Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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TL;DR: In this paper, the authors give some topological restrictions on spacetimes of de Sitter type, which admit a regular past and/or future conformal boundary, and show that if one of these boundaries is compact, it must have finite fundamental group and its conformal class must contain a metric of positive scalar curvature.
Abstract: Motivated by recent proposals for a de Sitter version of the AdS/CFT correspondence, we give some topological restrictions on spacetimes of de Sitter type, i.e., spacetimes with $\Lambda>0$, which admit a regular past and/or future conformal boundary. For example we show that if $M^{n+1}$, $n \ge 2$, is a globally hyperbolic spacetime obeying suitable energy conditions, which is of de Sitter type, with a conformal boundary to both the past and future, then if one of these boundaries is compact, it must have finite fundamental group and its conformal class must contain a metric of positive scalar curvature. Our results are closely related to theorems of Witten and Yau hep-th/9910245 pertaining to the Euclidean formulation of the AdS/CFT correspondence.
78 citations
TL;DR: In this article, the Berger-Nirenberg problem of prescribing the curvature on a Riemann surface (that is on an oriented surface equipped with a conformal class of metrics) is studied.
Abstract: In this article, we study the problem (sometimes called the Berger-Nirenberg problem) of prescribing the curvature on a Riemann surface (that is on an oriented surface equipped with a conformal class of Riemannian metrics).
78 citations
78 citations
TL;DR: In this article, the authors studied the thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential.
Abstract: We study thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we find that the divergence of scalar curvature is related to the divergence of specific heat with fixed chemical potential in the Weinhold metric and Ruppeiner metric, while in the Quevedo metric the divergence of scalar curvature is related to the divergence of specific heat with fixed number of colors and the vanishing of the specific heat with fixed chemical potential.
78 citations
TL;DR: The existence of global constant mean curvature (CMC) foliations of 3-dimensional maximal hyperbolic Lorentzian manifolds with genus(Σ> 1) is proved in this paper.
77 citations