Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes— An Overview

: A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classiﬁcations of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector ﬁeld, having ﬁnite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.


Introduction
Trapped surfaces were introduced by Sir Roger Penrose in [1] and play an important role in cosmology. From a purely differential geometric point of view, a marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector field is lightlike at every point, i.e., for every point p of the surface, the mean curvature vector H(p) satisfies H(p), H(p) = 0 and H(p) = 0, while every non-zero vector v tangent to the surface satisfies v, v > 0.
Since 2007, several authors have studied marginally trapped surfaces in spacetimes and their generalizations to higher signatures, dimensions and codimensions from a geometric point of view. Most of the results give a complete classification of marginally trapped surfaces in a specific spacetime under one or more additional geometric conditions, such as having positive relative nullity [2], having parallel mean curvature vector field [3], having finite type Gauss map [4], being invariant under certain 1-parameter groups of isometries [5][6][7] or being isotropic [8]. Several of these results and the above mentioned generalizations are due to Bang-Yen Chen and his collaborators and we should also mention his 2009 overview paper on the topic [9]. In 2015, Henri Anciaux and his collaborators gave a local description of any marginally trapped surface (and even codimension two submanifold) in a Lorentzian space form or a Robertson-Walker spacetime in [10,11], without requiring any additional properties. While this is the most general result, and in some sense the best one can hope for in this context, the previously mentioned results are still of great value, since they often give more explicit descriptions under the additional condition at hand. Also, it is not always easy to find the surfaces with a certain property from the general description, such as those with constant Gaussian curvature.
for vector fields X and Y tangent to S and a vector field ξ normal to S. Here, h is a symmetric (1, 2)-tensor field taking values in the normal bundle, called the second fundamental form, A ξ is a symmetric (1, 1)-tensor field, called the shape operator associated to ξ, and ∇ ⊥ is the normal connection.
The mean curvature vector at a point p ∈ S is defined by where {e 1 , . . . , e n } is an orthonormal basis of T p S. We say that S is marginally trapped if its mean curvature vector H(p) is null for every point p of S and we say that S has null second fundamental form if h(X, Y) is a null vector for all tangent vectors X and Y to S. It is clear that submanifolds with null second fundamental form are marginally trapped, but the converse is not necessarily true. Submanifolds for which the mean curvature vanishes at some points are not marginally trapped, since the zero vector is not a null vector. However, since we stay close to the original sources, some of the classifications in this paper include conditions preventing the vanishing of the mean curvature vector field, while some others don't (see also the remark after Theorem 1). For an explicit example of a submanifold, it is not hard to check whether the mean curvature vanishes at some points or not. We consider two types of ambient Lorentzian manifolds in this paper. The first type are those with the highest degree of symmetry, namely Lorentzian real space forms. We recall the definition of a (pseudo-)Riemannian space form of any index. Let R n s denote R n = {(x 1 , . . . , x n ) | x 1 , . . . , x n ∈ R} equipped with the inner product (x 1 , . . . , x n ), (y 1 , . . . , y n ) = −x 1 y 1 − . . . − x s y s + x s+1 y s+1 + . . . + x n y n .
For s = 0, the space R n 0 = R n is just the Euclidean space of dimension n and for s > 1, we call R n s the pseudo-Euclidean space of dimension n and index s. It is a flat manifold, i.e., a pseudo-Riemannian manifold with constant sectional curvature 0. We now define The manifolds S n s (x 0 , c) and H n s (x 0 , c), equipped with the induced metrics from R n+1 s and R n+1 s+1 respectively, are complete pseudo-Riemannian manifolds with constant sectional curvature c. We refer to R n s , S n s (x 0 , c) and H n s (x 0 , c) as real space forms of dimension n and index s. We simply denote S n s (x 0 , c) and H n s (x 0 , c) by S n s (c) and H n s (c) when x 0 is the origin. In the rest of this paper, we will often use the unified notation Q n s (c) to denote the following: If the index s = 0, we denote the Riemannian real space form Q n 0 (c) by Q n (c) and if the index s = 1, we call Q n 1 (c) a Lorentzian real space form. In particular, the four-dimensional Lorentzian real space forms R 4 1 , S 4 1 (1) and H 4 1 (−1) are known as the Minkowksi spacetime, the de Sitter spacetime and the anti-de Sitter spacetime.
The second type of ambient Lorentzian manifolds that we consider are warped products where t is a coordinate on I and · , · c is the metric of Q n−1 (c). If the warping function f is constant, then L n 1 ( f , c) is the Lorentzian product of (I, −dt 2 ) and a Riemannian real space form. For general f but n = 4, the Lorentzian manifolds L 4 1 ( f , c) are known as Robertson-Walker spacetimes. The next remark, which can for example be found in [12] in a slightly different form, determines when a Robertson-Walker spacetime has constant sectional curvature.

