RRecent developments in Ricci flows
Richard H Bamler ∗ February 26, 2021
The beginning of the new millennium markedan important point in the history of geomet-ric analysis. In a series of three very con-densed papers [Per02, Per03b, Per03], Perel-man presented a solution of the Poincar´e andGeometrization Conjectures. These conjec-tures — the former of which was 100 yearsold and a Millennium Problem — were fun-damental problems concerning the topology of3-manifolds, yet their solution employed Ricciflow, a technique introduced by Hamilton in1982, that combines methods of PDE and Rie-mannian geometry (see below for more de-tails). These spectacular applications werefar from coincidental as they provided a newperspective on 3-manifold topology using thegeometric-analytic language of Ricci flows.It took several years for the scientific com-munity to digest Perelman’s arguments andverify his proof. This was an exciting timefor geometric analysts, since the new tech-niques generated a large number of interestingquestions and potential applications. Amongstthese, perhaps the most intriguing goal was tofind further applications of Ricci flow to prob-lems in topology.This goal will form the central theme of thisarticle. I will first summarize the importantresults pertaining to Perelman’s work. Next, Iwill discuss more recent research that stands inthe same spirit; first in dimension 3, leading,amongst others, to the resolution of the Gen-eralized Smale Conjecture and, second, workin higher dimensions, where topological appli-cations may be forthcoming. There are also ∗ The author is a professor of mathematics at UCBerkeley. His email address is [email protected] thanks Robert Bamler, Paula Burkhardt-Guim,Bennett Chow, Bruce Kleiner, Yi Lai and John Lott forhelpful feedback on an early version of the manuscript. many other interesting recent applications ofRicci flows, which are more of geometric or an-alytic nature, and which I won’t have space tocover here, unfortunately. Among these are,for example, work on manifolds with positiveisotropic curvature [Bre18], smoothing con-structions for limit spaces of lower curvaturebounds [TS, BCRW19, Lai19], lower scalar cur-vature bounds for C -metrics [Bam16, BG19],as well as work on K¨ahler-Ricci flow. A Riemannian metric g on a smooth manifold M is given by a family of inner products onits tangent spaces T p M , p ∈ M , that can beexpressed in local coordinates ( x , . . . , x n ) as g = (cid:88) i,j g ij dx i dx j , where g ij = g ji are smooth coefficient func-tions. A metric allows us to define notionssuch as angles, lengths and distances on M .One can show that the metric g near any point p ∈ M looks Euclidean up to order 2, that iswe can find suitable local coordinates around p such that g ij = δ ij + O ( r ) . The second order terms can be described by aninvariant called the
Riemann curvature tensor Rm iklj , which is independent of the choice ofcoordinates up to a simple transformation rule.Various components of this tensor have localgeometric interpretations. A particularly in-teresting component is called the Ricci tensor
Ric ij , which arises from tracing the k, l indicesof Rm iklj . This tensor roughly describes the1 a r X i v : . [ m a t h . DG ] F e b econd variation of areas of hypersurfaces un-der normal deformations.A Ricci flow on a manifold M is given by asmooth family g ( t ), t ∈ [0 , T ), of Riemannianmetrics satisfying the evolution equation ∂ t g ( t ) = − g ( t ) . (2.1)Expressed in suitable local coordinates, thisequation roughly takes the form of a non-linearheat equation in the coefficient functions: ∂ t g ij = (cid:52) g ij + . . . (2.2)In addition, if we compute the time derivativeof the Riemannian curvature tensor Rm g ( t ) , weobtain ∂ t Rm g ( t ) = (cid:52) Rm g ( t ) + Q (Rm g ( t ) ) , (2.3)where the last term denotes a quadratic term;its exact form will not be important in the se-quel. Equations (2.2), (2.3) suggest that themetric g ( t ) becomes smoother or more homo-geneous as time moves on, similar to the evo-lution of a heat equation, under which heat isdistributed more evenly across its domain. Onthe other hand, the last term in (2.3) seemsto indicate that — possibly at larger scales orin regions of large curvature — this diffusionproperty may be outweighed by some othernon-linear effects, which could lead to singu-larities.But we are getting ahead of ourselves. Let usfirst state the following existence and unique-ness theorem, which was established by Hamil-ton [Ham82] in the same paper in which heintroduced the Ricci flow equation: Theorem 2.1.
Suppose that M is compactand let g be an arbitrary Riemannian metricon M (called the initial condition). Then:1. The evolution equation (2.1) combinedwith the initial condition g (0) = g hasa unique solution ( g ( t )) t ∈ [0 ,T ) for somemaximal T ∈ (0 , ∞ ] .2. If T < ∞ , then we say that the flow de-velops a singularity at time T and the cur-vature blows up: max M | Rm g ( t ) | −−−→ t (cid:37) T ∞ . The most basic examples of Ricci flows arethose in which g is Einstein, i.e. Ric g = λg . In this case the flow simply evolves by rescal-ing: g ( t ) = (1 − λt ) g (2.4)So for example, a round sphere ( λ >
0) shrinksunder the flow and develops a singularity in fi-nite time, where the diameter goes to 0. Onthe other hand, if we start with a hyperbolicmetric ( λ < g and hope that — at least in some cases —the flow is asymptotic to a solution of the form(2.4). This will then allow us to understand thetopology of the underlying manifold in termsof the limiting geometry. In dimension 2, Ricci flows are very well be-haved:
Theorem 3.1.
Any Ricci flow on a com-pact 2-dimensional manifold converges, mod-ulo rescaling, to a metric of constant curva-ture.
In addition, one can show that the flow indimension 2 preserves the conformal class, i.e.for all times t we have g ( t ) = f ( t ) g for somesmooth positive function f ( t ) on M . This ob-servation, combined with Theorem 3.1 can infact be used to prove of the UniformizationTheorem: Theorem 3.2.
