Geometrization of 3-Manifolds via the Ricci Flow Michael T. Anderson
Introduction results are already being used by others in research The classification of closed surfaces is a milestone on related topics. These circumstances serve to jus- in the development of topology, so much so that tify the writing of an article at this time, which oth- it is now taught to most mathematics undergrad- erwise might be considered premature. uates as an introduction to topology. Since the so- The work of Perelman builds on prior work of lution of the uniformization problem for surfaces Thurston and Hamilton. In the next two sections by Poincaré and Koebe, this topological classifica- we discuss the Thurston picture of 3-manifolds and tion is now best understood in terms of the the Ricci flow introduced and analyzed by Hamil- geometrization of 2-manifolds: every closed surface ton. For additional background, in particular on the Σ admits a metric of constant Gauss curvature +1, Poincaré Conjecture, see Milnor’s Notices survey [14] 0, or −1 and so is uniformized by one of the stan- and references therein. For much more detailed dard space-form geometries S2, R2, H2. Hence any commentary and discussion on Perelman’s work, surface Σ is a quotient of either the 2-sphere, the see [13]. Euclidean plane, or the hyperbolic disc by a discrete The Geometrization Conjecture group Γ acting freely and isometrically. The classification of higher-dimensional mani- While the Poincaré Conjecture has existed for about folds is of course much more difficult. In fact, due one hundred years, the remarkable insights of to the complexity of the fundamental group, a Thurston in the late 1970s led to the realistic pos- complete classification as in the case of surfaces sibility of understanding and classifying all closed is not possible in dimensions ≥ 4. In dimension 3 3-manifolds in a manner similar to the classifica- this argument does not apply, and the full classi- tion of surfaces via the uniformization theorem. fication of 3-manifolds has long been a dream of To explain this, we first need to consider what topologists. As a very special case, this problem in- are the corresponding geometries in 3-dimensions. cludes the Poincaré Conjecture. In terms of Riemannian geometry, a geometric In this article we report on remarkable recent structure on a manifold M is a complete, locally ho- work of Grisha Perelman [15]-[17], which may well mogeneous Riemannian metric g. Thus, M may be \ have solved the classification problem for 3- described as the quotient Γ G/H, where G is the manifolds (in a natural sense). Perelman’s work is isometry group of the universal cover (M,g) and currently under intense investigation and scrutiny by Γ, H are discrete and compact subgroups of the Lie many groups around the world. At this time, much group G respectively. Thurston showed that there of his work has been validated by experts in the area. are eight such simply connected geometries G/H 1 Although at the moment it is still too soon to declare in dimension 3 which admit compact quotients. a definitive solution to the problem, Perelman’s ideas As in two dimensions, the most important geome- are highly original and of deep insight. Morever, his tries are those of constant curvature: hyperbolic 1The Thurston classification is essentially a special case Michael T. Anderson is professor of mathematics at the of the much older Bianchi classification of homogeneous State University of New York, Stony Brook. His email space-time metrics arising in general relativity; cf. [3] address is [email protected]. for further remarks on the dictionary relating these Partially supported by NSF Grant DMS 0305865. classifications.
184 NOTICES OF THE AMS VOLUME 51, NUMBER 2 geometry H3 of curvature −1, Euclidean geometry M R3 of curvature 0, and spherical geometry S3 of cur- S2 vature +1. The remaining five geometries are prod- ucts or twisted products with the 2-dimensional 2 geometries. Trivial S1 bundles over a surface of S S2 genus g>1 have H2 × R geometry, while nontriv- S2 ial bundles have SL (2, R) geometry; nontrivial S1 bundles over T 2 have Nil geometry, while nontrivial T 2 bundles over S1 have Sol geometry (or Nil or R3 geometry); finally, S1 bundles over S2 have S2 × R (or S3) geometry. For example, any Seifert L fibered 3-manifold, a 3-manifold admitting a locally K2 1 1 free S action, has such a geometric structure. K1 Geometric 3-manifolds, that is 3-manifolds ad- mitting a geometric structure, are the building S2×S1 blocks of more complicated 3-manifolds. For sim- plicity, we assume throughout the article that all manifolds M are orientable. The building blocks are Figure 1. Sphere decomposition. then assembled along 2-spheres S2, via connected sum, and along tori T 2. As a simple example of such an assembly, let {Mi } be a finite collection of Seifert fibered 3-manifolds over surfaces Σi with non- K empty boundary, so that ∂Mi consists of tori. These H2 tori may then be glued together pairwise by dif- S feomorphisms to obtain a closed 3-manifold or a 2 T S1 3 3-manifold with toral boundary. A 