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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[1] K Akutagawa, M Ishida, C LeBrun, Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds, Arch. Math. \((\)Basel\()\) 88 (2007) 71 · Zbl 1184.53042 · doi:10.1007/s00013-006-2181-0
[2] G Anderson, B Chow, A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds, Calc. Var. Partial Differential Equations 23 (2005) 1 · Zbl 1082.53069 · doi:10.1007/s00526-003-0212-2
[3] M T Anderson, Scalar curvature and geometrization conjectures for \(3\)-manifolds, Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 49 · Zbl 0890.57026
[4] M T Anderson, Scalar curvature and the existence of geometric structures on 3-manifolds. I, J. Reine Angew. Math. 553 (2002) 125 · Zbl 1023.53020 · doi:10.1515/crll.2002.096
[5] M T Anderson, Canonical metrics on 3-manifolds and 4-manifolds, Asian J. Math. 10 (2006) 127 · Zbl 1246.53063 · doi:10.4310/AJM.2006.v10.n1.a8
[6] S B Angenent, D Knopf, Precise asymptotics of the Ricci flow neckpinch, Comm. Anal. Geom. 15 (2007) 773 · Zbl 1145.53049 · doi:10.4310/CAG.2007.v15.n4.a6
[7] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985) · Zbl 0591.53001
[8] W Beckner, Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999) 105 · Zbl 0917.58049 · doi:10.1515/form.11.1.105
[9] L Bessières, G Besson, M Boileau, S Maillot, J Porti, Weak Collapsing and Geometrisation of Aspherical 3-Manifolds · Zbl 1244.57003
[10] J P Bourguignon, Une stratification de l’espace des structures riemanniennes, Compositio Math. 30 (1975) 1 · Zbl 0301.58015
[11] D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc. (2001) · Zbl 0981.51016
[12] Y Burago, M Gromov, G Perelman, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3, 222 · Zbl 0802.53018 · doi:10.1070/RM1992v047n02ABEH000877
[13] E Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958) 45 · Zbl 0079.11801 · doi:10.1215/S0012-7094-58-02505-5
[14] H D Cao, B Chow, Recent developments on the Ricci flow, Bull. Amer. Math. Soc. \((\)N.S.\()\) 36 (1999) 59 · Zbl 0926.53016 · doi:10.1090/S0273-0979-99-00773-9
[15] H D Cao, X P Zhu, A complete proof of the Poincaré and geometrization conjectures-application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006) 165 · Zbl 1200.53057 · doi:10.4310/AJM.2006.v10.n2.a2
[16] I Chavel, Riemannian geometry-a modern introduction, Cambridge Tracts in Mathematics 108, Cambridge University Press (1993) · Zbl 0810.53001
[17] J Cheeger, D G Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library 9, North-Holland Publishing Co. (1975) · Zbl 0309.53035
[18] J Cheeger, D Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. \((2)\) 96 (1972) 413 · Zbl 0246.53049 · doi:10.2307/1970819
[19] J Cheeger, M Gromov, M Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15 · Zbl 0493.53035
[20] B L Chen, X P Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006) 119 · Zbl 1104.53032
[21] B Chow, D Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs 110, Amer. Math. Soc. (2004) · Zbl 1086.53085
[22] B Chow, P Lu, L Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics 77, Amer. Math. Soc. (2006) · Zbl 1118.53001
[23] T H Colding, W P Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005) 561 · Zbl 1083.53058 · doi:10.1090/S0894-0347-05-00486-8
[24] T H Colding, W P Minicozzi II, Width and Finite Extinction Time of Ricci Flow, Geom. Topol. 12 (2008) 2537 · Zbl 1161.53352 · doi:10.2140/gt.2008.12.2537
[25] J Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983) 703 · Zbl 0526.58047 · doi:10.1512/iumj.1983.32.32046
[26] J H Eschenburg, Local convexity and nonnegative curvature-Gromov’s proof of the sphere theorem, Invent. Math. 84 (1986) 507 · Zbl 0594.53034 · doi:10.1007/BF01388744
[27] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255 · Zbl 0504.53034
[28] R S Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, Amer. Math. Soc. (1988) 237 · Zbl 0663.53031