Remark 1.
A Robertson-Walker spacetime L 4 1 ( f , c) has constant sectional curvature K if and only if the warping function f satisfies We thus have the following: We end this section by defining the following family of functions on a space of type L n 1 ( f , c) which will appear in some of the results: for t 0 ∈ I.

Local Description of Codimension Two Marginally Trapped Submanifolds
In this section, we give a local description of any codimension two marginally trapped submanifold of a Lorentzian space form or a space of type L n 1 ( f , c). In particular, these descriptions hold for marginally trapped surfaces in Minkowski spacetime, de Sitter spacetime, anti-de Sitter spacetime and Robertson-Walker spacetimes. The results were proven in [10,11].
The results use the notion of the Gauss map of a hypersurface of a Riemannian space form.
Definition 1. Let S be an immersed hypersurface of a Riemannian real space from Q n+1 (c) and denote by ν a unit normal vector field along the immersion. Such a vector field always exists locally and it exists globally if S is orientable.
This map is called the Gauss map of the hypersurface.

Local Description in Lorentzian Space Forms
The flat and non-flat cases are treated separately. In both theorems, a distinction is made between the submanifolds with null second fundamental form and the other marginally trapped submanifolds. Theorem 1. [10] (1) Let Ω be an open domain of R n and τ ∈ C 2 (Ω), such that ∆τ is never zero, where ∆ is the Laplacian of R n . Then, the immersion φ : is flat and its second fundamental form is given by h(X, Y) = Hess τ (X, Y)(1, 0, . . . , 0, 1).
In particular, φ has null second fundamental form and is therefore marginally trapped.
Conversely, any n-dimensional submanifold of R n+2 1 with null second fundamental form is locally congruent to the image of such an immersion. (2) Let ϕ be an immersion of class C 4 of an n-dimensional manifold S into R n+1 and denote by ν the Gauss map of ϕ. Assume that ϕ admits p ≥ 2 distinct, non-vanishing principal curvatures κ 1 , . . . , κ p with multiplicities m 1 , . . . , m p respectively and denote by τ 1 , . . . , τ p−1 the p − 1 roots of the polynomial .
Conversely, any n-dimensional marginally trapped submanifold of R n+2 1 whose second fundamental form is not null is locally congruent to the image of such an immersion.

Remark 2.
Recall that the zero vector is by definition spacelike. The condition that ∆τ is nowhere vanishing in case (1) of the above theorem ensures that the mean curvature of the corresponding immersion φ is nowhere zero. However, the submanifolds described in case (2) do include examples with zero mean curvature. For example, if ϕ is minimal, then and the immersion defined by φ(x) = (0, ϕ(x)) is minimal. Similar remarks apply to all the following theorems in this section.
In particular, φ has null second fundamental form and is therefore marginally trapped.
Conversely, any n-dimensional spacelike submanifold with null second fundamental form is locally congruent to the image of such an immersion. (2) Let ϕ be an immersion of class C 4 of an n-dimensional manifold S into S n+1 (1) (respectively, H n+1 (−1)) and denote by ν the S n+1 (1)-valued (respectively S n+1
Conversely, any n-dimensional marginally trapped submanifold of S n+2

1
(1) (respectively, of H n+2 1 (−1)), whose second fundamental form is not null is locally congruent to the image of such an immersion.