3-manifold as- T 1 T2 sembled in this way is called a graph manifold. (One H1 assigns a vertex to each Seifert fibered space and T S an edge to each torus connecting two such Seifert 4 3 spaces). A torus bundle over S1 is a graph mani- fold, since it is the union of two Seifert fibered spaces over S1 × I. Graph manifolds were intro- duced, and their structure completely analyzed, by Figure 2. Torus decomposition, (Si Siefert fibered, Hj torus- Waldhausen. irreducible). Conversely, let M be an arbitrary closed (replacing regions S2 × I by two copies of B3); see 3-manifold, as above always orientable. One then Figure 1 for a schematic representation. decomposes or splits it into pieces according to The K factors in (1) may also contain topologi- the structure of the simplest surfaces embedded cally essential tori. A torus T 2 embedded in M is in M, namely spheres and tori. Topologically, this called incompressible if the inclusion map induces is accomplished by the following classical results an injection on π1. A 3-manifold N is called torus- in 3-manifold topology. irreducible if every embedded incompressible torus Sphere (or Prime) Decomposition (Kneser, may be deformed to a torus in ∂N. Hence, if Milnor) ∂N =∅, then N has no incompressible tori. Let M be a closed 3-manifold. Then M admits a Torus Decomposition (Jaco-Shalen, Johannsen) finite connected sum decomposition Let M be a closed, irreducible 3-manifold. Then there is a finite collection, possibly empty, of dis- (1) M = (K #...#K )#(L #...#L )#(#r S2 × S1). 1 p 1 q 1 joint incompressible tori in M that separate M into The K and L factors here are closed irreducible a finite collection of compact 3-manifolds (with 3-manifolds; i.e. every embedded 2-sphere S2 toral boundary), each of which is torus-irreducible bounds a 3-ball. The K factors have infinite fun- or Seifert fibered. damental group and are aspherical 3-manifolds A coarser, but essentially equivalent, decompo- (K(π,1)’s), while the L factors have finite funda- sition is given by tori separating M into torus- mental group and have universal cover a homotopy irreducible and graph manifold components; see 3-sphere. Since M#S3 = M, we assume no L factor Figure 2. is S3 unless M = L = S3. The factors in (1) are then With the simple exceptions of S2 × S1 and its ori- Z 2 × 1 RP3 RP3 unique up to permutation and are obtained from ented 2 quotient S Z2 S # , essential M by performing surgery on a collection of essen- 2-spheres are obstructions to the existence of a geo- tial, i.e. topologically nontrivial, 2-spheres in M metric structure on a 3-manifold. The same is true
FEBRUARY 2004 NOTICES OF THE AMS 185 for essential tori, unless M happens to be a Seifert Independently and around the same time as fibered or a Sol 3-manifold. Thus, the sphere and Thurston, Gromov [6], [7] also studied the defor- torus decompositions divide M topologically into mation and degeneration of more general Rie- pieces where these known obstructions are re- mannian metrics with merely bounded curvature moved. in place of constant curvature. The idea is that Geometrization Conjecture (Thurston). Let M be one can control the behavior of a metric, or of a a closed, oriented 3-manifold. Then each component family of metrics, given a uniform bound on the 2 of the sphere and torus decomposition admits a Riemann curvature tensor Riem of the metric. geometric structure. This leads to the important Gromov compactness theorem, the structure theory of almost flat man- The geometrization conjecture gives a complete ifolds, and the theory of collapsing Riemannian and effective classification of all closed 3-manifolds, manifolds, worked out in detail with Cheeger and closely resembling in many respects the classifi- Fukaya. cation of surfaces. More precisely, it reduces the One version of these results is especially rele- classification to that of geometric 3-manifolds. vant for our purposes. Let (M,g) be a closed The classification of geometric 3-manifolds is rather Riemannian manifold, normalized to unit volume, simple and completely understood, except for the and suppose case of hyperbolic 3-manifolds, which remains an active area of research. (2) |Riem|≤Λ, As an illustration of the power of the Thurston for some arbitrary constant Λ < ∞. The metric g Conjecture, let us see how it implies the Poincaré Conjecture. If M is a simply connected 3-manifold, provides a natural decomposition of M into thick = ν ∪ then the sphere decomposition (1) implies that M and thin parts, M M Mν , where must be an L factor. The geometrization conjecture ν (3) M ={x ∈ M : volBx(1) ≥ ν}, implies that L is geometric, and so L = S3/Γ. Hence, ={ ∈ } M = L = S3. Mν x M : volBx(1) <ν ; Thurston’s formulation and work on the here B (1) is the geodesic ball about x of radius 1 and geometrization conjecture revolutionized the field x ν>0 is an arbitrary but fixed small number. Now of 3-manifold topology; see [18], [19] and further consider the class of all Riemannian n-manifolds of references therein. He recognized that in the class unit volume satisfying (2), and consider the corre- of all (irreducible) 3-manifolds, hyperbolic 3- sponding decompositions (3). Then the geometry manifolds are overwhelmingly the most prevalent, and topology of Mν is a priori controlled. For any given as is the case with surfaces, and developed a vast array of new ideas and methods to understand the ν>0, there are only a finite number, (depending on ν structure of 3-manifolds. Thurston and a number Λ and ν), of possible topological types for M . More- ν of other researchers proved the geometrization over, the space of metrics on M is compact in a nat- conjecture in several important cases, the most ural sense; any sequence has a subsequence con- 1,α celebrated being the Haken manifold theorem: if M verging in the C topology, α<1 (modulo is an irreducible Haken 3-manifold, i.e. M contains diffeomorphisms). For ν sufficiently small, the com- an incompressible surface of genus ≥ 1, then the plementary thin part Mν admits an F-structure in the geometrization conjecture is true for M. sense of Cheeger-Gromov; in dimension 3 this just An important ingredient in the Thurston ap- means that Mν is a graph manifold with toral (or empty) proach is the deformation and degeneration of boundary. In particular, the topology of Mν is strongly hyperbolic structures on noncompact manifolds (or restricted. A metric on Mν is highly collapsed in the the deformation of singular hyperbolic structures sense that the circles in the Seifert fibered pieces of on compact manifolds). The eight geometric struc- Mν and the tori glueing these pieces together have tures are rigid in that there are no geometries very small diameter, depending on ν; see Figure 3 for which interpolate continuously between them. a schematic picture. Moreover, for any fixed ν>0, Hence, on a composite 3-manifold M, the geomet- the distance between Mν and the arbitrarily thin part ric structure on each piece must degenerate in Mν becomes arbitrarily large as ν /ν → 0. passing from one piece to the next; there is no sin- We point out that similar results hold locally and gle structure or metric giving the geometrization for complete noncompact manifolds; thus the unit of all of M. For example, in Figure 2 the H pieces volume normalization above is not essential. may be hyperbolic 3-manifolds separated by tori from Seifert fibered pieces S. Although this split- 2The curvature tensor is a complicated (3,1) tensor ting is topologically well defined, the geometries expressed in terms of the second derivatives of the do not match in the glueing region, and metrically metric; in a local geodesic normal coordinate system there is no natural region in which to perform the at a given point, the components of Riem are given by l =−1 + − − glueing. Rijk 2 (∂i ∂kgjl ∂j ∂l gik ∂j ∂kgil ∂i ∂l gjk).
186 NOTICES OF THE AMS VOLUME 51, NUMBER 2 The Thurston approach to geometrization has made a great deal of progress on the “hyperbolic M ν M ν part” of the conjecture. In comparison with this, relatively little progress has been made on the M ν “positive curvature part” of the conjecture, for example the Poincaré Conjecture. It is worth Mν pointing out that among the eight geometries, the constant curvature geometries H3 and S3 are by far the most important to understand (in terms of Figure 3. Thick-thin decomposition. characterizing which manifolds are geometric); the other (mixed) geometries are much simpler in example, rescaling the Ricci flow (5) so that the comparison. volume of (M,g(t)) is preserved leads to the flow From the point of view of Riemannian geome- equation (4) with λ = 2 R, twice the mean value try, the Thurston conjecture essentially asserts the of the scalar curvature R. existence of a “best possible” metric on an arbitrary In a suitable local coordinate system, equation closed 3-manifold. In the case that M is not itself (5) has a very natural form. Thus, at time t, choose geometric, one must allow the optimal metric to local harmonic coordinates so that the coordinate have degenerate regions. The discussion and fig- functions are locally defined harmonic functions ures above suggest that the degeneration should in the metric g(t). Then (5) takes the form be via the pinching off of 2-spheres (sphere de- d composition) and collapse of graph manifolds along (6) gij = ∆gij + Qij(g,∂g), circles and tori (torus decomposition). dt where ∆ is the Laplace-Beltrami operator on func- The Ricci Flow tions with respect to the metric g = g(t) and Q is One method to find a best metric on a manifold is a lower-order term quadratic in g and its first- to find a natural evolution equation, described by order partial derivatives. This is a nonlinear a vector field on the space of metrics, and try to heat-type equation for gij. From the analysis of prove that the flow lines exist for all time and con- such PDE, one obtains existence and uniqueness of verge to a geometric limit. In case a flow line does solutions to the Ricci flow on some time interval, not converge, the corresponding metrics degener- starting at any smooth initial metric. This is the ate, and one then needs to relate the degeneration reason for the