[29] R S Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995) 545 · Zbl 0840.53029 · doi:10.2307/2375080
[30] R S Hamilton, The formation of singularities in the Ricci flow, Int. Press, Cambridge, MA (1995) 7 · Zbl 0867.53030
[31] R S Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1 · Zbl 0892.53018
[32] R S Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999) 695 · Zbl 0939.53024
[33] H Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac. 38 (1995) 101 · Zbl 0833.35053
[34] V Kapovitch, Perelman’s Stability Theorem, Int. Press, Cambridge, MA (2007) 103 · Zbl 1151.53038
[35] B Kleiner, J Lott, Locally Collapsed \(3\)-Manifolds, to appear
[36] B Kleiner, J Lott, Notes and commentary on Perelman’s Ricci flow papers
[37] P Lu, G Tian, Uniqueness of standard solutions in the work of Perelman
[38] S Matveev, Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics 9, Springer (2003) · Zbl 1048.57001
[39] W H Meeks III, S T Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. \((2)\) 112 (1980) 441 · Zbl 0458.57007 · doi:10.2307/1971088
[40] J W Morgan, Recent progress on the Poincaré conjecture and the classification of 3-manifolds, Bull. Amer. Math. Soc. \((\)N.S.\()\) 42 (2005) 57 · Zbl 1100.57016 · doi:10.1090/S0273-0979-04-01045-6
[41] J Morgan, G Tian, Completion of Perelman’s proof of the geometrization conjecture
[42] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, Amer. Math. Soc. (2007) · Zbl 1179.57045
[43] G D Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press (1973) · Zbl 0265.53039
[44] L Ni, The entropy formula for linear heat equation, J. Geom. Anal. 14 (2004) 87 · Zbl 1062.58028 · doi:10.1007/BF02922078
[45] L Ni, A note on Perelman’s LYH-type inequality, Comm. Anal. Geom. 14 (2006) 883 · Zbl 1116.58031 · doi:10.4310/CAG.2006.v14.n5.a3
[46] G Perelman, The entropy formula for the Ricci flow and its geometric applications · Zbl 1130.53001
[47] G Perelman, Ricci flow with surgery on three-manifolds · Zbl 1130.53002
[48] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds · Zbl 1130.53003
[49] G Prasad, Strong rigidity of \(\mathbfQ\)-rank \(1\) lattices, Invent. Math. 21 (1973) 255 · Zbl 0264.22009 · doi:10.1007/BF01418789
[50] M Reed, B Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press (1978) · Zbl 0401.47001
[51] O S Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981) 110 · Zbl 0471.58025 · doi:10.1016/0022-1236(81)90050-1
[52] R Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Ann. of Math. Stud. 103, Princeton Univ. Press (1983) 111 · Zbl 0532.53042
[53] R Schoen, S T Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Ann. of Math. Stud. 102, Princeton Univ. Press (1982) 209 · Zbl 0481.53036
[54] P Scott, The geometries of \(3\)-manifolds, Bull. London Math. Soc. 15 (1983) 401 · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[55] V Sharafutdinov, The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to \(\mathbbR^n\), Siberian Math. J. 18 (1977) 649 · Zbl 0374.53018
[56] W X Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223 · Zbl 0676.53044
[57] W X Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303 · Zbl 0686.53037
[58] T Shioya, T Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann. 333 (2005) 131 · Zbl 1087.53033 · doi:10.1007/s00208-005-0667-x
[59] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. \((\)N.S.\()\) 6 (1982) 357 · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0
[60] P Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series 325, Cambridge University Press (2006) · Zbl 1105.58013
[61] R Ye, On the Uniqueness of 2-Dimensional \(\kappa\)-Solutions
[62] R Ye, On the \(l\)-function and the reduced volume of Perelman. I, Trans. Amer. Math. Soc. 360 (2008) 507 · Zbl 1130.53047 · doi:10.1090/S0002-9947-07-04405-4
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