Each compact surface M ad-mits a metric of constant curvature in eachconformal class. This is our first topological consequence ofRicci flow. Of course, the Uniformization The-orem has already been known for about 100years. So in order to obtain any new topolog-ical results, we will need to study the flow inhigher dimensions. The proof of Theorem 3.1, which was establishedby Chow and Hamilton [Ham88, Cho91], relied on theUniformization Theorem. This dependence was laterremoved by Chen, Lu, Tian [CLT06]. r , r , r . The flows depicted on the top are the corresponding singu-larity models. These turn out to be the only singularity models, even in the non-rotationallysymmetric case (see Subsection 4.3). In dimension 3, the behavior of the flow — andits singularity formation — becomes far morecomplicated. In this section, I will first discussan example and briefly review Perelman’s anal-ysis of singularity formation and the construc-tion of Ricci flows with surgery. I will keep thispart short and only focus on aspects that willbecome important later; for a more in-depthdiscussion and a more complete list of refer-ences see the earlier Notices article [And04].Next, I will focus on more recent work byKleiner, Lott and the author on singular Ricciflows and their uniqueness and continuous de-pendence, which led to the resolution of severallongstanding topological conjectures.
To get an idea of possible singularity forma-tion of 3-dimensional Ricci flows, it is usefulto consider the famous dumbbell example (see Figure 1). In this example, the initial manifold(
M, g ) is the result of connecting two roundspheres of radii r , r by a certain type of rota-tionally symmetric neck of radius r (see Fig-ure 1). So M ≈ S and g = f ( s ) g S + ds is awarped product away from the two endpoints.It can be shown that any flow starting froma metric of this form must develop a singularityin finite time. The singularity type, however,depends on the choice of the radii r , r , r .More specifically, if the radii r , r , r are com-parable (Figure 1, left), then the diameterof the manifold converges to zero and, af-ter rescaling, the flow becomes asymptoticallyround — just as in Theorem 3.1. This caseis called extinction . On the other hand, if r (cid:28) r , r (Figure 1, right), then the flowdevelops a neck singularity , which looks like around cylinder ( S × R ) at small scales. Notethat in this case the singularity only occurs in acertain region of the manifold, while the met-ric converges to a smooth metric everywhereelse. Lastly, there is also an intermediate case(Figure 1, center), in which the flow develops3igure 2: A Ricci flow M × [0 , T ) that developsa singularity at time T and a sequence of points( x i , t i ) that “run into a singularity”. The ge-ometry in the parabolic neighborhoods around( x i , t i ) (rectangles) is close to the singularitymodel modulo rescaling if i (cid:29) Perelman’s work implied that the previous ex-ample is in fact prototypical for the singularityformation of 3-dimensional Ricci flows start-ing from general, not necessarily rotationallysymmetric, initial data. In order to make thisstatement more precise, let us first recall amethod called blow-up analysis, which is usedfrequently to study singularities in geometricanalysis.Suppose that ( M, ( g ( t )) t ∈ [0 ,T ) ) is a Ricciflow that develops a singularity at time T < ∞ (see Figure 2). Then we can find a sequence ofspacetime points ( x i , t i ) ∈ M × [0 , T ) such that λ i := | Rm | ( x i , t i ) → ∞ and t i (cid:37) T ; we willsay that the points ( x i , t i ) “run into a singular-ity”. Our goal will be to understand the localgeometry at small scales near ( x i , t i ), for large i . For this purpose, we consider the sequenceof pointed, parabolically rescaled flows (cid:0) M, ( g (cid:48) i ( t ) := λ i g ( λ − i t + t i )) t ∈ [ − λ i t i , , ( x i , (cid:1) . Geometrically, the flow ( g (cid:48) i ( t )) is the result ofrescaling distances by λ / i , times by λ i and anapplication of a time-shift such that the point( x i ,
0) in the new flow corresponds the point( x i , t i ) in the old flow. The new flow ( g (cid:48) i ( t ))still satisfies the Ricci flow equation and it is defined on larger and larger backwards time-intervals of size λ i t i → ∞ . Moreover, we have | Rm | ( x i ,
0) = 1 on this new flow. Observealso that the geometry of the original flow near( x i , t i ) at scale λ − / i (cid:28) g (cid:48) i ( t )) near ( x i ,
0) at scale 1.The hope is now to apply a compactness the-orem (`a la Arzela-Ascoli) such that, after pass-ing to a subsequence, we have convergence (cid:0) M, ( g (cid:48) i ( t )) t ∈ [ − λ i t i , , ( x i , (cid:1) −−−→ i →∞ (cid:0) M ∞ , ( g ∞ ( t )) t ≤ , ( x ∞ , (cid:1) . (4.1)The limit is called a blow-up or singularitymodel , as it gives valuable information on thesingularity near the points ( x i , t i ). This modelis a Ricci flow that is defined for all times t ≤ ancient. So in summary,a blow-up analysis reduces the study of singu-larity formation to the classification of ancientsingularity models.
The notion of convergence in (4.1) is a gen-eralization of Cheeger-Gromov convergence toRicci flows, due to Hamilton. Instead of de-manding global convergence of the metric ten-sors, as in Theorem 3.1, we only require con-vergence up to diffeomorphisms here. Weroughly require that we have convergence φ ∗ i g (cid:48) i ( t ) C ∞ loc −−−→ i →∞ g ∞ (4.2)on M ∞ × ( −∞ ,
0] of the pull backs of g (cid:48) i ( t ) via(time-independent) diffeomorphisms φ i : U i → V i ⊂ M that are defined over larger and largersubsets U i ⊂ M ∞ and satisfy φ i ( x ∞ ) = x i . Wewill see later (in Section 5) that this notion ofconvergence is too strong to capture the moresubtle singularity formation of higher dimen-sional Ricci flows and we will discuss necessaryrefinements. Luckily, in dimension 3 the cur-rent notion is still sufficient for our purposes,though. Arguably one of the most groundbreaking dis-coveries of Perelman’s work was the classifi-cation of singularity models of 3-dimensionalRicci flows and the resulting structural de-scription of the flow near a singularity. The4ollowing theorem summarizes this classifica-tion. Theorem 4.1.