minus sign in (5); a plus sign leads with the topology of M. to a backwards heat-type equation, which has no There is essentially only one simple and natural solutions in general. vector field (or more precisely family of vector Here are a few simple examples of explicit so- fields) on the space of metrics. It is given by lutions to the Ricci flow. If the initial metric g(0) is of constant Ricci curvature, Ric = a · g, then the d (4) g(t) =−2Ricg(t) + λ(t) · g(t). evolution g(t) is just a rescaling of g(0): dt g(t) = (1 − 2at)g(0). Note that if a>0, then the flow contracts the metric, while if a<0, the flow Here Ric is the Ricci curvature, given in local co- = = k expands the metric, uniformly in all directions. ordinates by Rij (Ric)ij k Rikj , so that Ric is a trace of the Riemann curvature. The constant 2 Hence, if one rescales g(t) to have constant volume, is just for convenience and could be changed by the resulting curve is constant. The stationary rescaling the time parameter; λ(t) is a constant de- points of the volume-normalized Ricci flow are ex- pending on time t. The Ricci flow, introduced by actly the class of Einstein metrics, i.e. metrics of Hamilton [11], is obtained by setting λ = 0, i.e. constant Ricci curvature. In dimension 3, Einstein metrics are of constant curvature and so give the d H3, R3 and S3 geometries. (5) g(t) =−2Ric . dt g(t) More generally, if Ric(x, t) > 0, then the flow contracts the metric g(t) near x, to the future, The reason (4) is the only natural flow equation is while if Ric(x, t) < 0, then the flow expands g(t) essentially the same as that leading to the Einstein near x. At a general point, there will be directions field equations in general relativity. The Ricci of positive and negative Ricci curvature along which curvature is a symmetric bilinear form, as is the the metric will locally contract or expand. metric. Besides multiples of the metric itself, it is Suppose g(0) is a product metric on S1 × Σ, the only such form depending on at most the where Σ is a surface with constant curvature met- second derivatives of the metric, and invariant ric. Then g(t) remains a product metric, where the under coordinate changes, i.e. a (2, 0) tensor length of the S1 factor stays constant while the sur- formed from the metric. By rescaling the metric and face factor expands or contracts according to the time variable t, one may transform (5) into (4). For sign of its curvature.
FEBRUARY 2004 NOTICES OF THE AMS 187 Finally, the Ricci flow commutes with the action a general lower bound Ric ≥−λ, for λ>0. For the of the diffeomorphism group and so preserves all remainder of the paper, we assume then that isometries of an initial metric. Thus, geometric 3- dimM = 3. manifolds remain geometric. For the “nonpositive” In the Gromov compactness result and thick/thin mixed geometries H2 × R, SL (2,R), Nil, and Sol, decomposition (3), the hypothesis of a bound on the volume-normalized Ricci flow contracts the S1 |Riem| can now also be replaced by a bound on or T 2 fibers and expands the base surface factor, |Ric| (since we are in dimension 3). Further, on time while for the positive mixed geometry S2 × R, the intervals [0,t] where |Ric| is bounded, the metrics volume-normalized flow contracts the S2 and g(t) are all quasi-isometric to each other: expands in the R-factor. cg(0) ≤ g(t) ≤ Cg(0) as bilinear forms, where c,C Now consider the Ricci flow equation (5) in gen- depend on t. Hence, the arbitrarily thin region Mν, eral. From its form it is clear that the flow g(t) will ν<<1, can only arise, under bounds on |Ric|, in continue to exist if and only if the Ricci curvature arbitrarily large times. remains bounded. This suggests one should con- The discussion above shows that the Ricci flow sider evolution equations for the curvature, in- is very natural and has many interesting proper- duced by the flow for the metric. The simplest of ties. One can see some relations emerging with the these is the evolution equation for the scalar Thurston picture for 3-manifolds. However, the ij curvature R = trgRic = g Rij: first real indication that the flow is an important d new tool in attacking geometric problems is the (7) R = ∆R + 2|Ric|2. following result of Hamilton: dt • Space-form Theorem [8]. If g(0) is a metric of
Evaluating (7) at a point realizing the minimum Rmin positive Ricci curvature on a 3-manifold M, then of R on M gives the important fact that Rmin is the volume-normalized Ricci flow exists for all monotone nondecreasing along the flow. In par- time and converges to the round metric on S3/Γ, ticular, the Ricci flow preserves positive scalar cur- where Γ is a finite subgroup of SO(4) acting freely 3 vature (in all dimensions). Moreover, if Rmin(0) > 0, on S . d ≥ 2 2 Thus the Ricci flow “geometrizes” 3-manifolds then the same argument gives dt Rmin n Rmin , n = dimM, by the Cauchy-Schwarz inequality of positive Ricci curvature. Since this ground- | |2 ≥ 1 2 breaking result, it has been an open question Ric n R . A simple integration then implies whether it can be generalized to initial metrics n (8) t ≤ . with positive scalar curvature. 