite complex space form of complex dimension n and complex index i. The complex index is defined as the complex dimension of the largest complex negative definite subspace of ˜ n the tangent space. If i = 1, we say that Mi (4ε) is Lorentzian. ˜ ˜ n The curvature tensor R of Mi (4ε) is given by R˜(X,Y)Z = ε{hY,ZiX−hX,ZiY+hJY,ZiJX−hJX,ZiJY+2hX,JYiJZ}.

n n Let C denote the complex number n-space with complex coordinates z1,...,zn. The C endowed with gi,n, i.e., the real part of the Hermitian form i n n bi,n(z,w) = − ∑ z¯kwk + ∑ z¯jw j, z,w ∈ C , k=1 j=i+1 n defines a flat indefinite complex space form with complex index i. We simply denote the pair (C ,gi,n) n 3 by Ci . Let h , i denote the inner product on Ci induced from gi,3 for i = 1 or 2. Let us put εv = 1 (respectively, εv = −1) whenever v is a spacelike vector (respectively, v is timelike). Consider the differentiable manifold: 2n+1 n+1 S2 = {z ∈ C1 ; b1,n+1(z,z) = 1}, which is an indefinite real space form of constant sectional curvature one. The Hopf fibration 2n+1 n ∗ π : S2 → CP1 (4) : z 7→ z · C n is a submersion and there exists a unique pseudo-Riemannian metric of complex index one on CP1 (4) n such that π is a Riemannian submersion. The pseudo-Riemannian manifold CP1 (4) is a Lorentzian complex space form of positive holomorphic sectional curvature 4. 2n+1 n+1 Analogously, consider H3 = {z ∈ C2 ; b2,n+1(z,z) = −1}, which is an indefinite real space form of constant sectional curvature −1. The Hopf fibration 2n+1 n ∗ π : H3 (−1) → CH1 (−4) : z 7→ z · C n is also a submersion and there is a unique pseudo-Riemannian metric of complex index 1 on CH1 (−4) n such that π is a Riemannian submersion. The CH1 (−4) is a Lorentzian complex space form of negative holomorphic sectional curvature −4. ˜ n An n-dimensional submanifold M of M1(4ε) is called Lagrangian if the almost complex structure J of ˜ n ˜ n M1(4ε) interchanges the tangent and the normal spaces of M. Lagrangian submanifolds of M1(4ε) are Lorentzian. Similar to marginally surfaces in Lorentzian space forms, a surface in a Lorentzian Kahler¨ surface (or more generally, in a pseudo-Riemannian manifold) is called marginally trapped if its mean curvature vector is light-like at each point on the surface. 54 B.Y. Chen 3 Some existence results

Recall the following Existence and Uniqueness theorems from [19].

Theorem 1 Let (M,g) be a simply connected Lorentzian surface and TM be the tangent bundle of M. If σ be a TM-valued symmetric bilinear form on M satisfying

(1) hσ(X,Y),Zi is totally symmetric,

(2) (∇σ)(X,Y,Z) = ∇X σ(Y,Z) − σ(∇XY,Z) − σ(Y,∇X Z) is totally symmetric,

(3) R(X,Y)Z = ε(hY,ZiX − hX,ZiY) + σ(σ(Y,Z),X) − σ(σ(X,Z),Y), then there exists a Lagrangian isometric immersion L from (M,g) into a complete simply-connected ˜ 2 Lorentzian complex space form M1(4c) whose second fundamental form h is given by h(X,Y) = Jσ(X,Y).

˜ 2 Theorem 2 Let L1,L2 : M → M1(4c) be Lagrangian isometric immersions of a Lorentzian surface M with second fundamental forms h1 and h2, respectively. If

1 2 h (X,Y),JL1?Z = h (X,Y),JL2?Z ,

˜ 2 for all vector ﬁelds X,Y,Z tangent to M, then there exists an isometry φ of M1(4c) such that L1 = L2 ◦φ.

In order to state the classiﬁcation of marginally trapped Lagrangian surfaces in non-ﬂat complex Lorentzian space forms, we need the following results from [10] whose proofs are based on Theorems 1 and 2.

Proposition 1 Let ψ(x,y) be a solution of the third order differential equation:

2 ψψxxy − ψxxψy − 3ψ ψx + ψy = 0.

2 with ψx,ψy 6= 0 and let Dψ be a simply-connected domain in R such that ψ is nowhere zero on Dψ. Denote the surface (Dψ,gψ) endowed with Lorentzian metric gψ = −ψdxdy by Mψ. Then, up to rigid 2 2 motions of CP1 (4), there exists a unique Lagrangian isometric immersion ψˆ : Mψ → CP1 (4) whose second fundamental form satisﬁes ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ ψxψy 2 ∂ ∂ h , = J , h , = J , h , = ψxy − − ψ J + J . ∂x ∂x ψ ∂y ∂x ∂y ∂x ∂y ∂y ψ ∂x ∂y

Proposition 2 Let φ(x,y) be a solution of the third order differential equation:

2 φφxxy − φxxφy + 3φ φx + φy = 0.

