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Non-singular solutions of the Ricci flow on three-manifolds. (English) Zbl 0939.53024
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 \(i=1,\dots,n\).

MSC:
53C20 Global Riemannian geometry, including pinching
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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