Any singularity model ( M ∞ , ( g ∞ ( t )) t ≤ ) obtained as in (4.1) isisometric, modulo rescaling, to one thefollowing:1. A quotient of the round shrinking sphere ( S , (1 − t ) g S ) .2. The round shrinking cylinder ( S × R , (1 − t ) g S + g R ) or its quotient ( S × R ) / Z .3. The Bryant soliton ( M Bry , ( g Bry ( t )))Note that these three models correspond tothe three cases in the rotationally symmet-ric dumbbell example from Subsection 4.1 (seeFigure 1). The Bryant soliton in 3. is a ro-tationally symmetric solution to the Ricci flowon R with the property that all its time-slicesare isometric to a metric of the form g Bry = f ( r ) g S + dr , f ( r ) ∼ √ r. The name soliton refers to the fact that alltime-slices of the flow are isometric, so the flowmerely evolves by pullbacks of a family of dif-feomorphisms (we will encounter this solitonagain at the end of this article).It turns out that we can do even better thanTheorem 4.1: we can describe the structureof the flow near any point of the flow thatis close to a singularity — not just near thepoints used to construct the blow-ups. In or-der to state the next result efficiently, we willneed to consider the class of κ -solutions. Thisclass simply consists of all solutions listed inTheorem 4.1, plus an additional compact, el-lipsoidal solution (the details of this solutionwon’t be important here ). Then we have: Theorem 4.2 (Canonical Neighborhood The-orem) . If ( M, ( g ( t )) t ∈ [0 ,T ) ) , T < ∞ , is a 3-dimensional Ricci flow and ε > , then there Perelman proved a version of Theorem 4.1 thatcontained a more qualitative characterization inCase 3., which was sufficient for most applications.Later, Brendle [Bre20] showed that the only possibilityin Case 3. is the Bryant soliton. This solution does not occur as a singularity of asingle flow, but can be observed as some kind of tran-sitional model in families of flows that interpolate be-tween two different singularity models.
Figure 3: A schematic depiction of a Ricciflow with surgery. The almost-singular parts M almost-sing , i.e. the parts that are discardedunder each surgery construction, are hatched. is a constant r can ( g (0) , T, ε ) > such that forany ( x, t ) ∈ M × [0 , T ) with the property that r := | Rm | − / ( x, t ) ≤ r can the geometry of the metric g ( t ) restricted to theball B g ( t ) ( x, ε − r ) is ε -close to a time-slice ofa κ -solution. Let us digest the content of this theorem.Recall that the norm of the curvature tensor | Rm | , which can be viewed as a measure of“how singular the flow is near a point”, hasthe dimension of length − , so r = | Rm | − / can be viewed as a “curvature scale”. So, inother words, Theorem 4.2 states that regionsof high curvature locally look very much likecylinders, Bryant solitons, round spheres etc. Our understanding of the structure of the flownear a singularity now allows us to carry out aso-called surgery construction.
Under this con-struction (almost) singularities of the flow are Similar to (4.2), this roughly means that there isa diffeomorphism between an ε − -ball in a κ -solutionand this ball such that the pullback of r − g ( t ) is ε -closein the C [ ε − ] -sense to the metric on the κ -solution. (3-dimensional)Ricci flow with surgery (see Figure 3) consistsof a sequence of Ricci flows( M , ( g ( t )) t ∈ [0 ,T ] ) , ( M , ( g ( t )) t ∈ [ T ,T ] ) , ( M , ( g ( t )) t ∈ [ T ,T ] ) , . . . , which live on manifolds M , M , . . . of pos-sibly different topology and are parame-terized by consecutive time-intervals of theform [0 , T ] , [ T , T ] , . . . whose union equals[0 , ∞ ). The time-slices ( M i , g i ( T i )) and( M i +1 , g i +1 ( T i )) are related by a surgery pro-cess, which can be roughly summarized asfollows. Consider the set M almost-sing ⊂ M i of all points of high enough curvature, suchthat they have a canonical neighborhood asin Theorem 4.2. Cut M i open along ap-proximate cross-sectional 2-spheres of diame-ter r surg ( T i ) (cid:28) M almost-sing , discard most of the high-curvature components (including the closed,spherical components of M almost-sing ), and gluein cap-shaped 3-disks to the cutting sur-faces. In doing so we have constructed anew, “less singular”, Riemannian manifold( M i +1 , g i +1 ( T i )), from which we can restartthe flow. Stop at some time T i +1 > T i , shortlybefore another singularity occurs and repeatthe process.The precise surgery construction is quitetechnical and more delicate than presentedhere. The main difficulty in this constructionis to ensure that the surgery times T i do notaccumulate, i.e. that the flow can be extendedfor all times. It was shown by Perelman thatthis and other difficulties can indeed be over-come: Theorem 4.3.
