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 article, the convergence of the Yamabe flow was shown to hold if the dimension of the initial metric is locally conformally flat and the curvature of the scalar curvature is known.
Abstract: We consider the Yamabe flow $\frac{\partial g}{\partial t} = -(R_g - r_g) \, g$ where $g$is a Riemannian metric on a compact manifold $M, R_g$ denotes its scalar curvature, and $r_g$ denotes the mean value of the scalar curvature. We prove convergence of the Yamabe flow if the dimension $n$ satisfies $3 \leq n \leq 5$ or the initial metric is locally conformally flat.
199 citations
TL;DR: In this paper, the existence of K-Einstein metrics on K-stable Fano manifolds is discussed and the relation between the K-stability and the energy of the Fano manifold is discussed.
Abstract: This is an expository paper on Kahler metrics of positive scalar curvature. It is for my Takagi Lectures at RIMS in November of 2013. In this paper, I first discuss the Futaki invariants, the K-stability and its relation to the K-energy. Next I will outline my work in 2012 on the existence of Kahler–Einstein metrics on K-stable Fano manifolds. Finally, I will present S. Paul’s work on stability of pairs with some modifications of mine.
198 citations
198 citations
TL;DR: In this article, the Yamabe number of a smooth closed manifold M is defined as the supremum of all conformal classes C of Riemannian metrics on M, defined as a function of the curvature of the manifold.
Abstract: The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] and references cited in it). In this paper we shall consider the same problem under a constraint on the volume. For this purpose it is useful to introduce an invariant p(M) of a smooth closed manifold M, which will be called the Yamabe number of M, defined as the supremum of #(M, C) of all conformal classes C of Riemannian metrics on M,
196 citations
01 Jan 1996
TL;DR: In this article, a macroscopic view of Riemannian manifolds with positive scalar curvature and the proof of the homotopy invariance of some Novikov higher signatures of non-simply connected manifolds is presented.
Abstract: Our journey starts with a macroscopic view of Riemannian manifolds with positive scalar curvature and terminates with a glimpse of the proof of the homotopy invariance of some Novikov higher signatures of non-simply connected manifolds. Our approach focuses on the spectra of geometric differential operators on compact and non-compact manifolds V where the link with the macroscopic geometry and topology is established with suitable index theorems for our operators twisted with almost flat bundles over V. Our perspective mainly comes from the asymptotic geometry of infinite groups and foliations.
196 citations