arXiv research

The paper studies marginally trapped submanifolds in Lorentzian manifolds under null energy condition.

We give explicit representation formulas for marginally trapped submanifolds of co-dimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature. This paper is dedicated to the memory of Franki Dillen, 1963-2013.

We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.

Generic singularities found in spacetimes with weakly trapped submanifolds.

We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null accel…

The purpose of the present work is to study (marginally) trapped submanifolds lying in a null hypersurface. Let $(M,g,N)\to\Bm(c)$ be a null hypersurface of a space-time with constant sectional curvature $c$, endowed with a Screen Integrable and Conformal rigging $N$. The (Marginally) Trapped Submanifolds we are intere…

The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we i…

In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds i…

Study shows no closed trapped submanifolds can be tangent to certain spacelike hypersurfaces.

Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.

The paper introduces canonical parameters for marginally trapped surfaces in Minkowski space.

Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…

Overview of marginally trapped surfaces in various spacetimes.

Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.

In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situatio…

We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data c…

We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfa…

New integrable systems for marginally trapped surfaces in 4D Lorentz-Minkowski space.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their …

The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using t…

New rigidity results for submanifolds in GRW spacetimes.

Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …

Study stability and rigidity of axisymmetric marginally outer trapped surfaces.

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.

In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable margi…

Study shows genericity of singularities in spacetimes with weakly trapped submanifolds.

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem…

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our…

We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initi…

A spacelike surface $S\subset \mathbb{S}^4_1$ is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface $S \subset \mathbb{S}^4_1$ we show that a choice of orientation of the normal bundle $ν(S)$ determines a smooth map $G: S \to \mathbb{S}^3$ which we call the null Gauss map of…

In a recent paper (gr-qc/0509107) the author and Rick Schoen obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. It was proved that, apart from certain exceptional circumstances, cross sections of the event horizon, in the stationary case, and 'weakly …

The paper defines marginal tubes and proves their null nature.

Study proves existence of MOTTs in de Sitter spacetime.

We construct an example of a spherically symmetric black hole interior in which there is NO (spherically symmetric) marginally trapped tube asymptotic to the event horizon. The construction uses a self-gravitating massive scalar field matter model, and the key condition we impose is that the scalar field $φ$ be bounded…

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy momentum term. In this note we cons…

We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay p…

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider th…

Small deformations of marginally outer trapped surfaces (MOTS) are studied by using the stability operator introduced by Andersson-Mars-Simon. Novel formulae for the principal eigenvalue are presented. A characterization of the many marginally outer trapped tubes (MOTT) passing through a given MOTS is given, and the po…

Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static situations. In this paper we study an important aspect of this equivalence, namely wheth…

Rigidity results for initial data sets related to the positive mass theorem.

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smoo…

Study shows how charged MOTS restrict spacetime configurations.

Study of mean curvature flow on null hypersurfaces leading to MOTS.

Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.