Let ( M, g ) be a closed, 3-dimensional Riemannian manifold. If thesurgery scales r surg ( T i ) > are chosen suf-ficiently small (depending on ( M, g ) and T i ),then a Ricci flow with surgery with initial con-dition ( M , g (0)) = ( M, g ) can be constructed. Note that the topology of the underlyingmanifold M i may change in the course of a surgery, but only in a controlled way. In par-ticular, it is possible to show that for any i theinitial manifold M is diffeomorphic to a con-nected sum of components of M i and copies ofspherical space forms S / Γ and S × S . Soif the flow goes extinct in finite time, meaningthat M i = ∅ for some large i , then M ≈ kj =1 ( S / Γ j ) m ( S × S ) . (4.3)Perelman moreover showed that if M is simplyconnected, then the flow has to go extinct andtherefore M must be of the form (4.3). Thisimmediately implies the Poincar´e Conjecture— our first true topological application of Ricciflow: Theorem 4.4 (Poincar´e Conjecture) . Anysimply connected, closed 3-manifold is diffeo-morphic to S . On the other hand, Perelman showed that ifthe Ricci flow with surgery does not go extinct,meaning if it exists for all times, then for largetimes t (cid:29) t ) into a thick and a thin part: M thick ( t ) ·∪ M thin ( t ) , (4.4)such that the metric on M thick ( t ) is asymp-totic to a hyperbolic metric and the metricon M thin ( t ) is collapsed along fibers of type S , S or T . A further topological analy-sis of these induced fibrations implied the Ge-ometrization Conjecture — our second topo-logical application of Ricci flow: Theorem 4.5 (Geometrization Conjecture) . Every closed 3-manifold is a connected sumof manifolds that can be cut along embedded,incompressible copies of T into pieces whicheach admit a locally homogeneous geometry. Despite their spectacular applications, Ricciflows with surgery have one major drawback:their construction is not canonical. In other The precise description of these fibrations is a bitmore complex and omitted here. In addition, M thick ( t )and M thin ( t ) are separated by embedded 2-tori thatare incompressible, meaning that the inclusion into theambient manifold is an injection on π . ∂ t .words, each surgery step depends on a numberof auxiliary parameters, for which there doesnot seem to be a canonical choice, such as: • The surgery scales r surg ( T i ), i.e. the diam-eters of the cross-sectional spheres alongwhich the manifold is cut open. Thesescales need to be positive and small. • The precise locations of these surgeryspheres.Different choices of these parameters may in-fluence the future development of the flow sig-nificantly (as well as the space of future surgeryparameters). Hence a Ricci flow with surgeryis not uniquely determined by its initial metric.This disadvantage was already recognized inPerelman’s work, where he conjectured thatthere should be another flow, in which surg-eries are effectively carried out automaticallyat an infinitesimal scale (think “ r surg = 0”), orwhich, in other words, “flows through singu-larities”.Perelman’s conjecture was recently re-solved by Kleiner, Lott and the author[KL17, BK17b]: The “Existence” part is due to Kleiner and Lottand the “Uniqueness” part is due to Kleiner and theauthor. Part 2. of Theorem 4.6 follows from a combi-nation of both papers.
Theorem 4.6.
There is a notion of singularRicci flow (through singularities) such that:1. For any compact, 3-dimensional Rieman-nian manifold ( M, g ) there is a uniquesingular Ricci flow M whose initial time-slice ( M , g ) is ( M, g ) .2. Any Ricci flow with surgery starting from ( M, g ) can be viewed as an approximationof M . More specifically, if we considera sequence of Ricci flows with surgerystarting from ( M, g ) with surgery scales max t r surg ( t ) → , then these flows con-verge to M in a certain sense. In addition, the concept of a singular Ricciflow is far less technical than that of a Ricciflow with surgery — in fact, I will be able tostate its full definition here. To do this, I willfirst define the concept of a Ricci flow space-time. In short, this is a smooth 4-manifold thatlocally looks like a Ricci flow, but which mayhave non-trivial global topology (see Figure 4).
Definition 4.7. A Ricci flow spacetime con-sists of:1. A smooth 4-dimensional manifold M withboundary, called spacetime.
2. A time-function t : M → [0 , ∞ ). Its levelsets M t := t − ( t ) are called time-slices and we require that M = ∂ M .3. A time-vector field ∂ t on M with ∂ t · t ≡ ∂ t are called worldlines.
4. A family g of inner products on ker d t ⊂ T M , which induce a Riemannian metric g t on each time-slice M t . We require thatthe Ricci flow equation holds: L ∂ t g t = − g t . By abuse of notation, we will often write M instead of ( M , t , ∂ t , g ).A classical, 3-dimensional Ricci flow( M, ( g ( t )) t ∈ [0 ,T ) ) can be converted into a Ricciflow spacetime by setting M := M × [0 , T ),letting t , ∂ t be the projection onto the secondfactor and the pullback of the unit vector fieldon the second factor, respectively, and letting g t be the metric corresponding to g ( t ) on7 × { t } ≈ M . Hence worldlines correspondto curves of the form t (cid:55)→ ( x, t ).Likewise, a Ricci flow with surgery,given by flows ( M , ( g ( t )) t ∈ [0 ,T ] ) , ( M , ( g ( t )) t ∈ [ T ,T ] ) , . . . can be convertedinto a Ricci flow spacetime as follows.Consider first the Ricci flow spacetimes M × [0 , T ] , M × [ T , T ] , . . . arising fromeach single flow. We can now glue theseflows together by identifying the set of points U − i ⊂ M i × { T i } and U + i ⊂ M i +1 × { T i } that survive each surgery step via maps φ i : U − i → U + i . The resulting space has aboundary that consists of the time-0-slice M × { } and the points S i = ( M i × { T i } \ U − i ) ∪ ( M i +1 × { T i } \ U + i ) , which were removed and added during eachsurgery step. After removing these points, weobtain a Ricci flow spacetime of the form: M = ( M × [0 , T ] ∪ φ M × [ T , T ] ∪ φ . . . ) \ ( S ∪ S ∪ . . . ) . (4.5)Note that for any regular time t ∈ ( T i − , T i )the time-slice M t is isometric to ( M i , g i ( t )).On the other hand, the time-slices M T i corre-sponding to surgery times are incomplete; theyhave cylindrical open ends of scale ≈ r surg ( T i ).The following definition captures this incom-pleteness: Definition 4.8.
A Ricci flow spacetime is r -complete, for some r ≥
0, if the following holds.Consider a smooth path γ : [0 , s ) → M withthe property thatinf s ∈ [0 ,s ) | Rm | − / ( γ ( s )) > r and:1. γ ([0 , l )) ⊂ M t is contained in a singletime-slice and its length measured with re-spect to the metric g t is finite, or2. γ is a worldline, i.e. a trajectory of ± ∂ t .Then the limit lim s (cid:37) s γ ( s ) exists.So M being r -complete roughly means thatit has only “holes” of scale (cid:46) r . For example,the flow from (4.5) is C max t r surg ( t )-completefor some universal C < ∞ .In addition, Theorem 4.2 motivates the fol-lowing definition: Definition 4.9.
