##
**Geometrisation of 3-manifolds.**
*(English)*
Zbl 1244.57003

EMS Tracts in Mathematics 13. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-082-1/hbk). x, 237 p. (2010).

The Kneser decomposition and JSJ-splitting theorems imply that every closed, orientable \(3\)-manifold can be canonically split by a (possibly empty) collection of essential \(2\)-spheres and tori into \(3\)-submanifolds that are atoroidal or Seifert fibered. The Thurston Geometrization Conjecture is that (after capping off the boundary spheres with 3-balls) each piece of the Kneser-JSJ splitting is geometric. Here a compact \(3\)-manifold is geometric, if its interior is the quotient of a geometry \(X\) (a homogeneous, simply connected, unimodular Riemann manifold) by a free action of a discrete subgroup of the isometry group of \(X\). Thurston showed that there are eight relevant model geometries for compact orientable \(3\)-manifolds (six of these yield Seifert fiber spaces, one is hyperbolic geometry, one is Euclidean geometry) and proved the Conjecture for Haken-manifolds (i.e. the irreducible \(3\)-manifolds that have either non-empty boundary or contain a closed essential surface); in particular, any atoroidal Haken-manifold is hyperbolic or Seifert fibered.

The full Geometrization Conjecture was proved by Grigori Perelman in 2003. A consequence is the Elliptization Conjecture, that a closed 3-manifold with finite fundamental group is spherical, with the Poincaré Conjecture as a corollary. Perelman’s proof is based on Hamilton’s theory of Ricci flow with surgery. The main goal of the book under review is to present a proof that is more attractive to topologists by accepting Thurston’s proof for the the case of Haken manifolds and concentrating on the proof of the following Theorem: Let \(M\) be a closed, orientable, irreducible, atoroidal \(3\)-manifold. Then (i) If \(\pi_1 (M)\) is finite, \(M\) is spherical. (ii) If \(\pi_1 (M)\) is infinite, \(M\) is hyperbolic or Seifert fibered. The strategy for the proof is well-outlined in the first chapter of the book with the definitions and proofs occupying the remaining chapters that form parts I–IV of the book. The authors make a special effort to simplify and streamline Perelman’s arguments, and make the various parts independent of one another.

Hamilton introcuced the Ricci flow equation \(dg/dt = -2 Ric_{g(t)}\), where \(g\) is the Riemannian metric and \(Ric_g\) its Ricci tensor. Some important results of Hamilton are (for \(M\) a closed, orientable \(3\)-manifold):

(i) For any metric \(g_0\) on \(M\) there exists \(T\in (0,\infty]\), such that the Ricci flow equation has a unique solution on \([0,T)\), satisfying \(g(0)=g_0\).

(ii) If \(g_0\) is a Riemann metric of positive Ricci curvature, then \(T\) is finite and the volume-rescaled Ricci flow converges (as \(t\rightarrow T\)) to a metric of positive constant sectional curvature; i.e. \(M\) is spherical.

(ii) If \(g(\cdot)\) is a Ricci flow defined on \([0, \infty )\) and the sectional curvature of the rescaled flow is bounded independently of \(t\), then the rescaled flow converges (as \(t\rightarrow \infty\)) to a hyperbolic metric or collapses with bounded sectional curvature, or \(M\) contains an incompressible torus.

In the general case, when \(T\) is finite, one says that the Ricci flow has a singularity at \(T\). Singularities occur only when the scalar curvature tends somewhere to \(+\infty\). Perelman showed that at large scalar curvature, points have canonical neighborhoods, in particular such neighborhoods as \(\epsilon\)-necks \(S^2\times (-\epsilon^{-1} , \epsilon^{-1})\) (where \(S^2\) is round of scalar curvature \(1\)), or \(\epsilon\)-caps ( 3-balls with collar neighborhood an \(\epsilon\)-neck). The authors’ strategy is to do metric surgery before a singularity arises by choosing a fixed large number \(\Theta\) that serves as a curvature threshold; if the Ricci flow reaches \(\Theta\) at some time \(t_0\) then either the minimum of the scalar curvature of the metric at time \(t_0\) is large enough, in which case every point has a canonical neighborhood and the manifold is shown to be spherical. Otherwise \(g(t_0 )\) is modified so that the maximum of the scalar curvature of the new metic \(g_{+}(t_0 )\) is at most \(\Theta /2\), in which case the metric in some \(3\)-balls containing regions of high curvature (big bubbles) is replaced by special types of \(\epsilon\)-caps. Then the Ricci flow is started again with \(g_{+}(t_0 )\) as initial metric and it is shown that surgery times can not accumulate. To show that it is possible to choose \(\Theta\) and iterate this process, the authors define a Ricci flow with bubbling-off and show (with the proof occupying Parts I and II of the book) that a closed, orientable, irreducible \(3\)-manifold is either spherical or for every \(T>0\) and every Riemann metric \(g_0\) on \(M\), there exists a Ricci flow with bubbling-off \(g(\cdot )\) on \([0,T]\), with \(g(0)=g_0 \).

