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.

Ricci Flow and the Sphere Theorem

Abstract: In 1982, R Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture Furthermore, various convergence theorems have been established This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow The proofs rely mostly on maximum principle arguments Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold whose sectional curvatures all lie in the interval (1,4] is diffeomorphic to a spherical space form This question has a long history, dating back to a seminal paper by H E Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen This text originated from graduate courses given at ETH Zurich and Stanford University, and is directed at graduate students and researchers The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required

Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes, and spacetime horns

Abstract: An exhaustive classification of a certain class of static solutions for the five-dimensional Einstein-Gauss-Bonnet theory in vacuum is presented. The class of metrics under consideration is such that the spacelike section is a warped product of the real line with a nontrivial base manifold. It is shown that for generic values of the coupling constants the base manifold must be necessarily of constant curvature, and the solution reduces to the topological extension of the Boulware-Deser metric. It is also shown that the base manifold admits a wider class of geometries for the special case when the Gauss-Bonnet coupling is properly tuned in terms of the cosmological and Newton constants. This freedom in the metric at the boundary, which determines the base manifold, allows the existence of three main branches of geometries in the bulk. For the negative cosmological constant, if the boundary metric is such that the base manifold is arbitrary, but fixed, the solution describes black holes whose horizon geometry inherits the metric of the base manifold. If the base manifold possesses a negative constant Ricci scalar, two different kinds of wormholes in vacuum are obtained. For base manifolds with vanishing Ricci scalar, a different class of solutions appears resembling ``spacetime horns.'' There is also a special case for which, if the base manifold is of constant curvature, due to a certain class of degeneration of the field equations, the metric admits an arbitrary redshift function. For wormholes and spacetime horns, there are regions for which the gravitational and centrifugal forces point towards the same direction. All of these solutions have finite Euclidean action, which reduces to the free energy in the case of black holes, and vanishes in the other cases. The mass is also obtained from a surface integral.

Constant mean curvature tori in terms of elliptic functions.

Abstract: Based on a numerical approximation of such a solution, we could produce plots of one ff-torus. In these Computer generated pictures the curvature lines for the smaller principal curvature λ^ looked almost planar. We then decided to restrict ourselves to fftori with one family of planar curvature lines. This condition translates into a second partial differential equation which induces a Separation of variables in the sinh-Gordon equation. Therefore the overdetermined System can be solved explicitly in terms of elliptic functions. We obtain a classification of all ff-tori in E which have one family of planar curvature lines.

The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature

Abstract: In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.