A Ricci flow spacetime issaid to satisfy the (cid:15) -canonical neighborhoodassumption at scales ( r , r ) if for any point x ∈ M t with r := | Rm | − / ( x ) ∈ ( r , r ) themetric g t restricted to the ball B g t ( x, ε − r ) is ε -close, after rescaling by r − , to a time-sliceof a κ -solution.We can finally define singular Ricci flows(through singularities), as used in Theo-rem 4.6: Definition 4.10. A singular Ricci flow is aRicci flow spacetime M with the following twoproperties:1. It is 0-complete.2. For any ε > T < ∞ there is an r ( ε, T ) > M re-stricted to [0 , T ) satisfies the ε -canonicalneighborhood assumption at scales (0 , r ).See again Figure 4 for a depiction of a singu-lar Ricci flow. The time-slices M t for t < T sing develop a cylindrical region, which collapses tosome sort of topological double cone singular-ity in the time- T sing -slice M T sing . This singu-larity is immediately resolved and the flow issmooth for all t > T sing .Let us digest the definition of a singularRicci flow a bit more. It is tempting to thinkof the time function t as a Morse functionand compare critical points with infinitesimalsurgeries. However, this comparison is flawed:First, by definition t cannot have critical pointssince ∂ t t = 1. In fact, a singular Ricci flowis a completely smooth object. The “singularpoints” of the flow are not part of M , but canbe obtained after metrically completing eachtime-slice by adding a discrete set of points.Second, it is currently unknown whether theset of singular times, i.e. the set of times whosetime-slices are incomplete, is discrete.Similar notions of singular flows have beendeveloped for the mean curvature flow (a closecousin of the Ricci flow). These are called levelset flows and Brakke flows. However, their def-initions differ from singular Ricci flows in thatthey characterize the flow equation at singularpoints via barrier and weak integral conditions,respectively. This is possible, in part, becausea mean curvature flow is an embedded objectand its singular set has an analytic meaning.8y contrast, the definition of a singular Ricciflow only characterizes the flow on its regu-lar part. In lieu of a weak formulation of theRicci flow equation on the singular set, we haveto impose the canonical neighborhood assump-tion, which serves as an asymptotic character-ization near the incomplete ends.Finally, I will briefly explain how singularRicci flows are constructed and convey themeaning of Part 2. of Theorem 4.6. Fixan initial time-slice ( M, g ) and consider a se-quence of Ricci flow spacetimes M j that corre-spond to Ricci flows with surgery starting from( M, g ), with surgery scale max t r surg ,j ( t ) →
0. It can be shown that these flows are C max t r surg ,j ( t )-complete and satisfy the ε -canonical neighborhood assumption at scales( C ε max t r surg ,j ( t ) , r ε ), where C, C ε , r ε do notdepend on j . A compactness theorem impliesthat a subsequence of the spacetimes M j con-verges to a spacetime M , which is a singularRicci flow. This implies the existence of M ;the proof of uniqueness uses other techniques,which are outside the scope of this article. The proof of the uniqueness property in The-orem 4.6, due to Kleiner and the author, im-plies an important continuity property, whichwill lead to further topological applications.To state this property, let M be a compact3-manifold and for every Riemannian metric g on M let M g be the singular Ricci flow withinitial condition ( M g , g ) = ( M, g ). Theorem 4.11.
The flow M g depends con-tinuously on g . Recall that the topology of the flow M g maychange as we vary g . We therefore have tochoose an appropriate sense of continuity inTheorem 4.11 that allows such a topologicalchange. This is roughly done via a topologyand lamination structure on the disjoint union (cid:70) g M g , transverse to which the variation ofthe flow can be studied locally.Instead of diving into these technicalities, letus discuss the example illustrated in Figure 5.In this example ( g s ) s ∈ [0 , denotes a contin-uous family of metrics on S such that thecorresponding flows M s := M g s interpolatebetween a round and a cylindrical singularity. For s ∈ [0 , ) the flow M s can be describedin terms of a conventional, non-singular Ricciflow ( g st ) on M and the continuity statementin Theorem 4.11 is equivalent to continuousdependence of this flow on s . Likewise, theflows M s restricted to [0 , T sing ) can again bedescribed by a continuous family of conven-tional Ricci flows. The question is now whathappens at the critical parameter s = , wherethe type of singularity changes. The unique-ness property guarantees that the flows M s for s (cid:37) and s (cid:38) must limit to the same flow M . The convergence is locally smooth, butthe topology of the spacetime manifold M s may still change. Theorem 4.11 provides us a tool to deduce thefirst topological applications of Ricci flow sincePerelman’s work.The first example of such an application con-cerns the space of metrics of positive scalarcurvature Met
PSC ( M ) ⊂ Met( M ) on a mani-fold M , which is a subset of the space of allRiemannian metrics on M (both spaces areequipped with the C ∞ -topology). Since thepositive scalar curvature condition is preservedby Ricci flow, Theorem 4.11 roughly impliesthat — modulo singularities and the associatedtopological changes — Ricci flow is a “continu-ous deformation retraction” of Met PSC ( M ) tothe space of round metrics on M . This heuris-tic was made rigorous by Kleiner and the au-thor [BK19] and implied: Theorem 4.12.
For any closed 3-manifold M the space Met
PSC ( M ) is either contractible orempty. The study of the spaces Met
PSC ( M ) was ini-tiated by Hitchin in the 70s and has led tomany interesting results — based on index the-ory — which show that these spaces have non-trivial topology when M is high dimensional.Theorem 4.12 provides first examples of mani-folds of dimension ≥ PSC ( M ) is completely understood;see also prior work by Marques [Mar12].A second topological application concernsthe diffeomorphism group Diff( M ) of a mani-fold M , i.e. the space of all diffeomorphisms9igure 5: A family of singular Ricci flows starting from a continuous family of initial conditions. φ : M → M (again equipped with the C ∞ -topology). The study of these spaces was ini-tiated by Smale, who showed that Diff( S ) ishomotopy equivalent to the orthogonal group O (3), i.e. the set of isometries of S . More gen-erally, we can fix an arbitrary closed manifold M , pick a Riemannian metric g and considerthe natural injection of the isometry groupIsom( M, g ) −→ Diff( M ) . (4.6)The following conjecture allows us to under-stand the homotopy type of Diff( M ) for manyimportant 3-manifolds. Conjecture 4.13 (Generalized Smale Conjec-ture) . Suppose that ( M , g ) is closed and hasconstant curvature K ≡ ± . Then (4.6) is ahomotopy equivalence. This conjecture has had a long history andmany interesting special cases were establishedusing topological methods, including the case M = S by Hatcher and the hyperbolic caseby Gabai. For more background see the firstchapter of [HKMR12].An equivalent version of Conjecture 4.13 isthat Theorem 4.12 remains true if we replaceMet PSC ( M ) by the space Met K ≡± ( M ) of con-stant curvature metrics. This was verified byKleiner and the author [BK17, BK19], whichled to: Theorem 4.14.