Local Description in Robertson-Walker Spacetimes
Local representation formulas for codimension two marginally trapped submanifolds of the Lorentzian product of the real line and a space form, corresponding to the warping function f of L n 1 ( f , c) being constant, were proven in [10]. These results were further generalized to marginally trapped submanifolds of spaces of type L n 1 ( f , c) for arbitrary positive smooth f in [11]. We state the results below. Theorem 3. [10] (1) There are no n-dimensional submanifolds of R 1 × S n+1 (1) with null second fundamental form.
(2) Let ϕ be an immersion of class C 4 of an n-dimensional manifold S into S n+1 (1). Denote by ν the Gauss map of ϕ and by κ 1 , . . . , κ p its p distinct principal curvatures with multiplicities m 1 , . . . , m p respectively. Then, the polynomial Conversely, any n-dimensional marginally trapped submanifold of R 1 × S n+1 (1) is locally congruent to the image of such an immersion.
It is useful to introduce following notation: The local description of codimension two marginally trapped submanifolds of L n 1 ( f , c) is as follows.
Theorem 5. [11] (1) Let φ : S → L n+2 1 ( f , c) be an immersion of an n-dimensional submanifold with null second fundamental form. Then there are two possibilities: where t 0 ∈ I is a constant and ϕ defines a totally umbilical hypersurface of Q n+1 (c) with a mean curvature vector of constant length | f (t 0 )|; where θ is defined in (1), is constant, say C 0 , and φ is locally congruent to an immersion of the form where τ : S → R is a real function of class C 2 and ϕ defines a totally umbilical hypersurface of Q n+1 (c) with Gauss map ν and with a mean curvature vector of constant length |C 0 |. Consider the immersion φ : where τ is a real function in C 2 (S) and θ is defined as in equation (1). The immersion φ is marginally trapped if and only if τ : Conversely, any marginally trapped codimension two submanifold of L n+2 1 ( f , c) is locally congruent to the image of such an immersion.

Marginally Trapped Surfaces with Positive Relative Nullity
Let S be a spacelike submanifold of a pseudo-Riemannian manifold and denote the second fundamental form by h. The relative null space at a point p ∈ S is defined by The dimension of N p (S) is called the relative nullity of S at p and the submanifold S is said to have positive relative nullity if dim N p (S) > 0 for all p ∈ S.

Classification in Lorentzian Space Forms
The following result classifies all marginally trapped surfaces with positive relative nullity in the Minkowski spacetime R 4 1 .

Theorem 6.
[2] Up to isometries, there are two families of marginally trapped surfaces with positive relative nullity in the Minkowski spacetime R 4 1 : where f is an arbitrary differentiable function such that f vanishes nowhere; (2) a surface parametrized by where q and r are defined on an open interval I containing 0, such that q + q vanishes nowhere on I.
Conversely, every marginally trapped surface with positive relative nullity in the Minkowski spacetime R 4 1 is congruent to an open part of a surface in one of the two families.
The next corollary follows immediately from the two explicit parametrizations in Theorem 6 since the first and fourth components are equal to each other in both paramatrizations. However, one can also prove it directly by using the Erbacher-Magid reduction theorem from [13], see for example [14].

Corollary 1.
Every marginally trapped surface with positive relative nullity in R 4 1 is contained in a null hyperplane of R 4 1 .
The following two theorems give classifications of marginally trapped surfaces with positive relative nullity in the de Sitter and anti-de Sitter spacetimes. Theorem 7. [2] Up to isometries, there are two families of marginally trapped surfaces with positive relative nullity in the de Sitter spacetime S 4 1 (1): (1) a surface parametrized by where f is an arbitrary differentiable function such that f + f vanishes nowhere; (2) a surface parametrized by where b is a real number, p and r are defined on an open interval I containing 0 such that r is non-constant, Conversely, every marginally trapped surface with positive relative nullity in the de Sitter spacetime S 4 1 (1) is congruent to an open part of a surface in one of the two families. (1) a surface parametrized by where f is a differentiable function such that f − f vanishes nowhere; (2) a surface parametrized by a surface parametrized by φ(x, y) = x 2 e y , 3e y 2 − 2 sinh y, e y − 2 sinh y, xe y , x 2 e y − e y 2 ; (4) A surface parametrized by where f is a differentiable function such that f vanishes nowhere; where b is a real number, p and r are defined on an open interval I containing 0 such that r is non-constant, Conversely, every marginally trapped surface with positive relative nullity in the anti-de Sitter spacetime H 4 1 (−1) is congruent to an open part of a surface in one of the five families.
Also for de Sitter and anti-de Sitter spacetimes, we have corollaries similar to Corollary 1.