2R (0) min Although the evolution of the curvature along Thus, the Ricci flow exists only up to a maximal time the Ricci flow is very complicated for general ini- tial metrics, a detailed analysis of (9) leads to the T ≤ n/2Rmin(0) when Rmin(0) > 0. In contrast, in regions where the Ricci curvature stays negative following important results: definite, the flow exists for infinite time. • Curvature pinching estimate [10], [12]. Let g(t) The evolution of the Ricci curvature has the be a solution to the Ricci flow on a closed same general form as (7): 3-manifold M. Then there is a nonincreasing func- tion φ :(−∞, ∞) → R, tending to 0 at ∞, and a con- d (9) R = ∆R + Q . stant C, depending only on g(0), such that dt ij ij ij (10) Riem(x, t) ≥−C − φ(R(x, t)) ·|R(x, t)|. The expression for Q is much more complicated than the Ricci curvature term in (7) but involves only This statement means that all the sectional curva- quadratic expressions in the curvature. However, tures Rijji of g(t), where ei is any orthonormal Q involves the full Riemann curvature tensor Riem basis at (x, t), are bounded below by the right side of g and not just the Ricci curvature (as (7) in- of (10). volves Ricci curvature and not just scalar curvature). This estimate does not imply a lower bound on An elementary but important feature of dimen- Riem(x, t) uniform in time. However, when com- sion 3 is that the full Riemann curvature Riem is bined with the fact that the scalar curvature R(x, t) determined algebraically by the Ricci curvature. is uniformly bounded below, it implies that This implies that, in general, Ricci flow has a much |Riem|(x, t) >> 1 only where R(x, t) >> 1. Hence, better chance of “working” in dimension 3. For ex- to control the size |Riem| of the full curvature, it ample, an analysis of Q shows that the Ricci flow suffices to obtain just an upper bound on the scalar preserves positive Ricci curvature in dimension 3: curvature R. This is remarkable, since the scalar if Ricg(0) > 0, then Ricg(t) > 0, for t>0. This is not curvature is a much weaker invariant of the the case in higher dimensions. On the other hand, metric than the full curvature. Moreover, at points in any dimension > 2, the Ricci flow does not pre- where the curvature is sufficiently large, (10) shows serve negative Ricci curvature, nor does it preserve that Riem(x, t)/R(x, t) ≥−δ, for δ small. Thus,
188 NOTICES OF THE AMS VOLUME 51, NUMBER 2 if one scales the metric to make R(x, t) = 1, then A singularity can form for the Ricci flow Riem(x, t) ≥−δ. In such a scale, the metric then only where the curvature becomes unbounded. 2 has almost nonnegative curvature near (x, t). Suppose then that one has λi =|Riem|(xi ,ti ) →∞, • Harnack estimate [9]. Let (N,g(t)) be a solution on a sequence of points xi ∈ M , and times to the Ricci flow with bounded and nonnegative cur- ti → T<∞. It is then natural to consider the vature Riem ≥ 0, and suppose g(t) is a complete rescaled metrics and times ≤ Riemannian metric on N. Then for 0
FEBRUARY 2004 NOTICES OF THE AMS 189 noncompact, then N is diffeomorphic to R3, S2 × R, metrics of constant Ricci curvature.3 It is natural to or a quotient of these spaces. If N is compact, then try to relate the Ricci flow with R; for instance, is a slightly stronger form of Hamilton’s theorem the Ricci flow the gradient flow of R (with respect above implies N is diffeomorphic to S3/Γ, S2 × S1 to a natural L2 metric on the space M)? However, 2 × 1 or S Z2 S . Moreover, the Harnack estimate (11) while rather close to being true, it has long been rec- holds on the limit. ognized that this is not the case. In fact, the gra- R These general features of singularity models dient flow of does not even exist, since it implies are certainly encouraging. Nevertheless, there are a backwards heat-type equation for the scalar cur- many problems to overcome to obtain any real vature R (similar to (7) but with a minus sign be- benefit from this picture. fore ∆). I. One needs to prove noncollapse at the scale Consider now the following functional enhanc- ing R: of the curvature to obtain a singularity model. II. In general, the curvature may blow up at 2 −f (14) F(g,f) = (R +|∇f | )e dVg. many different rates or scales, and it is not nearly M sufficient to understand just the structure of the singularity models at points of (space-time) maxi- This is a functional on the larger space M × ∞ R mal curvature. Somewhat analogous phenomena C (M, ), or equivalently a family of func- M ∞ R 4 (usually called bubbling) arise in many other tionals on , parametrized by C (M, ). Fix any geometrical variational problems, for instance smooth measure dm on M and define the Perelman harmonic maps, Yang-Mills fields, Einstein met- coupling by requiring that (g,f) satisfy rics, and others. (In such elliptic contexts, these (15) e−f dV = dm. problems of multiple scales have been effectively g resolved.) The resulting functional III. Even if one can solve the two previous issues, this leaves the main issue. One needs to relate the (16) F m(g,f) = (R +|∇f |2)dm structure of the