2 with φx,φy 6= 0 and let Dφ be a simply-connected domain in R such that φ is nowhere zero on Dφ. Denote the surface (Dφ,gφ) endowed with Lorentzian metric gφ = −φdxdy by Pφ. Then, up to rigid 2 ˆ 2 motions of CH1 (−4), there exists a unique Lagrangian isometric immersion φ : Pφ → CH1 (−4) whose second fundamental form satisﬁes ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ φxφy 2 ∂ ∂ h , = J , h , = J , h , = φxy − + φ J + J . ∂x ∂x φ ∂x ∂x ∂y ∂x ∂y ∂y φ ∂x ∂y

We also have the following result from [9].

2 Proposition 3 Let U be a simply-connected domain of the Lorentzian plane E1 = {(x,y) : x,y ∈ R} with metric g = −dxdy. Let M denote the flat surface given by U together with the flat metric g = −dxdy. Assume that γ and δ are non-constant functions defined on M satisfying

γx + (lnδ)xγ = (lnβ)y, δx 6= 0, (γδ)x 6= 0, (lnδ)xy − (lnβ)xy = 2βγ Symposium Valenciennes 55

2 with β = δδx/(γγy − 2γ δx − γyδ − 2γγxδ). Then, up to rigid motions, there exists a unique marginally 4 trapped isometric immersion ψγδ : M → E2 whose second fundamental form h and normal connection D satisfy ∂ ∂ βγ ∂ ∂ ∂ ∂ h , = βe − e , h , = e , h , = γe + δe , ∂x ∂x 4 δ 3 ∂x ∂y 3 ∂y ∂y 3 4

δx βy δx βy D ∂ e3 = e3, D ∂ e3 = e3, D ∂ e4 = − e4, D ∂ e4 = − e4, ∂x δ ∂y β ∂x δ ∂y β for some pseudo-orthonormal normal frame {e3,e4} satisfyinghe3,e3i = he4,e4i = 0,he3,e4i = −1.

4 Marginally trapped surfaces with positive relative nullity

4 For marginally trapped spatial surfaces in the Minkowski space-time E1 , we have the following classi- ﬁcation result obtained in [16].

Theorem 3 Up to Monkowskian motions, there exist two families of marginally trapped spatial surfaces 4 with positive relative nullity in E1 : (1) A surface deﬁned by L(x,y) = f (x),x,y, f (x), where f (x) is an arbitrary differentiable function with f 00(x) being nowhere zero.

(2) A surface deﬁned by

Z x Z x L(x,y) = r(x)q0(x)dx + q(x)y,ycosx− r(x)sinxdx, 0 0

Z x Z x ! ysinx+ r(x)cosxdx, r(x)q0(x)dx + q(x)y , 0 0 where q and r are deﬁned on an open interval I 3 0 satisfying q00(x) + q(x) 6= 0 for each x ∈ I.

Conversely, every marginally trapped spatial surface with positive relative nullity in the Minkowski 4 space-time E1 is congruent to an open portion of a surface obtained from the two families.

We also have the following two classiﬁcation theorems from [16] for marginally trapped spatial surfaces 4 4 with positive relative nullity in the de Sitter space-time S1(1) and the anti-de Sitter space-time H1 (−1).

4 Theorem 4 Up to rigid motions of S1(1), there exist two families of marginally trapped spatial surfaces 4 5 with positive relative nullity in the de Sitter space-time S1(1) ⊂ E1 : (1) A surface given by L(x,y) = f (x)cosy,sinxcosy,siny,cosxcosy, f (x)cosy , where f is an arbitrary function deﬁned on an open interval I satisfying f 00 + f 6= 0 at each point in I.

(2) A surface given by L(s,y) = p(s),η1(s),η2(s),η3(s), p(s) cosy Z s Z s 0 0 − b − r(s)p (s)ds,ξ1(s),ξ2(s),ξ3(s),b − r(s)p (s)ds siny, 0 0 where b is a real number, p and r are deﬁned on an open interval I 3 0 such that r is non-constant, 2 3 η = (η1,η2,η3) is a unit speed curve in S (1) ⊂ E with geodesic curvature κg = r, and ξ = (ξ1,ξ2,ξ3) is the unit normal of η in S2(1).

Conversely, every marginally trapped spatial surface with positive relative nullity in the de Sitter space- 4 time S1 is congruent to an open portion of a surface obtained from the two families. 56 B.Y. Chen

4 Theorem 5 Up to rigid motions of H1 (−1), there exist five families of marginally trapped spatial 4 surfaces with positive relative nullity in the anti-de Sitter space-time H1 (−1): (1) L(x,y) = f (x)coshy,coshxcoshy,sinhy,sinhxcoshy, f (x)coshy , where f (x) is defined on an open interval I such that f 00(x) − f (x) 6= 0 at each x ∈ I. (2) L(x,y) = f (x)sinhy,coshy,cosxsinhy,sinxsinhy, f (x)sinhy , where f (x) is defined on an open interval I such that f 00(x) + f (x) 6= 0 at each x ∈ I.

