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Notes on Perelman’s papers. (English) Zbl 1204.53033
In the two remarkable preprints “The entropy formula for the Ricci flow and its geometric applications” [(1) arXiv e-print service, Cornell University Library, Paper No. 0211159 (2002; Zbl 1130.53001)] and “Ricci flow with surgery on three-manifolds” [(2) arXiv e-print service, Cornell University Library, Paper No. 0303109 (2003; Zbl 1130.53002)], G. Perelman announced a proof of the Poincaré conjecture and Thurston’s geometrization conjecture using the Ricci flow approach of Hamilton. The purpose of this paper is to present the details that are not seen in [(1), (2)].
Knowing some readers wish to take an abbreviated route that focuses on the main proofs, the authors of this paper provide the reading guide for the reader’s itinerary. The present reviewer follows this guide for his statements, beginning with an overview of Section 3, which describes the definition of Ricci flow and outlines two proofs.
Let \(M\) be a compact 3-manifold and let \(\{g(t)\}_{t\in [a,b]}\) be a smoothly varying family of Riemannian metrics on \(M\). Then \(g(\cdot)\) satisfies the Ricci flow equation if
\[ {\partial g(t)\over\partial t}= -2\,\text{Ric}(g(t)) \] holds for \(t\in [a,b]\). R. S. Hamilton [J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)] showed that for \(g(0)\) on \(M\), there is a \(T\in (0,\infty]\) with the property that there is a solution \(g(,)\) defined on \([0,T]\) with \(g(0)= g_0\), so that if \(T<\infty\), the curvature of \(g(t)\) becomes unbounded as \(t\to T\). Assuming for the moment that \(M\) is not simply-connected, suppose \(T<\infty\) and let \(\Omega\) be the set of points \(x\in M\) for which \(\lim_{t\to T^-}R(x,t)\) exists, and is finite.
Claim 3.4. The set \(\Omega\) is open and as \(t\to T\), the evolving \(g(,)\) converges on a compact subset \(\mathbb{T}\) of \(M-\Omega\) to a metric \(\overline g\). This claim serves to provide a topological description of J\(\mathbb{T}\) where the Ricci flow is giving singular metrics, and it gives rise to
Claim 3.5 (2). There is a well-defined way to perform surgery on \(M\), which yields a smooth post-surgery manifold \(M'\) with \(g'\).
Claim 3.7 [G. Perelman, arXiv e-print service, Cornell University Library, Paper No. 0307245 (2003; Zbl 1130.53003)]. If \(M\) is simply-connected, then any Ricci flow with surgery on \(M\) becomes extinct in finite time.
The proof of the Poincaré conjecture follows from these claims. From Claim 3.7, after some finite time, \(M\) is found to be diffeomorphic to a connected sum of factors that are each \(S^1\times S^2\) or \(S^3/\Gamma\), where \(\Gamma\) is a finite subgroup of \(\text{SO}(4)\). As we have assumed that \(M\) is simply-connected, van Kampen’s theorem asserts that \(M\) is diffeomorphic to a connected sum of \(S^{3'}\)s, and hence is diffeomorphic to \(S^3\). This is the outline of the proof of the Poincaré conjecture.
As to the outline of the proof of the geometrization conjecture, we drop the assumption of simple-connectedness and let \(M_t\) denote the time-\(t\) manifold in a Ricci flow with surgery. We consider the metric \(\widehat g(t)={1\over t} g(t)\) on \(M\). Given \(x\in M_t\), we define the scalar \(\rho(x,t)\) such that \(\text{inf}_{R(x,\rho)}R_m= -\rho^{-2}\), where \(R_m\) denotes the sectional curvature of \(\widehat g(t)\).
Claim 3.10 (2). There is a finite collection \(\{H_i, x_i\}^k_{i=1}\) of finite volume 3-manifolds with sectional curvature \(-{1\over 4}\) and for large \(t\), a decreasing function \(\alpha(t)\) tending to zero and a family of maps
\[ f_t: \bigcup^k_{i=1} H_i\supset\bigcup^k_{i=1} B\Biggl(x_i,{1\over \alpha(t)}\Biggr)\to M_t. \] Now, take a collection \(\{P_i\}^N_{i=1}\) of pairs of plants and a collection of closed 2-disks \(\{D^2_j\}^{N'}_{j= i}\). The 3-manifolds \(\{S^1\times P_i\}^N_{i=1}\cup \{S^1\times D^2_j\}^{N'}_{j=1}\) have toral boundary. By taking an even number of these tori, one matches and glues them by homeomorphisms to obtain a 3-manifold called graph-manifold. Then
Claim 3.12. Let \(Y_t\) be the truncation of \(\bigcup^k_{i=1} H_i\), obtained by removing horoballs at distance \({1\over 2\alpha(t)}\) from base point \(x_i\). Then, for large \(t\), \(M_t- f_t(Y_t)\) is a graph-manifold.
Claims 3.10 and 3.12 along with Claims 3.4, 3.5 imply the geometrization conjecture (Appendix I).
Now and in the sequel, we refer to Section X, Y of (1) as “I, X, Y” and Sections X, Y in (2) as “II, X, Y”. For the detailed proof of the Poincaré conjecture, the reading guide advises us to read I.7 (Sections 15–26) and proceed to I.11 (Sections 38–50) followed by II.1,2 and I,12,1 (Sections 51–52). At this stage, the readers should be ready for the overview of Perelman’s second paper in Section 5.7, and proceed with II,1–II,5 (Sections 58–80). Then, in conjunction with one of the finite extinction time results of T. H. Colding and W. P. II Minicozzi [J. Am. Math. Soc. 18, No. 3, 561–569 (2005; Zbl 1083.53058) and Geom. Topol. 12, No. 5, 2537–2586 (2008; Zbl 1161.53352)] and G. Perelman [loc. cit.; (Zbl 1130.53003)], this completes the Poincaré conjecture.
To proceed with the rest of the detailed proof of the geometrization conjecture, one may begin with the large time estimates for nonsingular Ricci flows, which are seen I.12,2–I.12.4 (Sections 53–55). The readers can then go to II.6 and II.7 (Sections 81–92).

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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