The Generalized Smale Con-jecture is true.
The proof of Theorem 4.14 provides a uni-fied treatment of all possible topological cases and it can also be extended to other manifolds M . In addition it is independent of Hatcher’sproof, so it gives an alternative proof in the S -case. n ≥ For a long time, most of the known results ofRicci flows in higher dimensions concerned spe-cial cases, such as K¨ahler-Ricci flows or flowsthat satisfy certain preserved curvature con-ditions.
General flows, on the other hand,were relatively poorly understood. Recently,however, there has been some movement onthis topic — in part, thanks to a slightlydifferent geometric perspective on Ricci flows[Bam20c, Bam20b, Bam20]. The goal of thissection is to convey some of these new ideasand to provide an outlook on possible geomet-ric and topological applications.
Let us start with the following basic question:What would be reasonable singularity mod-els in higher dimensions? One important classof such models are gradient shrinking solitons(GSS). The GSS equation concerns Rieman-nian manifolds (
M, g ) equipped with a poten-tial function f ∈ C ∞ ( M ) and reads:Ric + ∇ f − g = 0 . This generalization of the Einstein equation isinteresting, because it gives rise to an associ-10ted selfsimilar Ricci flow: g ( t ) := | t | φ ∗ t g, t < , where ( φ t : M → M ) t< is the flow of thevector field | t |∇ f .A basic example of a GSS is an Einstein met-ric (Ric = g ), for example a round sphere.In this case g ( t ) just evolves by rescaling andbecomes singular at time 0. A more inter-esting class of examples are round cylinders S k ≥ × R n − k , where g = 2( k − g S k + g R n − k , f = 14 n (cid:88) i = k +1 x i . In this case | t |∇ f generates a family of dila-tions on the R n − k factor and g ( t ) = 2( k − | t | g S k + g R n − k , which is isometric to | t | g . In dimensions n ≤ ) GSS are quotients of roundspheres or cylinders. However, more compli-cated GSS exist in dimensions n ≥ Conjecture 5.1.
For any Ricci flow “most”singularity models are gradient shrinking soli-tons.
This conjecture has been implicit in Hamil-ton’s work from the 90s and a similar result isknown to be true for mean curvature flow. Inthe remainder of this section I will present aresolution of a version of this conjecture.
Let us first discuss an example in order toadjust our expectations in regards to Conjec-ture 5.1. In [App19], Appleton constructs a Euclidean space R n equipped with f = r iscalled a trivial GSS. class of 4-dimensional Ricci flows that de-velop a singularity in finite time, which can bestudied via the blow-up technique from Sub-section 4.2 — this time we even allow therescaling factors to be any sequence of num-bers λ i → ∞ , not just λ i = | Rm | / ( x i , t i ).Appleton obtains the following classification ofall non-trivial blow-up singularity models:1. The Eguchi-Hanson metric, which is Ricciflat and asymptotic to the flat cone R / Z .2. The flat cone R / Z , which has an isolatedorbifold singularity at the origin.3. The quotient M Bry / Z of the Bryant soli-ton, which also has an isolated orbifoldsingularity at its tip.4. The cylinder R P × R .Here the models 1., 2. have to occur as singu-larity models, and it is unknown whether themodels 3., 4. actually do show up. The onlygradient shrinking solitons in this list are 2., 4.Note that the flow on R / Z is constant, buteach time-slice is a metric cone, and thereforeinvariant under rescaling. So we may also viewthis model as a (degenerate) gradient shrinkingsoliton (in this case f = r ).It is conceivable that there are Ricci flowsingularities whose only blow-up models areof type 1., 2. In addition, there are furtherexamples in higher dimensions [Sto19] whoseonly blow-up models that are gradient shrink-ing solitons must be singular and possibly de-generate. This motivates the following revisionof Conjecture 5.1. Conjecture 5.2.
For any Ricci flow “most”singularity models are gradient shrinking soli-tons that may be degenerate and may have asingular set of codimension ≥ . The previous example taught us that in higherdimensions it becomes necessary to consider non-smooth blow-up limits. The usual conver-gence and compactness theory of Ricci flows The flows are defined on non-compact manifolds,but the geometry at infinity is well controlled. compactness and partial regularity theory for Ricci flows, which will enable us to takelimits of arbitrary Ricci flows and study theirstructural properties. This theory was recentlyfound by the author and will lie at the heartof a resolution of Conjecture 5.2.Compactness and partial regularity theoriesare an important feature in many subfields ofgeometric analysis. To gain a better sensefor such theories, let us first review the com-pactness and partial regularity theory for Ein-stein metrics — an important special case,since every Einstein metric corresponds to aRicci flow. Consider a sequence ( M i , g i , x i )of pointed, complete n -dimensional Einsteinmanifolds, Ric g i = λ i g i , | λ i | ≤
1. Then, af-ter passing to a subsequence, these manifolds(or their metric length spaces, to be precise)converge in the Gromov-Hausdorff sense to apointed metric space( M i , d g i , x i ) GH −−−→ i →∞ ( X, d, x ∞ ) . If we now impose the following non-collapsingcondition: vol B ( x, r ) ≥ v > , (5.1)for some uniform r, v >
0, then the limit ad-mits is a regular-singular decomposition X = R ·∪ S such that the following holds:1. R can be equipped with the structure of aRiemannian Einstein manifold ( R , g ∞ ) insuch a way that the restriction d | R equalsthe length metric d g ∞ . In other words,( X, d ) is isometric to the metric comple-tion of ( R , d g ∞ ).2. We have the following estimate on theMinkowski dimension of the singular set:dim M S ≤ n − This theory is due to Cheeger, Colding, Gromov,Naber and Tian.