2 y 3ey y y 2 y ey (3) L(x,y) = x e , 2 − 2sinhy,e − 2sinhy,xe ,x e − 2 .

x2ey y y y y x2ey (4)L(x,y) = sinhy − 2 − e , f (x)e ,xe , f (x)e ,sinhy − 2 , where f (x) is deﬁned on an open interval I such that f 00(x) 6= 0 at each x ∈ I.

(5) A surface deﬁned by L(s,y) = (p(s),η1(s),η2(s),η3(s), p(s) coshy Z s Z s 0 0 − b − r(s)p (s)ds,ξ1(s),ξ2(s),ξ3(s),b − r(s)p (s)ds sinhy, 0 0 where b is a real number, p and r are deﬁned on an open interval I 3 0 such that r is non-constant, η = 2 3 (η1,η2,η3) is a unit speed curve in H (−1) ⊂ E1 with geodesic curvature κg = r, and ξ = (ξ1,ξ2,ξ3) is the unit normal of η in H2(−1).

Conversely, every marginally trapped spatial surfaces with positive relative nullity in the anti-de Sitter 4 space-time H1 is congruent to an open portion of a surface obtained from the ﬁve families.

Remark 1 If we do the construction described in case (2) of Theorem 4 using the curve

η(s) = (coss,0,sins)

2 3 in S ⊂ E , then we get case (1). Indeed, this curve has geodesic curvature κg = 0 and its unit normal in S2 is (0,−1,0). Similarly, we can obtain the surfaces of cases (1)–(4) of Theorem 5 by following the construction de- 2 3 scribed in case (5) starting from a curve η with constant geodesic curvature κg in H ⊂ E1 . In particular, 2 2 if κg is a constant satisfying κg < 1 (respectively, κg > 1), we get case (1) (respectively, case (2)). Also, 2 3 if we start with a horocycle, i.e., a curve of constant geodesic curvature one in H ⊂ E1 , we obtain a marginally trapped spatial surface which is congruent to case (3) or case (4).

Remark 2 A conformal representation formula of Weierstrass-Bryant type (in terms of meromorphic data) was obtained in [1] for the class of marginally trapped surfaces M in the Minkowski space-time 4 E1 which satisfy the following two conditions: 4 (a) M has ﬂat normal connection in E1 , and 3 3 (2) M is locally isometric either to a minimal surface in E or to a maximal surface in E1 .

5 Marginally trapped surfaces in Robertson-Walker space-times

One very important family of cosmological models in general relativity is the family of Robertson- Walker space-times:

4 c c 2 2 L1( f ,c) := (I × S,g f ), g f = −dt + f (t)gc, (12)

c equipped with a warped product Lorentzian metric g f , where the warping function f is a positive function defined on an open interval I and (S,gc) is a three-dimensional Riemannian manifold of constant Symposium Valenciennes 57 curvature c. The standard choices for S are the three-sphere S3(c), the Euclidean three-space E3, and the hyperbolic three-space H3(c), with curvature c > 0, c = 0, and c < 0, respectively. The family of Robertson-Walker space-times includes the de Sitter, Minkowski and open parts of the anti de Sitter space-time and also Friedmann’s cosmological models as well. Robertson-Walker space- times provide good descriptions of our Universe except in the earliest era and the final era, while the warping function f describes expanding or contracting of our Universe (cf. [20, 23]). Some fundamental study of surfaces in Robertson-Walker space-times have been obtained in [17]. It follows from remark 2.1 of [17] that if the warping function f satisfies f f 00 = ( f 0)2 + c for some real 4 number c, then L1( f ,c) is a Lorentzian space form. 00 0 2 4 The next result shows that if we have f f 6= ( f ) + c, the Robertson-Walker space-time L1( f ,c) does not admit marginally trapped surfaces with positive relative nullity.

4 Theorem 6 Let L1( f ,c) = I × f S be a Robertson-Walker space-time which contains no open subsets 4 of constant curvature. Then L1( f ,c) does not admit any marginally trapped surface M with positive relative nullity.

6 Marginally trapped Lagrangian surfaces in Lorentzian complex space forms

A curve z = z(t) defined on an open interval I ⊂ R in a pseudo-Riemannian manifold is called null if it velocity vector z0(x) is light-like for each x ∈ I. 5 5 0 A curve z in S2, (in H3 , or in LC) is called Legendre if hz ,izi = 0 holds identically. The squared 2 2 00 00 Legendre curvature κˆ of a unit speed Legendre curve z is defined by κˆ = εz0 hz ,z i; its Legendre 00 000 torsion is τˆ = εz0 hz ,iz i. 5 3 5 3 If z(s) is a unit speed Legendre curve in S2(1) ⊂ C1 or in H3 (−1) ⊂ C2), then z z ,i ,z0,iz0 |z| |z| are orthonormal vector fields defined along the curve. Thus, there exists a unit normal vector field Pz 0 0 such that z/|z|,iz/|z|,z ,iz ,Pz,iPz form an orthonormal frame. 0 0 00 00 00 By differentiating hz (s),iz(s)i = 0, hz (s),z(s)i = 0, we get hz ,izi = 0,hz ,zi = −εz0 , Thus, z can be expressed as

