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Classification of Marginally Trapped Surfaces

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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 curva- ture 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 classifi- cation 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.
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