Classification in Robertson-Walker Spacetimes
It turns out that marginally trapped surfaces with positive relative nullity in Robertson-Walker spacetimes (of non-constant sectional curvature) do not exist. Note that an open subset of L 4 1 ( f , c) of constant sectional is isometric to an open part of a Lorentzian space form, so marginally trapped surfaces with positive relative nullity in such a subset are classified in Theorem 6, Theorem 7 and Theorem 8.

Marginally Trapped Surfaces with Parallel Mean Curvature Vector Field
A submanifold S of a pseudo-Riemannian manifold is said to have parallel mean curvature vector field if ∇ ⊥ X H = 0 for every vector field X tangent to S and to have parallel second fundamental form or to be parallel for short, if ∇ X h = 0 for every vector field X tangent to S. Here, ∇ is the connection of Van der Waerden-Bortolotti, defined by for all vector fields X, Y and Z tangent to S. It is easy to see that a parallel submanifold has parallel mean curvature vector field, but the converse is not necessarily true.
In this section we give the classification of marginally trapped surfaces with parallel mean curvature vector field in four-dimensional Lorentzian space forms, which was proven in [3]. These surfaces are then related to marginally trapped surfaces with a 1-type Gauss map, as shown in [4]. The classification of marginally trapped surfaces with parallel mean curvature vector field uses the notion of a light cone.
We can see LC as a submanifold of de Sitter spacetime, respectively anti-de Sitter spacetime, by using the following natural embeddings: The following propositions show that marginally trapped surfaces with parallel mean curvature vector field arise naturally in the light cones of four-dimensional Lorentzian space forms.

Proposition 1. [3] Let S be a marginally trapped surface in Q
, then S has constant Gaussian curvature c and has parallel mean curvature vector field in Q 4 1 (c).

Classification in Lorentzian Space Forms
The following theorem classifies all marginally trapped surfaces with parallel mean curvature in R 4 1 , S 4 1 (1) and H 4 1 (−1).

Theorem 10.
[3] Let S be a marginally trapped surface with parallel mean curvature vector field in the Minkowski spacetime R 4 1 . Then S is one congruent to of the following six types of surfaces: (1) a flat parallel surface given by for some b ∈ R; (2) a flat parallel surface given by φ(x, y) = a cosh x, sinh x, cos y, sin y , with a > 0; (3) a flat surface given by where f is a smooth function on S such that ∆ f = a for some nonzero real number a; (4) a non-parallel flat surface lying in the light cone LC; (5) a non-parallel surface lying in the de Sitter spacetime S 3 1 (c) for some c > 0 such that the mean curvature vector field H of S in S 3 1 (c) satisfies H , H = −c; (6) a non-parallel surface lying in the hyperbolic space H 3 (c) for some c < 0 such that the mean curvature vector field H of S in H 3 (c) satisfies H , H = −c.
Conversely, every surface of types (1)-(6) above gives rise to a marginally trapped surface with parallel mean curvature vector in R 4 1 .

Theorem 11.
[3] Let S be a marginally trapped surface with parallel mean curvature vector field in the de Sitter spacetime S 4 1 (1). Then S is congruent to one of the following eight types of surfaces: (1) a parallel surface of Gaussian curvature 1 given by φ(x, y) = 1, sin x, cos x cos y, cos x sin y, 1 ; (2) a flat parallel surface defined by φ(x, y) = 1 2 2x 2 − 1, 2x 2 − 2, 2x, sin 2y, cos 2y ; (3) a flat parallel surface defined by (4) a flat parallel surface defined by Conversely, every surface of types (1)-(8) above gives rise to a marginally trapped surface with parallel mean curvature vector in S 4 1 (1).