00 0 z (s) = iψ(s)z (s) − εz0 cz(s) − a(s)Pz(s) + b(s)iPz(s) (13) for some real-valued functions ψ,a,b. The Legendre curve z(s) is called special Legendre if (13) reduces to

00 0 z (s) = iψ(s)z (s) − εz0 cz(s) − a(s)Pz(s), (14)

0 0 where Pz is a unit parallel normal vector ﬁeld, i.e., Pz(s) = µ(s)z (s) for µ = aεz0 εPz . It was proved in [8] that, for any given two functions ψ(s) 6= 0 and a(s) deﬁned on an open interval I, there exists a special Legendre curve satisfying (14). A Legendre curve z in the light cone LC is called special Legendre if hiz0,z00i = 0 holds identically. We have the following lemma from [8].

3 Lemma 1 If z : I → LC ⊂ Cs ,s = 1,2, is a unit speed special Legendre curve in the light cone LC, then it satisﬁes 000 2 0 1 2 0 z + ε 0 κˆ z + ε 0 (κˆ ) z − iτˆz = 0. (15) z 2 z 58 B.Y. Chen

Conversely, if a unit speed curve z(s) in LC satisfying (15) for the squared Legendre curvature κˆ 2 6= 0 and Legendre torsion τˆ, then it is special Legendre.

We have the following complete classiﬁcation of marginally trapped Lagrangian surfaces in Lorentzian complex plane from [10].

2 Theorem 7 Let M be a marginally trapped Lagrangian surface in the Lorentzian complex plane C1. (i) If M is flat, either M is an open part of a Lagrangian surfaces defined by (i.1) L(x,y) = eiyz(x) for some null curve z(x) in the light cone LC; or M is an open part of a Lagrangian surfaces defined by i f (t) (i.2) L(s,t) = c1se + z(t), where f (t) is a real-valued function, c1 is a light-like vector, and z(t) is a 2 0 i f 0 i f null curve in C1 satisfying hiz ,e c1i = 0 and hz ,e c1i = −1. (ii) If M is not flat, then it is an open part of a Lagrangian surface defined by Z y Z y iy 0 iy L(x,y) = z(x) e ψxdy − z (x) e ψdy, y0 y0 where ψ(x,y) with ψy 6= 0 is a solution of ψxx = q(x)ψ + 1 for some function q(x) and z(x) is curve in 2 00 the light cone LC of C1 satisfying z = qz.

For marginally trapped Lagrangian surfaces in non-ﬂat complex Lorentzian space forms, we have the following classiﬁcation results from [10].

Theorem 8 Let M be a marginally trapped Lagrangian surface in the Lorentzian complex projective 2 plane CP1 (4). (p) If M is of constant curvature K, then either K = 1 or K = 0. (p.1) If K = 1, then M is an open part of one of the following five types of Lagrangian surfaces: (p.1.1) A Lagrangian surface defined by √ √ √ √ √ √ √ ibu/ 2 6bu 6bu L(u,w) = e 3bw 3(i 2 + bw))sinh 2 −( 2+ibw)cosh 2 , √ √ √ √ √ √ 6bu 6bu −3ibu/ 2 3bwcosh 2 − (2 2 − ibw)sinh 2 ,( 2 + ibw)e , where b is a nonzero real number. (p.1.2) A Lagrangian surface defined by 1 √ √ √ √ L(u,v) = e 2iav 2 + ia(u − v) ,e 2iava(u − v), 2 + ia(u + v) , a(u + v) where a is a nonzero real number. (p.1.3) A Lagrangian surface defined by

2 √ L(u,v) = + 2i f (v) z(v) − z0(v), u + v where f (v) is a non-constant real-valued function and z(v) is a unit speed spacelike Legendre curve with Legendre squared curvature κˆ 2 = 6 f 2(v) in the light cone LC. (p.1.4) A Lagrangian surface defined by √ eibv/ 2 √ √ √ √ √ L(v,w) = 6 + i 3bwcosh( 6 bv) − 3 2i + bw)sinh( 6 bv), 3bw 2 2 √ √ √ √ √ √ ! 6 6 6 + i 3bw 3bwcosh( bv) + 3ibw − 2 2sinh( bv), √ , 2 2 e3ibv/ 2 Symposium Valenciennes 59 where b is a positive real number. z(v) z0(v) (p.1.5) A Lagrangian surface defined by L(u,v) = − , where z(v) is a unit speed timelike u + v 2 special Legendre curve in the light cone LC with zero Legendre curvature κˆ and nonzero Legendre torsion τˆ. (p.2) If K = 0, it is an open part of one of the following three types of flat Lagrangian surfaces:

(p.2.1) A Lagrangian surface deﬁned by

− 1 1 2 3 ix+ iy √ − 1 e 3 2 2 iy−3(2 3 )ix L(x,y) = √ 3(2 3 )x + 4y,8i + 3(2 3 )x + 4y,2 2e . 6 2