3. Every tangent cone (i.e. blow-up of (
X, d )pointed at the same point) is in fact a met-ric cone.4. We have a filtration of the singular set S ⊂ S ⊂ . . . ⊂ S n − = S such that dim H S k ≤ k and such that ev-ery x ∈ S k \ S k − has a tangent cone thatsplits off an R k -factor.Interestingly, compactness and partial reg-ularity theories for other geometric equations(e.g. minimal surfaces, harmonic maps, meancurvature flow, . . . ) take a similar form. Thereason for this is that these theories rely ononly a few basic ingredients (e.g. a mono-tonicity formula, an almost cone rigidity the-orem and an ε -regularity theorem), which canbe verified in each setting. A theory for Ricciflows, however, did not exist for a long time,because these basic ingredients are — at least a priori — not available for Ricci flows. Sothis setting required a different approach.Let us now discuss the new compactnessand partial regularity theory for Ricci flows.Before we begin, note that there is an addi-tional complication: Parabolic versions of no-tions like “metric space”, “Gromov-Hausdorffconvergence”, etc. didn’t exist until recently,so they — and a theory surrounding them —first had to be developed. I will discuss thesenew notions in more detail in Subsection 5.5.For now, let us try to get by with some morevague explanations.The first result is a compactness theorem.Consider a sequence of pointed, n -dimensionalRicci flows( M i , ( g i ( t )) t ∈ ( − T i , , ( x i , , where we imagine the basepoints ( x i ,
0) to livein the final time-slices, and suppose that T ∞ :=lim i →∞ T i >
0. Then we have:
Theorem 5.3.
After passing to a subsequence,these flows F -converge to a pointed metric flow: ( M i , ( g i ( t )) , ( x i , F −−−→ i →∞ ( X , ( ν x ∞ ; t )) . Here a “metric flow” can be thought of asa parabolic version of a “metric space”. It issome sort of Ricci flow spacetime (as in Def-inition 4.7) that is allowed to have singular12oints; think, for example, of isolated orbifoldsingularities in every time-slice as in Apple-ton’s example (see Subsection 5.2), or a singu-lar point where a round shrinking sphere goesextinct. The term “ F -convergence” can bethought of as a parabolic version of “Gromov-Hausdorff convergence”.The next theorem concerns the partial reg-ularity of the limit in the non-collapsed case,which we define as the case in which N x i , ( r ∗ ) ≥ − Y ∗ , (5.2)for some uniform constants r ∗ > Y ∗ < ∞ . This condition is the parabolic analogueto (5.1). The quantity N x,t ( r ) is the pointedNash-entropy, which is related to Perelman’s W -functional; it was rediscovered by work ofHein and Naber. Assuming (5.2), we have: Theorem 5.4.
There is a regular-singular de-composition X = R ·∪ S such that:1. The flow on R can be described by asmooth Ricci flow spacetime structure.Moreover, the entire flow X is uniquelydetermined by this structure.2. We have the following dimensional esti-mate dim M ∗ S ≤ ( n + 2) − .
3. Tangent flows (i.e. blow-ups based at afixed point of X ) are (possibly singular)gradient shrinking solitons.4. There is a filtration S ⊂ . . . ⊂ S n − = S with similar properties as in the Einsteincase. A few comments are in order here. First,note that the fact that X is uniquely deter-mined by the smooth Ricci flow spacetimestructure on R is comparable to what we haveobserved in dimension 3 (see Subsection 4.5),where we didn’t even consider the entire flow X . Second, Property 2. involves a parabolicversion of the Minkowski dimension that isnatural for Ricci flows; a precise definitionwould be beyond the scope of this article. Note that the time direction accounts for 2 dimen-sions, which is natural. An interesting case isdimension n = 3, in which we obtain that theset of singular times has dimension ≤ ; thisis in line with what is known in this dimen-sion. Lastly, in Property 3. the role of metriccones is now taken by gradient shrinking soli-tons; these are analogues of metric cones, asboth are invariant under rescaling.The dimensional bounds in Theorem 5.4 areoptimal. In Appleton’s example, the singularset S may consist of an isolated orbifold pointin every time-slice; so its parabolic dimensionis 2 = (4 + 2) −
4. On the other hand, a flow on S × T develops a singularity at a single timeand collapses to the 2-torus T , which againhas parabolic dimension 2. Theorems 5.3 and 5.4 finally enable us to studythe finite-time singularity formation and long-time behavior of Ricci flows in higher dimen-sions.Regarding Conjecture 5.2, we roughly ob-tain:
Theorem 5.5.
Suppose that ( M, ( g ( t )) t ∈ [0 ,T ) ) develops a singularity at time T < ∞ . Thenwe can extend this flow by a “singular time- T -slice” ( M T , d T ) such that the tangent flows atany ( x, T ) ∈ M T are (possibly singular) gradi-ent shrinking solitons. Regarding the long-time asymptotics, we ob-tain the following picture, which closely resem-bles that in dimension 3 compare with (4.4) inSubsection 4.4:
Theorem 5.6.