Theorem 12.
[3] Let S be a marginally trapped surface with parallel mean curvature vector field in the anti-de Sitter spacetime H 4 1 (−1). Then, S is congruent to one of the following eight types of surfaces: (1) a parallel surface of Gaussian curvature −1 given by φ(x, y) = 1, cosh x cosh y, sinh x, cosh x sinh y, 1 ; (2) a flat parallel surface defined by φ(x, y) = 1 2 2x 2 + 2, cosh 2y, 2x, sinh 2y, 2x 2 + 1 ; (3) a flat parallel surface defined by with |b| < 2; (4) a flat parallel surface defined by (5) a surface of constant curvature −1 given by where f is a smooth function satisfying ∆ f = a for some nonzero real number a; Conversely, every surface of types (1)-(8) above gives rise to a marginally trapped surface with parallel mean curvature vector in H 4 1 (−1).

Finite Type Gauss Map
In [4], marginally trapped surfaces with parallel mean curvature vector field were related to marginally trapped surfaces with a 1-type Gauss map. This notion of Gauss map is a little different from the one we used in Section 3, so we start by recalling the definition.
, be a spacelike oriented surface in a four-dimensional Lorentzian space form. For any p ∈ S, we denote ν(p) = e 1 ∧ e 2 , where (e 1 , e 2 ) is a positively oriented orthonormal basis of T p S. Then ν is a map from S to 2 R 4 1 ∼ = R 6 3 , respectively 2 R 5 1 ∼ = R 10 4 or 2 R 5 2 ∼ = R 10 6 , which we call the Gauss map of the surface.
It now makes sense to look at ∆ν, where ∆ is the Laplacian of S acting on every component of ν and we have the following definition, which extends the notion of having harmonic Gauss map.

Marginally Trapped Surfaces in R 4 1 Which Are Invariant under a 1-Parameter Group of Isometries
Marginally trapped surfaces in the Minkowski spacetime R 4 1 , satisfying the additional condition of being invariant under the action of a 1-parameter subgroup G of the isometry group of R 4 1 , are studied in [5][6][7]. The main results are the classifications of boost, rotation and screw invariant marginally trapped surfaces in R 4 1 respectively, which we discuss in this section.

Boost Invariant Marginally Trapped Surfaces
One says that a spacelike surface S in R 4 1 is invariant under boosts if it is invariant under the following group of linear isometries of R 4 1 : This means that B θ S = S for all θ ∈ R. A boost invariant surface S has an open and dense subset Σ α which can be parametrized by

Rotation Invariant Marginally Trapped Surfaces
One says that a spacelike surface S in R 4 1 is invariant under (spacelike) rotations if it is invariant under the following group of linear isometries of R 4 1 : which means that R θ S = S for all θ ∈ R. A rotation invariant surface S has an open and dense subset Σ α which can be parametrized by is a spacelike curve parametrized by arc length. Rotation invariant marginally trapped surfaces in R 4 1 are completely classified in the following theorem.

Screw Invariant Marginally Trapped Surfaces
One says that a spacelike surface S in R 4 1 is screw invariant if it is invariant under the following group of linear isometries of R 4 1 : where the matrices are written with respect to the ordered basis (k, l, e 3 , e 4 ), with k = (1, 1, 0, 0)/ √ 2, l = (1, −1, 0, 0)/ √ 2, e 3 = (0, 0, 1, 0) and e 4 = (0, 0, 0, 1). Note that k and l are null vectors with k, l = −1. A screw invariant surface S is contained in R + ∪ R − , defined as We will suppose that S is contained in R + , as similiar results can be obtained for surfaces lying in R − . A screw invariant surface S in R + has an open and dense subset Σ α which can be parametrized by is a spacelike curve parametrized by arc length. Note that α k , α l , 0 and α 4 are the coordinate functions of α with respect to the basis (k, l, e 3 , e 4 ). The next theorem classifies all screw invariant marginally trapped surfaces in R + . Similar results can also be obtained for surfaces lying in R − . Theorem 16. [7] Let S be a screw invariant marginally trapped surface in R + . Then, S is locally congruent to a surface Σ α whose profile curve α(s) = α k (s)k + α l (s)l + α 4 (s)e 4 is described in one of the following two cases.

Isotropic Marginally Trapped Surfaces
A complete classification of complete isotropic marginally trapped surfaces in Lorentzian space forms was obtained in [8].