singularities with the topology of M the underlying manifold. becomes a functional on M. At first sight this may The study of the formation of singularities in the appear much more complicated than (13); how- Ricci flow was initiated by Hamilton in [10]; cf. ever, for any g ∈ M there exists a large class of also [4] for a recent survey. Although there has been functions f (or measures dm) such that the L2 gra- further technical progress over the last decade, dient flow of F m exists at g and is given simply by the essential problems on the existence and struc- dg 2 ture of singularity models and their relation with (17) =−2(Ricg + D f ), topology remained unresolved until the appear- dt ance of Perelman’s work in 2002 and 2003. where D2f is the Hessian of f with respect to g . The evolution equation (17) for g is just the Ricci flow Perelman’s Work (5) modified by an infinitesimal diffeomorphism: Perelman’s recent work [15]-[17] (together with a 2 = ∗ =∇ D f (d/dt)(φt g) , where (d/dt)φt f . Thus, less crucial paper still to appear) implies a complete the gradient flow of F m is the Ricci flow, up to solution of the Geometrization Conjecture. This is diffeomorphisms. (Different choices of dm accomplished by introducing numerous highly correspond to different choices of diffeomorphism.) original geometric ideas and techniques to under- In particular, the functional F m increases along stand the Ricci flow. In particular, Perelman’s work the Ricci flow. completely resolves issues I–III above. We proceed What can one do with this more complicated by describing, necessarily very briefly, some of the functional? It turns out that, given any initial met- highlights. ric g(0) and t>0, the function f (and hence the measure dm) can be freely specified at g(t), where I. Noncollapse g(t) evolves by the Ricci flow (5). Perelman then uses Consider the Einstein-Hilbert action 3The action (13) leads to the vacuum Einstein field equations in general relativity for Lorentz metrics on a (13) R(g) = R(g)dVg 4-manifold. The term λ(t) in (4) is of course analogous to M the cosmological constant. 4The functional (14) arises in string theory as the low- as a functional on the space of Riemannian metrics energy effective action [5, §6]; the function or scalar field M R on a manifold M. Critical points of are Ricci- f is called the dilaton. It is interesting to note in this con- flat metrics (Ric = 0). The action may be adjusted, text that the gravitational field and the dilaton field arise for instance by adding a cosmological constant −2Λ, simultaneously from the low-energy quantization of the to give an action whose critical points are Einstein string world sheet (σ-model) [5, p. 837].
190 NOTICES OF THE AMS VOLUME 51, NUMBER 2 this freedom to probe the geometry of g(t) with suitable choices of f. For instance, he shows by a very simple study of the form of F m that the col- lapse or noncollapse of the metric g(t) near a point x ∈ M can be detected from the size of F m(g(t)) by choosing e−f to be an approximation to a delta function centered at x. The more collapsed g(t) is Ω near x, the more negative the value of F m(g(t)). The collapse of the metric g(t) on any scale in finite time is then ruled out by combining this with the fact that the functional F m is increasing along the Ricci flow. In fact, this argument is carried out Ω with respect to a somewhat more complicated Ω scale-invariant functional than F; motivated by certain analogies in statistical physics, Perelman calls this the entropy functional. Figure 4. Horns on singular limit. II. Singularity Models Harnack inequality (11). A second highlight of [15] is essentially a classifi- These results are of course rather technical, and cation of all complete singularity models (N,g(t)) the proofs are not simple. However, they are not that arise in finite time. Complete here means the exceptionally difficult and mainly rely on new in- metric g(0) is a complete Riemannian metric on N; sights and tools to understand the Ricci flow. A key we also drop the overbar from the notation from idea is the use of the noncollapse result above on now on. If N is smooth and compact, then it fol- all relevant scales. lows from Hamilton’s space-form theorem that N 3 2 1 2 1 is diffeomorphic to S /Γ, S × S or S ×Z S . In the 2 III. Relation with Topology more important and difficult case where N is com- The basic point now is the appearance of 2-spheres plete and noncompact, Perelman proves that the S2 near the singularities. Recall from (1) that one geometry of N near infinity is as simple and nat- first needs to perform the sphere decomposition ural as possible. At time 0 and at points x with on M before it can be geometrized. There is no r(x) = dist(x, x ) 1, for a fixed base point x , a 0 0 geometry corresponding to the sphere decompo- large neighborhood of x in the scale where R(x) = 1 sition.5 While the sphere decomposition is the is ε-close to a large neighborhood in the standard simplest operation to carry out topologically, geo- round product metric on S2 × R. Here ε may be metrically and analytically it is by far