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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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Book•
30 Jun 2006TL;DR: Riemannian geometry and singularity analysis of Ricci flow have been studied in this paper, where Ricci solitons and special solutions have been used for geometric flows.
Abstract: Riemannian geometry Fundamentals of the Ricci flow equation Closed 3-manifolds with positive Ricci curvature Ricci solitons and special solutions Isoperimetric estimates and no local collapsing Preparation for singularity analysis High-dimensional and noncompact Ricci flow Singularity analysis Ancient solutions Differential Harnack estimates Space-time geometry Appendix A Geometric analysis related to Ricci flow Appendix B Analytic techniques for geometric flows Appendix S Solutions to selected exercises Bibliography Index
646 citations
TL;DR: For Riemannian manifolds with a measure (M, g, edvolg) as mentioned in this paper showed that the Ricci curvature and volume comparison can be improved when the Bakry-Emery Ricci tensor is bounded from below.
Abstract: For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.
572 citations
TL;DR: In this article, a calculus for general relativity is developed in which the basic role of tensors is taken over by spinors, and the Riemann-Christoffel tensor is written in a spinor form according to a scheme of Witten.
559 citations
TL;DR: In this paper, the authors define and compute the energy of higher curvature gravity theories in arbitrary dimensions and show that these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant solutions with nontrivial energy properties.
Abstract: We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with nontrivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action is the square of the traceless Ricci tensor always has zero energy, unlike conformal (Weyl) gravity. We also study the string-inspired Einstein-Gauss-Bonnet model and show that both its flat and anti–de Sitter vacua are stable.
515 citations
Book•
01 Jan 2007
TL;DR: The Ricci flow with surgery has been studied in Riemannian geometry as discussed by the authors, where the standard solution is to perform surgery on a δ$-neck Ricci Flow with surgery.
Abstract: Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry Manifolds of non-negative curvature Basics of Ricci flow The maximum principle Convergence results for Ricci flow Perelman's length function and its applications: A comparison geometry approach to the Ricci flow Complete Ricci flows of bounded curvature Non-collapsed results $\kappa$-non-collapsed ancient solutions Bounded curvature at bounded distance Geometric limits of generalized Ricci flows The standard solution Ricci flow with surgery: Surgery on a $\delta$-neck Ricci flow with surgery: The definition Controlled Ricci flows with surgery Proof of non-collapsing Completion of the proof of Theorem 15.9 Completion of the proof of the Poincare conjecture: Finite-time extinction Completion of the proof of Proposition 18.24 3-manifolds covered by canonical neighborhoods Bibliography Index.
512 citations