Definition 5.
An isometric immersion of a Riemannian manifold S into a (pseudo-)Riemannian manifold is called isotropic if h(u, u), h(u, u) = λ(p) does not depend on the choice of the unit vector u ∈ T p S. The function λ : S → R is then called the isotropy function of the immersion. Definition 6. An isometric immersion of a Riemannian manifold S into a (pseudo-)Riemannian manifold is called pseudo-umbilical if there exists a function ρ : S → R such that h(X, Y), H = ρ X, Y for all vector fields X and Y tangent to S.
Remark that the function ρ in Definition 6 has to equal ρ = H, H . When the mean curvature vector of a spacelike surface in a four-dimensional Lorenztian manifold is null, the notions of isotropy and pseudo-umbilicity are equivalent. More precisely, the following proposition was proven in [8].
1 be a marginally trapped surface in a four-dimensional Lorentzian manifold. Then the following assertions are equivalent: φ has null second fundamental form.

Classification in Lorentzian Space Forms
The following theorem, which unifies Theorem 5.6, Theorem 5.10 and Theorem 5.13 from [8], classifies the isotropic marginally trapped surfaces in R 4 1 , S 4 1 (1) and H 4 1 (−1). Proposition 3 then allows the condition of being isotropic to be replaced by the stronger sounding condition of being isotropic with isotropy function 0 or by the condition of being pseudo-umbilical.
Theorem 17. [8] Let S be a complete connected spacelike surface in Q 4 1 (c), with c ∈ {−1, 0, 1}. Then S is an isotropic marginally trapped surface if and only if S has constant Gaussian curvature c and the immersion is congruent to where τ : Q 2 (c) → R is a smooth function such that ∆τ is nowhere zero, where ∆ is the Laplacian of Q 2 (c).
Remark that, since isotropic marginally trapped surfaces in four-dimensional Lorentzian space forms have null second fundamental form, Theorem 17 follows from the first cases of Theorem 1 and Theorem 2, which were historically proven after Theorem 17.

Classification in Robertson-Walker spacetimes
Isotropic marginally trapped submanifolds in Robertson-Walker spacetimes can now be classified as a corollary of Proposition 3 and the first cases of Theorem 5. (1) the immersion takes the form where t 0 ∈ I is constant and ϕ defines a totally umbilical surface in Q 3 (c) with a mean curvature vector of constant length | f (t 0 )|; (2) the function θ cos c θ + (θ ) 2 c sin c θ θ sin c θ − (θ ) 2 cos c θ , where θ is defined in (1), is constant, say C 0 , and φ is locally congruent to an immersion of the form where τ : S → R is a real function of class C 2 and ϕ defines a totally umbilical surface in Q 3 (c) with Gauss map ν and with a mean curvature vector of length |C 0 |.

Marginally Trapped Surfaces with Constant Gaussian Curvature
In this section, we look for the marginally trapped surfaces with constant Gaussian curvature in the classifications from the previous sections. Note that, in theory, it suffices to find the constant Gaussian curvature surfaces in the theorems of Section 3, but this seems to be a non-trivial task.

Surfaces with Positive Relative Nullity
The following result follows from the equation of Gauss and the definition of positive relative nullity.

Surfaces in Light Cones
Proposition 1 showed that marginally trapped surfaces lying in the light cone of Q 4 1 (c), with c ∈ {−1, 0, 1}, have constant Gaussian curvature c, while Proposition 2 proved the existence of such surfaces.

Surfaces with Parallel Mean Curvature Vector Field
Marginally trapped surfaces with parallel mean curvature vector field in four-dimensional Lorentzian space forms are classified in Theorem 10, Theorem 11 and Theorem 12. The surfaces with constant Gaussian curvature are already mentioned in the formulation of these theorems and we can summarize the situation as follows. (1) Surfaces of types (1)-(4) in Theorem 10, of types (2)-(4) in Theorem 11 and of types (2)-(4) in Theorem 12 are flat. (2) Surfaces of type (1) in Theorem 11 have constant Gaussian curvature K = 1.