the hardest made arbitrarily small by choosing r(x) sufficiently to understand. How does one detect 2-spheres in large. Such a region is called an ε-neck. Thus the M on which to perform surgery from the geome- geometry near infinity in N is that of a union of ε- try of a metric? We now see that such 2-spheres, necks, where the slowly varying radius of S2 may embedded in the ε-necks above, arise naturally either be uniformly bounded or diverge to infinity, near the singularities of the Ricci flow. but only at a rate much smaller than r(x). Moreover, The idea then is to surger the 3-manifold M this structure also holds on a long time interval to along the 2-spheres just before the first singular- the past of 0, so that on such regions the solution ity time T. Figure 4 gives a schematic picture of the is close to the (backwards) evolving Ricci flow on partially singular metric g(T ) on M. The metric S2 × R. Topologically, N is diffeomorphic to R3 or g(T ) is smooth on a maximal domain Ω ⊂ M, where (N,g) is isometric to S2 × R. the curvature is locally bounded but is singular, i.e. Perelman shows that this structural result for ill-defined, on the complement where the curvature the singularity models themselves also holds for blows up as t → T. the solution g(t) very near any singularity time T. Suppose first that Ω =∅, so that the curvature Thus, at any base point (x, t) where the curvature of g(t) blows up everywhere on M as t → T. One says is sufficiently large, the rescaling as in (12) of the that the solution to the Ricci flow becomes extinct space-time by the curvature is smoothly close, on at time T. Note that R(x, t) 1 for all x ∈ M and t large compact domains, to corresponding large near T (by the pinching estimate (10)). Given the un- domains in a complete singularity model. The derstanding of the singularity models above, it is “ideal” complete singularity models do actually not difficult to see that M is then diffeomorphic to describe the geometry and topology near any sin- gularity. Consequently, one has a detailed under- 5One might think that the S2 × R geometry corresponds standing of the small-scale geometry and topology to sphere decomposition, but this is not really correct; at everywhere on (M,g(t)), for t near T. In particular, best, this can be made sense of only in an idealized or this basically proves a general version of the limiting context.
FEBRUARY 2004 NOTICES OF THE AMS 191 3 2 × 1 2 × 1 ˆ ν S /Γ, S S , or S Z2 S . In this situation we are components). The smooth Ricci flow exists on M done, since M is then geometric. for infinite time, and the rescalings t−1g(t) con- If Ω ≠ ∅, then the main point is that small neigh- − 1 verge to the hyperbolic metric of curvature 4 as borhoods of the boundary ∂Ω consist of horns. A t →∞. (Since the Ricci flow exists for all time, it is 2 × 2 horn is a metric on S [0,δ] where the S factor reasonable to expect that the volume-normalized is approximately round of radius ρ(r), with ρ(r) flow converges to an Einstein metric, necessarily a → → small and ρ(r)/r 0 as r 0. Thus, a horn is a hyperbolic metric in our situation.) While there is union of ε-necks assembled on smaller and smaller less control on the ν-thin part Mˆ , there is enough scales. The boxed figure in Figure 4 represents a ν to conclude that Mˆ is diffeomorphic to a graph partially singular metric on the smooth manifold ν manifold G (with finitely many components). S2 × I, consisting of a pair of horns joined by a de- Although there may still be infinitely many surg- generate metric. At time T there may be infinitely many components of Ω, of arbitrarily small size, eries required to continue the Ricci flow for all ˆ = 6 containing such horns. However, all but finitely time, all further surgeries take place in Mν G. many of these components are doubled horns, Thus, the original 3-manifold M has been each topologically again of the form S2 × I. decomposed (at large finite time) topologically as In quantitative terms, there is a small constant = q 3 r 2 × 1 (18) M (K1#...#Kp)#(#1 S /Γi )#((#1S S ). ρ0 > 0 such that if Ω contains no horns with sphere S2 ×{δ} of radius ≥ ρ , then, as above when 0 Perelman has recently shown [17] that the S3/Γ =∅ 3 2 × 1 Ω , M is diffeomorphic to S /Γ, S S , or 2 × 1 2 × 1 and S S factors necessarily become extinct in S Z2 S , and we are done. If there are horns con- 2 bounded time (with bound depending on the taining a sphere S × δ of a definite size ρ0 in Ω, one then performs a surgery on each such initial metric), so that only the K factors exist after 2 a sufficiently long time. (This result is not needed, horn by truncating it along the S of radius ρ0 and glueing in a smooth 3-ball, giving then a disjoint however, for the geometrization conjecture.) = collection of 3-manifolds. Moreover, each K Ki decomposes via the Having now disconnected M by surgery on thick/thin decomposition as a union 2-spheres into a finite number of components, one (19) K = H ∪ G, then continues with the Ricci flow separately on each component. A conceptually simple, but tech- where H is a complete hyperbolic