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.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this article, a simple proof for the inequality that positive definite matrices constitute a Riemannian manifold of negative curvature is given, which leads to generalisations to non-Riemannians and connections with some well-known inequalities of mathematical physics.
93 citations
TL;DR: In this paper, the classification of translation surfaces with constant mean curvature or constant Gauss curvature in 3D Euclidean space E3 and 3D Minkowski space E 2 1 /ε 3 /ε 4 ) was given.
Abstract: We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E3 and 3-dimensional Minkowski space E
1
3
.
93 citations
TL;DR: In this paper, Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established.
Abstract: Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below
93 citations
TL;DR: In this paper, the authors considered the problem of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary of a compact Riemannian manifold with boundary.
Abstract: Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.
92 citations
TL;DR: In this paper, a conformal-anomaly driven inflation in the flat homogeneous and isotropic universe is studied, and it is shown that the Ricci scalar decreases during inflation and the standard evolution history of the universe is recovered at the small curvature regime.
Abstract: We explore conformal-anomaly driven inflation in $F(R)$ gravity without invoking the scalar-tensor representation. We derive the stress-energy tensor of the quantum anomaly in the flat homogeneous and isotropic universe. We investigate a suitable toy model of exponential gravity plus the quantum contribution due to the conformal anomaly, which leads to the de Sitter solution. It is shown that in $F(R)$ gravity model, the curvature perturbations with its enough amplitude consistent with the observations are generated during inflation. We also evaluate the number of $e$-folds at the inflationary stage and the spectral index $n_\mathrm{s}$ of scalar modes of the curvature perturbations by analogy with scalar tensor theories, and compare them with the observational data. As a result, it is found that the Ricci scalar decreases during inflation and the standard evolution history of the universe is recovered at the small curvature regime. Furthermore, it is demonstrated that in our model, the tensor-to-scalar ratio of the curvature perturbations can be a finite value within the $68\%\,\mathrm{CL}$ error of the very recent result found by the BICEP2 experiment.
92 citations