(p.2.2) A Lagrangian surface deﬁned by

2 2 2 2 i(p − 3q )cosh(px + 2bpqy) + 4pqsinh(px + 2bpqy) L(x,y) = eiqx+ib(q −p )y , pp2(p2 + 9q2)

p 2 2 p 2 2 ! p + q ib(p2+3q2)y−3iqx p + q p e , √ cosh(px + 2bpqy) , p2 + 9q2 2p √ where b is a real number either negative or greater than 3 4/3 and √ − 2 2 5 1 − 2 2 2 3 3η − 3 6 b 3 3 b + 2 3 η p 1 3 6 3 3 p = 2 q = 2 , η = (9b + 81b − 12b ) . 6 3 bη 6 3 bη

(p.2.3) A Lagrangian surface deﬁned by √ 2 2 2 1 √ √ (4kr − 3r(k + r )) 2 2 2 L(x,y) = √ e−i( 3k+r)x+ib(3k +2 3kr+r )y, 3k2 − 9kr2 √ 2 2 2ir(x+2bry) p √ ! (r − 3k )e 3k2r + 2 3kr2 + r3 2 2 √ , ei(3bk y+ 3k(x−2bry)−rx+br y) , 4 2 2 4 q √ 9k − 30k r + 9r 3k(r2 − k2) + 2 3k2r

2 4 1 2 4 1 √ √3 (4δ+2 3 δ2+2 3 ) 2 (4δ−2 3 δ2−2 3 ) 2 1 4 where k = √ , r = √ and δ = (2 − 27b3 + 3 81b6 − 12b3 ) 3 for some b ∈ (0, ). 2 6bδ 2 6bδ 3 2 (q) If M has non-constant curvature, then it is an open part of the Lagrangian surface ψˆ : Mψ → CP1 (4) deﬁned in Proposition 1.

Theorem 9 Let M be a marginally trapped Lagrangian surface in the Lorentzian complex hyperbolic 2 plane CH1 (−4). (h) If M is of constant curvature K, then K = −1 or K = 0. (h.1) If K = −1, then M is an open part of one of the following five types of Lagrangian surfaces: (h.1.1) A Lagrangian surface defined by √ √ 2 2 + ibw √ √ L(u,w) = e−ibu/ 2 √ sinh 6bu − cosh 6bu , 3bw 2 2 √ √ √ ! 2 − ibw √ 2 − ibw √ 2i − bw √ √ e3ibu/ 2, √ cosh 6bu + sinh 6bu , 3bw 3bw 2 bw 2 where b is a nonzero real number. (h.1.2) A Lagrangian surface defined by 1 √ √ √ √ L(u,v) = ae 2iav(u + v),a(u − v) + i 2,e 2iav(a(u + v) + 2i , a(u − v) 60 B.Y. Chen where a is a nonzero real number. 2 √ (h.1.3) A Lagrangian surface defined by L(u,v) = − 2iψ(v) z(v)+z0(v), where ψ(v) is a non- u − v constant real-valued function and z(v) is a unit speed timelike Legendre curve with non-constant Leg- endre squared curvature in the light cone LC. (h.1.4) A Lagrangian surface defined by √ e−ibv/ 2 √ √ √ √ L(v,w) = 3bw + i 2cosh( 6 bv) + 3 2 + ibwsinh( 6 bv), 3bw 2 2 √ √ √ √ √ √ √ 3ibv/ 2 6 6 e ( 6+i 3bw),3bwsinh( 2 bv)− 3 ibw−2 2 cosh( 2 bv) , where b is a nonzero real number. z(v) z0(v) (h.1.5) A Lagrangian surface defined by L(u,v) = − , where z(v) is a unit speed spacelike u − v 2 special Legendre curve in the light cone LC with zero Legendre curvature κˆ and nonzero Legendre torsion τˆ. (h.2) If K = 0, then M is an open part of one of the following three types of flat Lagrangian surfaces: (h.2.1) A Lagrangian surface defined by

i 2/3 (2y−3(2 )x) 1 e 6 5 iy+3ix/2 3 4 4 L(x,y) = 2 6 e ,3ix + 2 3 (2 − iy),3x − 2 3 y . 3(25/6)

(h.2.2) A ﬂat Lagrangian surface deﬁned by

2 2 iqx+ib(q2−p2)y 4pqcosh(px + 2bpqy) + i(p − 3q )sinh(px + 2bpqy) L(x,y) = e p , p 2p2 + 18q2 ! p 2 2 p 2 2 p + q i(bp2y+3bq2y−3qx) p + q p e , √ sinh(px + 2bpqy) , p2 + 9q2 2p √ where b is either a negative number or a positive number greater than 3 4/3 and

2 √ 5 2 1 − 3 2 6 − 3 2 3 1 2 3η − 3 b 2 η + 3 b p 6 3 3 3 p = 2 q = 2 , η = 81b − 12b − 9b 6 3 bη 6 3 bη

(h.2.3) A ﬂat Lagrangian surface deﬁned by √ p 2 2 3 √ 3k r − 2 3kr + r i (rx+r2y+3k2y+ 3k(x+2ry)) L(x,y) = q √ e b , 3 3k(k2 − r2) + 6k2r q √ ! 2 2 2ib−1r(2ry−x) 4kr2 + 3r(r2 + r2) √ (r − 3k )e i (rx+r2y+3k2y− 3k(x+2ry)) √ , p e b , 9k4 − 30k2r2 + 9r4 3k(k2 − 3r2)

√ √ q3 √ √ √ b(γ+1)2 b(γ−1)2 − 27b3−2+3 3b 27b4−4b where b ∈ (0, 3 4/3), k = √ , r = √ and γ = √ . 2 3γ 2 −3γ 3 2 ˆ ˜ 2 (g) If M has non-constant curvature, then it is an open part of the Lagrangian surface φ : Pφ → CH1 (−4) deﬁned in Proposition 2.