manifold of fi- nically hard, argument based on the decrease of nite volume (possibly disconnected) and G is a volume associated with each surgery shows that the graph manifold (possibly disconnected). The union surgery times are locally finite: on any finite time of H and G is along a collection of embedded tori. interval there are only finitely many times at which Perelman uses the proofs in [11] or [1], [2] to con- singularities form. clude that each such torus is incompressible in K. As a concrete example, suppose the initial met- This process gives then both the sphere and ric g(0) on M has positive scalar curvature. Then torus decomposition of the manifold M. Although the estimate (8) shows that Ricci flow completely terminates, i.e. becomes extinct, in finite time. it is not asserted that the Ricci flow detects the Hence only finitely many surgeries are applied to further decomposition of G into Seifert fibered M during the Ricci flow and it follows from the work components, this is comparatively elementary from above that M is diffeomorphic to a finite connected a topological standpoint. The torus-irreducible sum of S3/Γ and S2 × S1 factors. components of K have been identified as hyperbolic The upshot of this procedure is that if one suc- manifolds. cessively throws away or ignores such components This completes our brief survey of the which become extinct in finite times (and which have geometrization conjecture. Perelman’s work has already been identified topologically), the Ricci created a great deal of excitement in the mathe- flow with surgery then exists for infinite time [0, ∞). matical research community, as well as in the What then does the geometry of the remaining scientifically interested public at large. While at components {Mˆi } of M look like at a sufficiently the moment further evaluation of the details of large time T0? Here the thick-thin decomposition his work are still being carried out, the beauty and { ˆ}∈{ˆ } of Gromov-Thurston appears. Fix any M Mi depth of these new contributions are clear. = −1 and consider the rescaled metric gˆ(t) t g(t), for I am very grateful to Bruce Kleiner, John Lott, = t T0; it is easy to see from the Ricci flow equation and Jack Milnor for their many suggestions and ˆ that vol(M,gˆ(t)) is uniformly bounded. For ν suf- comments, leading to significant improvements in ficiently small, Perelman proves that there is suf- the paper. ficient control on the ν-thick part Mˆ ν, as defined ˆ ν 6 in (3), to see that M is diffeomorphic to a complete It is not asserted that the bound (2) holds on Mˆν for all t hyperbolic 3-manifold H (with finitely many large, for some Λ < ∞.
192 NOTICES OF THE AMS VOLUME 51, NUMBER 2 References [1] M. ANDERSON, Scalar curvature and geometrization con- jectures for 3-manifolds, in Comparison Geometry (Berkeley 1993–94), MSRI Publications, vol. 30, 1997, pp. 49–82. [2] ——— , Scalar curvature and the existence of geomet- ric structures on 3-manifolds. I, J. Reine Angew. Math. 553 (2002), 125–182. [3] L. ANDERSSON, The global existence problem in general relativity, preprint, 1999, gr-qc/9911032; to appear in 50 Years of the Cauchy Problem in General Relativity, P. T. Chrus´ciel and H. Friedrich, eds. [4] H.-D. CAO and B. CHOW, Recent developments on the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 59–74. [5] E. D’HOKER, String theory, in Quantum Fields and Strings: A Course for Mathematicians, Vol. 2, Amer. Math. Soc., Providence, RI, 1999. [6] M. GROMOV, Structures Métriques pour les Variétés Rie- manniennes, Cedic/Fernand Nathan, Paris, 1981. [7] ——— , Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 50 (1982), 5–100. [8] R. HAMILTON, Three manifolds of positive Ricci curva- ture, J. Differential Geom. 17 (1982), 255–306. [9] ——— , The Harnack estimate for the Ricci flow, J. Dif- ferential Geom. 37 (1993), 225–243. [10] ——— , Formation of singularities in the Ricci flow, Surveys in Differential Geometry, Vol. 2, International Press, 1995, pp. 7–136. [11] ——— , Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), 695–729. [12] T. IVEY, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), 301–307. [13] B. KLEINER and J. LOTT, Ricci flow website: http:// www.math.lsa.umich.edu/research/ricciflow/ perelman.html. [14] J. MILNOR, Towards the Poincaré Conjecture and the classification of 3-manifolds, preprint, 2003; Notices Amer. Math. Soc. 50 (2003), 1226–1233. [15] G. PERELMAN, The entropy formula for the Ricci flow and its geometric applications, preprint, 2002, math.DG/0211159. [16] ——— , Ricci flow with surgery on three-manifolds, preprint, 2003, math.DG/0303109. [17] ——— , Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, 2003, math.DG/0307245. [18] W. THURSTON, The Geometry and Topology of Three- Manifolds, preprint, 1978, Princeton University, avail- able online at http://www.msri.org/publications/ books/gt3m; and Three-Dimensional Geometry and Topology, Vol. 1, Princeton University Press, 1997. [19] ——— , Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N. S.) 6 (1982), 357–381.
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