In Part III it is shown that there is on \(M\) a long range (i.e., on \([0,+\infty)\)) Ricci flow with bubbling-off such that for given \(\epsilon >0\), \(t_n \rightarrow +\infty\), and \(x_n\) in the \(\epsilon\)-thick part of \((M, t_n^{-1}g(t_n ))\), the sequence of pointed manifolds \((M, t_n^{-1}g(t_n ), x_n )\) subconverges in the pointed smooth topology to a complete hyperbolic metric of finite volume and the sequence \((t_n^{-1}g(t_n ))\) has locally controlled curvature in the sense of Perelman. Here \(x\in M\) is \(\epsilon\)-thin in \((M,g)\), if for some \(0<\rho \leq 1\), there is a \(\rho\)-ball about \(x\) of volume \(< \epsilon \rho^3\), in which all sectional curvatures are bounded below by \(-\rho^{-2}\). Otherwise \(x\) is \(\epsilon\)-thick. If \(H\) is a hyperbolic limit as above, then \(vol(H)<V_0 (M)\), where \(V_0 (M)\) is the infimum of \(\{ vol(M\backslash L )\}\), over all hyperbolic links \(L\subset M\). The geometrisation conjecture for closed, orientable, irreducible \(3\)-manifolds follows then from the “Weak collapsing Theorem” whose proof uses Gromov’s vanishing theorem of the simplicial volume and is given in Part IV: Suppose that \(M\) is not simply-connected and there is a sequence \(g_n\) of Riemann metrics on \(M\) such that (i) \(\{vol(g_n )\}\) bounded, (ii) for every \(\epsilon >0\) and \(\epsilon\)-thick \(x_n\in (M, g_n )\), the sequence \((M, g_n, x_n )\) subconverges in the pointed smooth topology to a pointed hyperbolic \(3\)-manifold \(H\) of volume \(< V_0 (H )\), (iii) the sequence \(g_n\) has locally controlled curvature in the sense of Perelman. Then \(M\) is a Seifert fibered manifold or contains an incompressible torus.

The full Geometrization Conjecture was proved by Grigori Perelman in 2003. A consequence is the Elliptization Conjecture, that a closed 3-manifold with finite fundamental group is spherical, with the Poincaré Conjecture as a corollary. Perelman’s proof is based on Hamilton’s theory of Ricci flow with surgery. The main goal of the book under review is to present a proof that is more attractive to topologists by accepting Thurston’s proof for the the case of Haken manifolds and concentrating on the proof of the following Theorem: Let \(M\) be a closed, orientable, irreducible, atoroidal \(3\)-manifold. Then (i) If \(\pi_1 (M)\) is finite, \(M\) is spherical. (ii) If \(\pi_1 (M)\) is infinite, \(M\) is hyperbolic or Seifert fibered. The strategy for the proof is well-outlined in the first chapter of the book with the definitions and proofs occupying the remaining chapters that form parts I–IV of the book. The authors make a special effort to simplify and streamline Perelman’s arguments, and make the various parts independent of one another.

Hamilton introcuced the Ricci flow equation \(dg/dt = -2 Ric_{g(t)}\), where \(g\) is the Riemannian metric and \(Ric_g\) its Ricci tensor. Some important results of Hamilton are (for \(M\) a closed, orientable \(3\)-manifold):

(i) For any metric \(g_0\) on \(M\) there exists \(T\in (0,\infty]\), such that the Ricci flow equation has a unique solution on \([0,T)\), satisfying \(g(0)=g_0\).