7 Marginally trapped slant surfaces

Let M be a Riemannian n-manifold isometrically immersed in a Kahlerian¨ manifold (M˜ ,g,J) endowed with Kahler¨ metric g and almost complex structure J. For each vector X tangent to M, we put JX = PX + FX, (16) Symposium Valenciennes 61 where PX and FX are the tangential and normal components of JX. Then P is an endomorphism of the tangent bundle TM. π For any nonzero vector X tangent to M at a point p, the angle θ(X), θ(X) ∈ [0, 2 ], between JX and the tangent space TpM is called the Wirtinger angle of X. A submanifold M is called slant if its Wirtinger angle θ is constant, i.e., θ(X) is independent of the choice of the X in the tangent bundle TM. The Wirtinger angle θ of a slant immersion is called the slant angle. A slant submanifold with slant angle θ is simply called θ-slant (see, [6, 15] for details). ˜ 2 Let M be a Lorentzian surface in a Lorentzian Kahler¨ surface (M1,g,J). Then there exists an orthonormal local frame {eˆ1,eˆ2} on M such that

heˆ1,eˆ1i = 1, heˆ2,eˆ2i = −1, heˆ1,eˆ2i = 0. (17)

˜ 2 Similarly, one may deﬁne slant surfaces in a Lorentzian Kahler¨ surface (M1,g,J) as follows (cf. [14]).

˜ 2 Deﬁnition 1 A Lorentzian surface M in a Lorentzian Kahler¨ surface (M1,g,J) is called θ-slant if there exists a real number θ such that heˆ1,Jeˆ2i = sinhθ holds for some orthonormal basise ˆ1,eˆ2 satisfying (17) on M.

Without loss of generality, we may assume that θ ≥ 0 for every θ-slant surface (otherwise, we just replacee ˆ2 by −eˆ2). Since θ-slant surface with θ = 0 is a Lagrangian surface, we simply call a θ-slant surface with θ 6= 0 a proper slant surface. For slant surfaces in Lorentzian complex space forms we have the following results from [14].

Theorem 10 There do not exist marginally trapped proper slant Lorentzian surfaces in any Lorentzian ˜ 2 complex space form M1(4c) with c 6= 0.

2 2 This theorem implies that marginally trapped slant Lorentzian surfaces in CP1 and CH1 are always Lagrangian. 2 The next theorem together with Theorem 7 completely classify marginally trapped slant surfaces in C1.

Theorem 11 Let θ be a nonzero real number. Then we have: (I) If z(s) is a null curve in the light cone LC satisfying hz0,izi = 2sinhθ, then

1 (1−i csch ) L(s,t) = z(s)t 2 θ

2 defines a flat marginally trapped Lorentzian θ-slant surface in the Lorentzian complex plane C1. (II) If z(s) is a null curve lying in LC which satisfies hz,iz0i = 1, then

L(s,y) = z(s)e(i−sinhθ)y

2 defines a flat marginally trapped Lorentzian θ-slant surface in C1. (III) For any given function ϕ(t) defined on an open interval I containing 1,

Z t Z t 2sinhθ 1 (i csch −1) 1 (3−i csch ) i L(s,t) = tϕ(t)dt +t 2 θ ϕ(t)t 2 θ dt − 1 (i csch −1) t 2 θ 1 1 2 Z t Z t ! s 1 (i csch −1) 1 (3−i csch ) i s − cschθ, tϕ(t)dt +t 2 θ ϕ(t)t 2 θ dt + i − cschθ 2 1 1 2 2

2 deﬁnes a ﬂat marginally trapped Lorentzian θ-slant surface in C1. 62 B.Y. Chen

R t (IV) Let µ(t) and ϕ(t) be two functions deﬁned on an open interval I containing 0. Put F(t) = 0 µ(t)dt and Φ(t) = ϕ(t)e−2F(t)sinhθ. Then

Z t Z t L(s,t) = se(i−sinhθ)F(t) + (sinhθ − i) Φ(t) e(i+sinhθ)F(u)du dt 0 0 Z t 1 Z t + (1 + isinhθ) e(i+sinhθ)F(t)dt + i Φ(t)dt , 0 2 0 Z t Z t se(i−sinhθ)F(t) + (sinhθ − i) Φ(t) e(i+sinhθ)F(u)du dt 0 0 ! Z t i Z t + (i − sinhθ) e(i+sinhθ)F(t)dt + Φ(t)dt 0 2 0

2 defines a flat marginally trapped Lorentzian θ-slant surface in C1. (V) Let p(s) be a function defined on on open interval I 3 0, φ(s,y) be a solution of the second differential equation:

2 − 4 ysinhθ φss − p(s)φ = cosh θe 3 ,

2 00 00 0 00 2 and z be a null curve in C1 satisfying hz ,z i = 0 and hz ,iz i = cosh θ. If φ is not the product of two functions of single variable, then

00 0 2 − 4 ysinh Z y φz − φ z + cosh θe 3 θz L(s,y) = s dy + z(s)ey(i−sinhθ) −y(i+ 1 sinh ) 0 (sinhθ + i)e 3 θ

2 defines a non-flat marginally trapped Lorentzian θ-slant surface in C1. 2 Conversely, up to dilations and rigid motions of C1, every marginally trapped proper slant Lorentzian 2 surface in C1 is locally obtained from one of the five families given above.

4 8 Flat marginally trapped surfaces in E2

4 The following result from [9] classiﬁes marginally trapped Lorentzian ﬂat surfaces in E2.

4 4 Theorem 12 If L : M → E2 is a marginally trapped Lorentzian ﬂat surface in E2, then L is congruent to a surface of the following eight types:

(1) A surface deﬁned by L(x,y) = √1 ϕ(x,y),x + y,x − y,ϕ(x,y), where ϕ(x,y) is a function satisfying 2 2 ϕxy > 0 on an open domain U ⊂ E1. (2) A surface deﬁned by L(x,y) = z(x)y+w(x), where z(x) is a null curve in the light-cone LC and w(x) is a null curve satisfying hz0,w0i = 0 and hz,w0i = −1.

(3) A surface deﬁned by

1 L = 2abcosaxcosby − sinaxsinby,2abcosaxsinby + sinaxcosby, 2ab 2abcosaxcosby + sinaxsinby,2abcosaxsinby − sinaxcosby , a,b > 0.

(4) A surface deﬁned by

1 L = 2abcosaxcoshby − sinaxsinhby,2abcosaxsinhby + sinaxcoshby, 2ab 2abcosaxcoshby + sinaxsinhby,2abcosaxsinhby − sinaxcoshby Symposium Valenciennes 63 for some positive numbers a,b.

(5) A surface deﬁned by

1 L = 2abcoshaxcoshby − sinhaxsinhby,2abcoshaxsinhby + sinhaxcoshby, 2ab 2abcoshaxcoshby + sinhaxsinhby,2abcoshaxsinhby − sinhaxcoshby for some positive numbers a,b.

(6) A surface deﬁned by L(x,y) = z(y)cosax + w(y)sinax, where a is a positive number and z,w are null curves lying in the light cone LC satisfying hz,wi = z00 + δz = w00 + δw = 0 and hz,w0i = a−1 for some non-constant function δ(y).

(7) A surface deﬁned by L(x,y) = z(y)coshax + w(y)sinhax, where a is a positive number and z,w are null curves lying in the light cone LC satisfying hz,wi = z00 + δz = w00 + δw = 0 and hz,w0i = a−1 for some non-constant function δ(y).

4 (8) A surface ψγδ : M → E2 deﬁned in Proposition 3.

9 Biharmonic marginally trapped surfaces

4 Let (x1,x2,x3,x4) be the canonical coordinates of the semi-Euclidean 4-space Es . Let us assume that 4 4 L : M → Es is an isometric immersion from a semi-Riemannian surface M into Es . Denote by ∆ the 4 Laplacian on M acting on smooth functions. The Laplacian ∆ can be extended to act on Es -valued maps P = (P1,...,P4) by ∆P = (∆P1,...,∆P4).

4 It is well-known that the position vector of M in Es satisﬁes (cf. [6, 7])

∆L = −2H. (18)

4 4 where H is the mean curvature vector of M in Es . It follows from (18) that M is minimal in Es if and only if the immersion L is harmonic, i.e., ∆L = 0. 4 An isometric immersion L : M → Es is called biharmonic if and only if we have (cf. [7, 12, 13, 3])

∆2L = 0. (19)

It is known that biharmonic submanifolds are closely related with the critical points of the bi-energy functional (see [4, 29] for details). 4 For spatial biharmonic marginally surfaces in Es we have the following result from [12].

4 4 Theorem 13 Let L : M → Es be a spatial surface in Es with ﬂat normal connection. Then L is a biharmonic marginally trapped surface if and only if we have (a) s = 1; (b) M is an open portion of the Euclidean plane E2 with ﬂat metric g = du2 + dv2; and 4 4 (c) up to rigid motions of E1 , the immersion L : M → Es is congruent to

x(u,v) = (ϕ(u,v),ϕ(u,v),u,v), where ϕ = ϕ(u,v) is function on M satisfying ∆ϕ 6= 0 and ∆2ϕ = 0.

