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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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