(ii) If \(g_0\) is a Riemann metric of positive Ricci curvature, then \(T\) is finite and the volume-rescaled Ricci flow converges (as \(t\rightarrow T\)) to a metric of positive constant sectional curvature; i.e. \(M\) is spherical.

(ii) If \(g(\cdot)\) is a Ricci flow defined on \([0, \infty )\) and the sectional curvature of the rescaled flow is bounded independently of \(t\), then the rescaled flow converges (as \(t\rightarrow \infty\)) to a hyperbolic metric or collapses with bounded sectional curvature, or \(M\) contains an incompressible torus.

In the general case, when \(T\) is finite, one says that the Ricci flow has a singularity at \(T\). Singularities occur only when the scalar curvature tends somewhere to \(+\infty\). Perelman showed that at large scalar curvature, points have canonical neighborhoods, in particular such neighborhoods as \(\epsilon\)-necks \(S^2\times (-\epsilon^{-1} , \epsilon^{-1})\) (where \(S^2\) is round of scalar curvature \(1\)), or \(\epsilon\)-caps ( 3-balls with collar neighborhood an \(\epsilon\)-neck). The authors’ strategy is to do metric surgery before a singularity arises by choosing a fixed large number \(\Theta\) that serves as a curvature threshold; if the Ricci flow reaches \(\Theta\) at some time \(t_0\) then either the minimum of the scalar curvature of the metric at time \(t_0\) is large enough, in which case every point has a canonical neighborhood and the manifold is shown to be spherical. Otherwise \(g(t_0 )\) is modified so that the maximum of the scalar curvature of the new metic \(g_{+}(t_0 )\) is at most \(\Theta /2\), in which case the metric in some \(3\)-balls containing regions of high curvature (big bubbles) is replaced by special types of \(\epsilon\)-caps. Then the Ricci flow is started again with \(g_{+}(t_0 )\) as initial metric and it is shown that surgery times can not accumulate. To show that it is possible to choose \(\Theta\) and iterate this process, the authors define a Ricci flow with bubbling-off and show (with the proof occupying Parts I and II of the book) that a closed, orientable, irreducible \(3\)-manifold is either spherical or for every \(T>0\) and every Riemann metric \(g_0\) on \(M\), there exists a Ricci flow with bubbling-off \(g(\cdot )\) on \([0,T]\), with \(g(0)=g_0 \).

In Part III it is shown that there is on \(M\) a long range (i.e., on \([0,+\infty)\)) Ricci flow with bubbling-off such that for given \(\epsilon >0\), \(t_n \rightarrow +\infty\), and \(x_n\) in the \(\epsilon\)-thick part of \((M, t_n^{-1}g(t_n ))\), the sequence of pointed manifolds \((M, t_n^{-1}g(t_n ), x_n )\) subconverges in the pointed smooth topology to a complete hyperbolic metric of finite volume and the sequence \((t_n^{-1}g(t_n ))\) has locally controlled curvature in the sense of Perelman. Here \(x\in M\) is \(\epsilon\)-thin in \((M,g)\), if for some \(0<\rho \leq 1\), there is a \(\rho\)-ball about \(x\) of volume \(< \epsilon \rho^3\), in which all sectional curvatures are bounded below by \(-\rho^{-2}\). Otherwise \(x\) is \(\epsilon\)-thick. If \(H\) is a hyperbolic limit as above, then \(vol(H)<V_0 (M)\), where \(V_0 (M)\) is the infimum of \(\{ vol(M\backslash L )\}\), over all hyperbolic links \(L\subset M\). The geometrisation conjecture for closed, orientable, irreducible \(3\)-manifolds follows then from the “Weak collapsing Theorem” whose proof uses Gromov’s vanishing theorem of the simplicial volume and is given in Part IV: Suppose that \(M\) is not simply-connected and there is a sequence \(g_n\) of Riemann metrics on \(M\) such that (i) \(\{vol(g_n )\}\) bounded, (ii) for every \(\epsilon >0\) and \(\epsilon\)-thick \(x_n\in (M, g_n )\), the sequence \((M, g_n, x_n )\) subconverges in the pointed smooth topology to a pointed hyperbolic \(3\)-manifold \(H\) of volume \(< V_0 (H )\), (iii) the sequence \(g_n\) has locally controlled curvature in the sense of Perelman. Then \(M\) is a Seifert fibered manifold or contains an incompressible torus.

Reviewer: Wolfgang Heil (Tallahassee)