Suppose that ( M, ( g ( t )) t ≥ ) isimmortal. Then for t (cid:29) we have a thick-thindecomposition M = M thick ( t ) ·∪ M thin ( t ) such that the flow on M thick ( t ) converges, af-ter rescaling, to a singular Einstein metric (Ric g ∞ = − g ∞ ) and the flow on M thin ( t ) iscollapsed in the opposite sense of (5.2). Theorems 5.5 and 5.6 essentially generalizePerelman’s results to higher dimensions.13 .5 Metric flows
So what precisely is a metric flow? To answerthis question, we will imitate the process ofpassing from a (smooth) Riemannian manifold(
M, g ) to its metric length space (
M, d g ). Herea new perspective on the geometry of Ricciflows will be key.So our goal will be to turn a Ricci flow( M, ( g ( t )) t ∈ I ) into a synthetic object, whichwe call “metric flow”. To do this, let us firstconsider the spacetime X := X × I and thetime-slices X t := X × { t } equipped with thelength metrics d t := d g ( t ) . It may be temptingto retain the product structure X × I on X ,i.e. to record the set of worldlines t (cid:55)→ ( x, t ).However, this turns out to be unnatural. In-stead, we will view the time-slices ( X t , d t ) asseparate, metric spaces, whose points may noteven be in 1-1 correspondence to some givenspace X .It remains to record some other relation be-tween these metric spaces ( X t , d t ). This will bedone via the conjugate heat kernel K ( x, t ; y, s )— an important object in the study of Ricciflows. For fixed ( x, t ) ∈ M × I and s < t thiskernel satisfies the backwards conjugate heatequation on a Ricci flow background:( − ∂ s − (cid:52) g ( s ) + R g ( s ) ) K ( x, t ; · , s ) = 0 , (5.3)centered at ( x, t ). This kernel has the propertythat for any ( x, t ) and s < t ˆ M K ( x, t ; · , s ) dg ( s ) = 1 , which motivates the definition of the followingprobability measures: dν ( x,t ); s := K ( x, t ; · , s ) dg ( s ) , ν ( x,t ); t = δ x . This is the additional information that we willrecord. So we define:
Definition 5.7. A metric flow is (essen-tially ) given by a pair (cid:0) ( X t , d t ) t ∈ I , ( ν x ; s ) x ∈X t ,s Figure 6: The Bryant soliton and a conjugateheat kernel starting at ( x Bry , x ∈ X t , y ∈ X s at two times s < t , it is not possible to say whether “ y cor-responds to x ”. Instead, we only know that “ y belongs to the past of x with a probability den-sity of dν x ; s ( y )”. This definition is surprisinglyfruitful. For example, it is possible to use themeasures ν x ; s to define a natural topology on X and to understand when and in what sensethe geometry of time-slices X t depends contin-uously on t .The concept of metric flows also allows thedefinition of a natural notion of geometric con-vergence — F -convergence — which is similarto Gromov-Hausdorff convergence. Even bet-ter, this notion can be phrased on terms ofa certain d F -distance, which is similar to theGromov-Hausdorff distance, and the Compact-ness Theorem 5.3 can be expressed as a state-ment on the compactness of a certain subsetof metric flow (pairs), just as in the case ofGromov-Hausdorff compactness.Lastly, I will sketch an example that il-lustrates why it was so important that wehave divorced ourselves from the conceptof worldlines. Consider the Bryant soli-ton ( M Bry , ( g Bry ( t )) t ≤ ) (see Figure 6) fromSubsection 4.1. Recall that every time-slice( M Bry , g Bry ( t )) is isometric to the same rota-tionally symmetric model with center x Bry . ByTheorems 5.3 and 5.4 any pointed sequence of Strictly speaking, F -convergence and d F -distanceconcern metric flow pairs , ( X , ( ν x ; t )), where the secondentry serves as some kind of substitute of a basedpoint. downs ( λ i → M Bry , ( λ i g Bry ( λ − i t )) t ≤ , ( x Bry , , F -converges to a pointed metric flow X thatis regular on a large set. What is this F -limit X ? For any fixed time t < M Bry , λ i g Bry ( λ − i t ) , x Bry ) converges toa pointed ray of the form ([0 , ∞ ) , x Bry , t ) corresponding to the “official” base-point ( x Bry , 0) at time t . Instead, we have tofocus on the “past” of ( x Bry , M Bry , λ i g Bry ( λ − i t )) where the conjugateheat kernel ν ( x Bry , λ − i t is concentrated. Thisregion is cylindrical of scale ∼ (cid:112) | t | , becausethe conjugate heat kernel “drifts away from thetip” at an approximate linear rate. In fact, onecan show that the blow-down limit X is isomet-ric to a round shrinking cylinder that developsa singularity at time 0. While this may seemslightly less intuitive at first, it turns out to bea much more natural way of looking at it. This new theory demonstrates that, at least onan analytical level, Ricci flows behave similarlyin higher dimension as they do in dimension3. However, while there are only a handfulof possible singularity models in dimension 3,gaining a full understanding of all such modelsin higher dimensions (e.g. classifying gradientshrinking solitons) seems like an intimidatingtask. Some past work in dimension 4 (e.g. byMunteanu and Wang) has demonstrated thatmost non-compact gradient shrinking solitonshave ends that are either cylindrical or conical.This motivates the following conjecture: Conjecture 5.8. Given a closed Riemannian4-manifold ( M, g ) there is a certain kind of“Ricci flow through singularities” in whichtopological change occurs along cylinders orcones and in which time-slices are allowed tohave isolated orbifold singularities. Showing the existence of such a flow wouldbe an analytical challenge, given that the con-struction in dimension 3 already filled several hundred pages. However, I currently don’t seea reason why such a flow should not exist.Even more exciting would be the questionwhat such a flow (or an analogue in higher di-mensions) would accomplish on a topologicallevel. It is unlikely that it would allow us toprove the smooth Poincar´e Conjecture in di-mension 4, because even in the best possiblecase, such a flow seems to provide insufficienttopological information. A more feasible appli-cation would the -Conjecture, which statesthat for every closed spin 4-manifold we have b ( M ) ≥ | σ ( M ) | . (5.4)This conjecture is the missing piece in the clas-sification of closed, simply connected, smooth4-manifolds up to homeomorphy (due to Don-aldson, Freedman and Kirby). At least on aheuristic level, it would be suited for a Ricciflow approach since (closed) Einstein mani-folds and spin gradient shrinking solitons au-tomatically satisfy (5.4) — due to the Hitchin-Thorpe inequality in the Einstein case or thefact that gradient shrinking solitons have posi-tive scalar curvature. Other potential applica-tions would be questions concerning the topol-ogy of 4-manifolds that admit metrics of posi-tive scalar curvature. Lastly, there also seemsto be potential in K¨ahler geometry, for exam-ple towards the Minimal Model Program andthe Abundance Conjecture.Time will tell how far Ricci flow methodswill take us precisely. Almost 20 years ago,geometric analysts and topologists were busydigesting Perelman’s work on the Poincar´e andGeometrization Conjectures, thus closing animportant chapter in the field. Today, wehave good reasons to be optimistic that fur-ther topological applications are on the hori-zon. 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