Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurface Σ n immersed with constant mean curvature H in the anti-de Sitter space H n 1 1 , when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space H n . Our estimate depends only on the value of H and on the geometry of ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of H n 1 1 and nonexistence results in connection with such types of hypersurfaces

Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurface Σn immersed with constant mean curvature H in the anti-de Sitter space ℍ1n+1, when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space ℍn. Our estimate depends only on the value of H and on the geometry of ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of ℍ1n+1 and nonexistence results in connection with such types of hypersurfaces

Generalized Alexandrov theorems in spacetimes with integral conditions

We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $\Sigma$ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in \cite{wang2017Minkowski} by relaxing the curvature conditions on $\Sigma$ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for $\Sigma$ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.Comment: 19 page

Spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean spaces

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Generalized Minkowski inequality via degenerate Hessian equations on exterior domains

In this paper, we prove a generalized Minkowski inequality holds for any smooth, $(k-1)$-convex, starshaped domain $\Omega.$ Our proof relies on the solvability of the degenerate $k$-Hessian equation on the exterior domain $\mathbb R^n\setminus\Omega.

Deformation and Contraction of Symmetries in Special Relativity

This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The first approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifications of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specific Hopf algebraic deformation of the PoincareI algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries

Explorations in black hole chemistry and higher curvature gravity

This thesis has two goals. The primary goal is to communicate two results within the framework of black hole chemistry, while the secondary goal is concerned with higher curvature theories of gravity. Super-entropic black holes will be introduced and discussed. These are new rotating black hole solutions that are asymptotically (locally) anti de Sitter with horizons that are topologically spheres with punctures at the north and south poles. The basic properties of the solutions are discussed, including an analysis of the geometry, geodesics, and black hole thermodynamics. It is found that these are the first black hole solutions to violate the reverse isoperimetric inequality, which was conjectured to bound the entropy of anti de Sitter black holes in terms of the thermodynamic volume. Implications of this result for the inequality are discussed. The second main result is a new phase transition in black hole thermodynamics: the $\lambda$-line. This is a line of second order (continuous) phase transitions with no associated first order phase transition. The result is illustrated for black holes in higher curvature gravity --- cubic Lovelock theory coupled to real scalar fields. The properties of the black holes exhibiting the transition are discussed and it is shown that there are no obvious pathologies associated with the solutions. The features of the theory that allow for the transition are analyzed and then applied to obtain a further example in cubic quasi-topological gravity. The secondary goal of the thesis is to discuss higher curvature theories of gravity. This serves as a transition between the discussion of super-entropic black holes and $\lambda$-lines but also provides an opportunity to discuss recent work in the area. Higher curvature theories are introduced through a study of general theories on static and spherically symmetric spacetimes. It is found that there are three classes of theories that have a single independent field equation under this restriction: Lovelock gravity, quasi-topological gravity, and generalized quasi-topological gravity, the latter being previously unknown. These theories admit natural generalizations of the Schwarzschild solution, a feature that turns out to be equivalent to a number of other remarkable properties of the field equations and their black hole solutions. The properties of these theories are discussed and their applicability as toy models in gravity and holography is suggested, with emphasis on the previously unknown generalized quasi-topological theories

Asymptotic behaviour of spacetimes with positive cosmological constant

Tesis por compendio de publicaciones[EN] In this thesis we study the asymptotic Cauchy problem of general relativity with positive cosmological constant in arbitrary (n + 1)-dimensions. Our aim is to provide geometric characterizations of Kerr-de Sitter and related spacetimes by means of their initial data at conformally flat (n-dimensional) I . In our setting, the conformal Killing vector fields (CKVFs) of I become very relevant because of their relation with the symmetries of the spacetime. In the first part of the thesis, we study the CKVFs ξ of conformally flat n-metrics γ, as well as their equivalence classes [ξ] up to conformal transformations of γ. We do that by analyzing in detail SkewEnd(M1,n+1), the skew-symmetric endomorphisms of the Minkowski space M1,n+1. The cases n = 2, 3 are worked out in special detail. A canonical form that fits every element in SkewEnd(M1,n+1) is obtained along with several applications. Of relevance for the study of asymptotic data is that it gives a canonical form for CKVFs which allows us to determine the conformal classes [ξ] and study the quotient topology associated to these clases. In addition, the canonical form for CKVFs is applied to the n = 3 case to obtain a set of coordinates adapted to an arbitrary CKVF. With these coordinates we provide the set of asymptotic data which generate all conformally extendable spacetimes solving the (Λ > 0)-vacuum field equations and admitting two commuting symmetries, one of which axial. From this, a characterization of Kerr-de Sitter and related spacetimes follows. Our study provides in principle a good arena to test definitions of mass and angular momentum for positive cosmological constant. In the second part of this thesis we focus in the asymptotic Cauchy problem in arbitrary dimensions. For this we use the Fefferman-Graham formalism. We carry out an study of the asymptotic initial data in this picture and extend an existing geometric characteri- zation of them, in the conformally flat I case, to arbitrary signature and cosmological constant. We discuss the validity of this geometric characterization of data beyond the conformally flat I case. We provide a KID equation for asymptotic analytic data (which comprise Kerr-de Sitter). This equation being satisfyied by the data amounts to the existence of a Killing vector field in the corresponding spacetime. With the above results in hand we provide a geometric characterization of Kerr-de Sitter by means of its asymptotic initial data, which happen to be determined by the conformally flat class of metrics [γ] and one particular conformal class of CKVFs [ξ] of [γ]. These data admit a generalization, keeping [γ] conformally flat, by allowing [ξ] to be an arbitrary confor- mal class. This extends the so-called Kerr-de Sitter-like class with conformally flat I , defined in previous works in four spacetime dimensions, to arbitrary dimensions. We study this class and prove that the corresponding spacetimes are contained in the set of (Λ > 0)-vacuum Kerr-Schild spacetimes, which share (conformally flat) I with their background metric (de Sitter). We name these Kerr-Schild-de Sitter spacetimes. The proof largely relies on our study of the space of classes of CKVFs and in particular on the properties of its quotient topology. In addition, we prove the converse inclusion, providing a full characterization of the Kerr-de Sitter-like class as the Kerr-Schild-de Sitter spacetimes

Holography, Locality and Symmetries of the Universe

It is an interesting question that, with a well tested duality between the quantum gravity in anti de Sitter space and a quantum field theory in one lower dimension, whether quantum gravity in a cosmological background has a well defined dual description. In large 1/N limit, this duality could be a correspondence between an approximately local gravity theory describing cosmology and a quantum field theory. In dS/CFT, the quantum field theory is a Euclidean CFT living at the conformal boundary of de Sitter space, in large N limit, we should expect the local observables in de Sitter cosmology be recovered from the CFT. We explicitly develop this construction for scalar fields and derive the operator map at lowest order of 1/N expansion. Having addressed the fundamental question of how local fields in de Sitter cosmology arise via holography, we focus on the theory of cosmological perturbations that is described in terms of local field theory. The curvature perturbations during inflation, which originated from quantum fluctuations of inflaton and which induced the CMB inhomogeneity we see today, as well as the large scale structure, can be described as Goldstone boson fields which nonlinearly realize a subset of general coordinate transformations as residual symmetries. This fact puts strong constraints on the behavior of the cosmological correlation functions, and a series of consistency relations constraining the soft limits of these correlation functions can be derived as Ward identities