Non-singular solutions of the Ricci flow on three-manifolds
type: Item
instance of: scholarly article
title: Non-singular solutions of the Ricci flow on three-manifolds
zbMATH Open document ID: 0939.53024
author: Richard S. Hamilton
published in: Communications in Analysis and Geometry
publication date: 2000-07-11T00:00:00Z
review text: The behaviour of nonsingular solutions of the Ricci flow \(\partial g(X,Y)/\partial t=2({r\over 3}g(X,Y)-\text{Ric}(X,Y))\), \(0\leq t< \infty\), on a compact three-manifold \({\mathcal M}\) is classified. The flow is normalized by employing the cosmological constant \(r=\int R/\int 1\) assuming that the curvature remains bounded all the time. Let \(\rho(P)\) be the injectivity radius at a point \(P\in{\mathcal M}\) (i.e., the largest radius of open balls of the tangent space which inject under the exponential map), and \(\widehat\rho\) be the maximum of \(\rho(P)\) on \({\mathcal M}\). Then, either (i) \(\widehat\rho(t)\to 0\) as \(t\to\infty\), or (ii) the solution converges to a metric of constant sectional curvature, or (iii) there is a finite family of complete non-compact hyperbolic three-manifolds \({\mathcal H}_1,\dots,{\mathcal H}_n\) and diffeomorphisms \(\varphi_i(t)\): \({\mathcal H}_i\to {\mathcal M}\) \((t>\text{const} .)\) such that the pull-back of the solution metrics \(g(t)\) by \(\varphi_i(t)\) converges to the hyperbolic metric as \(t\to\infty\). Moreover, the volume of the exceptional part of \(M\) not lying in the images of any \(\varphi_i\) converges to zero and \(\varphi_i\) injects \(\pi_i({\mathcal H}_i)\) into \(\pi_i({\mathcal M})\) for any \(i=1,\dots,n\).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 1384292
zbMATH Keywords: hyberbolic manifold
zbMATH Keywords: Ricci flow
zbMATH Keywords: constant sectional curvature
Wikidata QID: Q56430481
reviewed by: Jan Chrastina
MaRDI profile type: Publication
DOI: 10.4310/CAG.1999.V7.N4.A2
description: scientific article; zbMATH DE number 1384292