By applying Theorem 12, we have the next theorem from [9] which completely classiﬁes biharmonic 4 marginally trapped Lorentzian surface in Es . 64 B.Y. Chen

4 Theorem 14 A marginally trapped Lorentzian surface in Es is biharmonic if and only if s = 2 and M is congruent to one of the following surfaces:

(a) A surface deﬁned by 1 L(x,y) = √ ϕ(x,y),x + y,x − y,ϕ(x,y), 2 2 where ϕ(x,y) is a function satisfying ϕxy 6= 0 and ϕxxyy = 0 on an open domain U ⊂ E1. (b) A surface deﬁned by L(x,y) = z(x)y + w(x), where z(x) is a null curve in the light-cone LC and w(x) is a null curve satisfying hz0,w0i = 0 and hz,w0i = −1.

Remark 3 It was pointed out in [27] that the only marginally trapped biharmonic Lagrangian surface 2 in Lorentzian complex space form is the one in C1 given by case (i.2) of Theorem 7 (which is a special case of case (b) of Theorem 14).

10 Open problem.

Problem 1: Classify all marginally trapped surfaces in Lorentzian real and complex space forms. Problem 2: Classify all marginally trapped surfaces in Robertson-Walker space-times. 4 4 4 Problem 3: Classify all marginally trapped surfaces in E2 , in S2(1) and in H2 (−1).

References

[1] J. A. Aledo, J. A. Galvez´ and P. Mira. Marginally trapped surfaces in L4 and an extended Weierstrass-Bryant representation. Ann. Global Anal. Geom. 28, 395–415, 2005.

[2] L. Andersson, M. Mars and W. Simon. Local existence of dynamical and trapping horizons, Phys. Rev. Letters 95, 111102-(1-4), 2005.

[3] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and B. J. Papantoniou. Biharmonic Lorentz hy- 4 persurfaces in E1. Paciﬁc J. Math. 229, 293–305, 2007. [4] R. Caddeo, S. Montaldo and C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math. 130, 109–123, 2002.

[5] B. Y. Chen. Total mean curvature and submanifolds of ﬁnite type. World Scientiﬁc, New Jersey, 1984.

[6] B. Y. Chen. Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, 1990.

[7] B. Y. Chen. A report on submanifolds of ﬁnite type. Soochow J. Math. 22, 117–337, 1996.

[8] B. Y. Chen. Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes. Math. Nachr. 278, 1242–1281, 2005.

4 [9] B. Y. Chen. Classiﬁcation of marginally trapped Lorentzian ﬂat surfaces in E2 and its application to biharmonic surfaces. (preprint).

[10] B. Y. Chen and F. Dillen. Classiﬁcation of marginally trapped Lagrangian surfaces in Lorentzian complex space forms. J. Math. Phys. 48, no. 1, 013509, 23 pages, 2007.

[11] B. Y. Chen and J. Fastenakels. Classiﬁcation of ﬂat Lagrangian Lorentzian surfaces in complex Lorentzian plane. Acta Math. Sinica (Engli. Ser.) 23, 2007. Symposium Valenciennes 65

[12] B. Y. Chen and S. Ishikawa. Biharmonic surfaces in pseudo-Euclidean spaces. Memoirs Fac. Sci. Kyushu Univ. Ser. A, Math. 45, 325–349, 1991. [13] B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu J. Math. 52, 1–18, 1998. [14] B. Y. Chen and I. Mihai. Classification of marginally trapped slant surfaces in Lorentzian complex space forms. preprint. [15] B. Y. Chen and Y. Tazawa. Slant submanifolds of complex projective and complex hyperbolic spaces. Glasgow Math. J. 42, 439–454, 2000. [16] B. Y. Chen and J. Van der Veken. Marginally trapped surfaces in Lorentzian space forms with positive relative nullity. Classical Quantum Gravity 24, 551–563, 2007. [17] B. Y.Chen and J. Van der Veken. Spatial and Lorentzian surfaces in Robertson-Walker space-times. J. Math. Phys. 48, no. 7, 12 pages, 2007. [18] B. Y. Chen and J. Van der Veken. Classification of parallel surfaces in Lorentzian space forms. (in preparation). [19] B. Y. Chen and L. Vrancken. Lagrangian minimal isometric immersions of a Lorentzian real space form into a Lorentzian complex space form. Tohoku Math. J. 54, 121–143, 2002. [20] S. W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-time. Cambridge Univ. Press, Cambridge, 1973. [21] S. W. Hawking and R. Penrose. The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. London Ser. A 314, 529–548, 1970. [22] A. Krolak.´ Black holes and the weak cosmic censorship. Gen. Relativity Gravitation 16, 365–373, 1984. [23] B. O’Neill. Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York, 1982. [24] R. Penrose. Gravitational collapse and space-time singularities. Phys. Rev. Letters 14, 57– 59, 1965. [25] R. Penrose. The question of cosmic censorship. In Black holes and relativistic stars 103–122, Univ. Chicago Press, Chicago, 1998. [26] H. Reckziegel. Horizontal lifts of isometric immersions into the bundle space of a pseudo- Riemannian submersion. Lect. Notes Math. 1156, 264–279, 1985. [27] T. Sasahara. Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms. preprint. [28] L. Vrancken. Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms. Proc. Amer. Math. Soc. 130, 1459–1466, 2002. [29] The Bibliography of Biharmonic Maps, http://beltrami.sc.unica.it/biharmonic/. 66 B.Y. Chen