Boost Invariant Surfaces
A boost invariant surface S in Minkowski spacetime is locally congruent to a surface Σ α with unit speed profile curve α = (α 1 , 0, α 3 , α 4 ), where α 1 is positive, and has Gaussian curvature Theorem 14 describes all boost invariant marginally trapped surfaces in R 4 1 and the next proposition determines all such surfaces with constant Gaussian curvature. Proposition 5. [5,14] A boost invariant marginally trapped surface S in Minkowski spacetime has constant Gaussian curvature if and only it is locally congruent to a surface Σ α whose unit speed profile curve α = (α 1 , 0, α 3 , α 4 ), with α 1 > 0, is given by one of the following cases.

Rotation Invariant Surfaces
A rotation invariant surface S in Minkowski spacetime is locally congruent to a surface Σ α with unit profile curve α = (α 1 , α 2 , α 3 , 0), where α 3 is positive, and has Gaussian curvature Theorem 15 classifies all rotation invariant marginally trapped surfaces in R 4 1 and the next proposition determines all such surfaces with constant Gaussian curvature. Proposition 6. [6] A rotation invariant marginally trapped surface S in Minkowski spacetime has constant Gaussian curvature if and only it is locally congruent to a surface Σ α whose unit speed profile curve α = (α 1 , α 2 , α 3 , 0), with α 3 > 0, is given by one of the following cases.
(1) If S is flat, then α is given by one of the following curves:

Screw Invariant Surfaces
A screw invariant surface S in Minkowski spacetime is locally congruent to a surface Σ α with unit speed profile curve α = (α k , α l , 0, α 4 ), where α k is positive, and has a Gaussian curvature described by Theorem 16 classifies all screw invariant marginally trapped surfaces in R 4 1 and the next proposition determines all such surfaces with constant Gaussian curvature. Proposition 7. [7] A screw invariant marginally trapped surface S in Minkowski spacetime has constant Gaussian curvature if and only it is locally congruent to a surface Σ α whose unit speed profile curve α = (α k , α l , 0, α 4 ), with α k positive, is given by one of the following cases. (1) If S is flat, then α is a curve of type (1) in Theorem 16. (2) If S has constant Gaussian curvature K > 0, then α is a curve of type (2) in Theorem 16 with ρ(s) = 2c 2 1 K cos 2 √ Ks + c 2 , with c 1 , c 2 ∈ R.
If S has constant Gaussian curvature K < 0, then α is a curve of type (2) in Theorem 16 with with c 1 , c 2 ∈ R.

Isotropic surfaces
Theorem 17 implies that an isotropic marginally trapped surface in a Lorentzian space form Q 4 1 (c) has constant Gaussian curvature c. All isotropic marginally trapped surfaces in these spacetimes therefore provide examples of constant Gaussian curvature marginally trapped surfaces.

Conclusions and Open Questions
While trapped and marginally trapped surfaces are concepts from physics, they have very natural geometric definitions. In particular, a marginally trapped surface is a Riemannian surface in a spacetime whose mean curvature vector field, one of the most important invariants in submanifold geometry, is lightlike at every point. It is hence no surprise that this family of surfaces and their generalizations were studied intensively from a purely geometric point of view. In this paper we gave an overview of this study when the ambient space is Minkowski space, de Sitter space, anti-de Sitter space or a Robertson-Walker spacetime. Most results are classification theorems under additional geometric conditions. The local descriptions in Section 3 provide in some sense a complete classification without additional assumptions, but it is not always easy to find surfaces with particular properties from this general description, such as constant Gaussian curvature surfaces, see Section 8.
We finish the paper with some open questions regarding marginally trapped surfaces in spacetimes, which arise naturally form the current overview article. (1) What are the marginally trapped surfaces with parallel mean curvature vector in a Robertson-Walker spacetime? (2) For any one-parameter group of isometries of de Sitter, anti-de Sitter or a Robertson-Walker spacetime: what are the invariant marginally trapped surfaces?
(3) What are the marginally trapped surfaces with constant Gaussian curvature in Minkowski, de Sitter, anti-de Sitter or a Robertson-Walker spacetime? In particular, what are the flat marginally trapped surfaces? (4) What are the marginally trapped surfaces satisfying any of the additional conditions appearing in this paper in other (four-dimensional) Lorentzian manifolds, such as Kerr spacetime and Schwarzschild spacetime?
Remark that one could in principle start from the general descriptions given in Section 3 to tackle questions (1)-(3).