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"Hyperspace."
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Nonexistence of non-Hopf Ricci-semisymmetric real hypersurfaces in and
In this paper, we solved an open problem raised by Cecil and Ryan (2015, Geometry of Hypersurfaces , Springer Monographs in Mathematics, Springer, New York, p. 531) by proving the nonexistence of non-Hopf Ricci-semisymmetric real hypersurfaces in$\\mathbb {C}P^{2}$and$\\mathbb {C}H^{2}$.
Journal Article
Black holes in higher dimensions
\"Black holes are one of the most remarkable predictions of Einstein's general relativity. Now widely accepted by the scientific community, most work has focussed on black holes in our familiar four spacetime dimensions. But in recent years, ideas in brane-world cosmology, string theory, and gauge/gravity duality have all motivated a study of black holes in more than four dimensions, with surprising results. In higher dimensions, black holes exist with exotic shapes and unusual dynamics. Edited by leading expert Gary Horowitz, this exciting book is the first devoted to this new field. The major discoveries are explained by the people who made them: RobMyers describes theMyers-Perry solutions that represent rotating black holes in higher dimensions; Ruth Gregory describes the Gregory-Laflamme instability of black strings; and Juan Maldacena introduces gauge/gravity duality, the remarkable correspondence that relates a gravitational theory to nongravitational physics. There are two additional chapters on this duality describing how black holes can be used to describe relativistic fluids and aspects of condensed matter physics\"-- Provided by publisher.
Book
On nonnegatively curved hypersurfaces in H n + 1
In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
Journal Article
Transverse expansion at null hypersurfaces and applications to Killing horizons
In this report we present a novel framework to obtain the full transverse expansion of the metric at a general null hypersurface in terms of the tower of derivatives of the ambient Ricci tensor. Particularizing the identities to horizons, we recover and generalize in several directions known results both in the homothetic horizon case [1] and the Killing horizon one [9, 3, 2]. As an application, we prove general existence and uniqueness results when data is posed on a single null hypersurface. In the specific case of Killing horizons with spherical cross-sections and integrable connection one-form, we show existence and uniqueness of Λ-vacuum spacetimes from initial data at a non-degenerate Killing horizon of any topology, dimension and possibly including a bifurcation surface.
Journal Article
Generalized -Einstein Real Hypersurfaces in and
In this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in$\\mathbb{C}P^{2}$and$\\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in$\\mathbb{C}P^{2}$or$\\mathbb{C}H^{2}$with constant mean curvature is generalized${\\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in$\\mathbb{C}P^{2}$with constant scalar curvature is generalized${\\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.
Journal Article
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
Journal Article
The Cosmological Constant and Trapped Surfaces
The cosmological constant is a model of dark energy that is one explanation for the accelerated expansion of the universe. The concept of a trapped surface provides a precise characterization of gravitational collapse that has proceeded beyond the point of no return. Therefore, because of its importance, we examined spherically-symmetric spacetimes with a cosmological constant to determine its impact on the characteristic development of trapped surfaces. The principal result of this study is a relationship between the mass and the cosmological constant that is a necessary and suf icient condition for trapped surfaces to develop to the future of a branch of a marginally trapped hypersurface.
Journal Article