perelman ricci flows y conj poincaré

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in dimension four converge, modulo scaling, to metrics of constant positive

curvature.Without assumptions on curvature the long time behavior of the metricevolving by Ricci flow may be more complicated. In particular, as t ap-proaches some finite time T , the curvatures may become arbitrarily large insome region while staying bounded in its complement. In such a case, it isuseful to look at the blow up of the solution for t close to T at a point wherecurvature is large (the time is scaled with the same factor as the metric ten-sor). Hamilton [H 9] proved a convergence theorem , which implies that asubsequence of such scalings smoothly converges (modulo diffeomorphisms)to a complete solution to the Ricci flow whenever the curvatures of the scaledmetrics are uniformly bounded (on some time interval), and their injectivity

radii at the origin are bounded away from zero; moreover, if the size of thescaled time interval goes to infinity, then the limit solution is ancient, thatis defined on a time interval of the form (−∞, T ). In general it may be hardto analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton[H 4] proved that in dimension three, at the points where scalar curvatureis large, the negative part of the curvature tensor is small compared to thescalar curvature, and therefore the blow-up limits have necessarily nonneg-ative sectional curvature. On the other hand, Hamilton [H 3] discovered aremarkable property of solutions with nonnegative curvature operator in ar-bitrary dimension, called a differential Harnack inequality, which allows, in

particular, to compare the curvatures of the solution at different points anddifferent times. These results lead Hamilton to certain conjectures on thestructure of the blow-up limits in dimension three, see [H 4,§26]; the presentwork confirms them.

The most natural way of forming a singularity in finite time is by pinchingan (almost) round cylindrical neck. In this case it is natural to make a surgeryby cutting open the neck and gluing small caps to each of the boundaries, andthen to continue running the Ricci flow. The exact procedure was describedby Hamilton [H 5] in the case of four-manifolds, satisfying certain curvatureassumptions. He also expressed the hope that a similar procedure wouldwork in the three dimensional case, without any a priory assumptions, and

that after finite number of surgeries, the Ricci flow would exist for all timet → ∞, and be nonsingular, in the sense that the normalized curvaturesRm(x, t) = tRm(x, t) would stay bounded. The topology of such nonsingularsolutions was described by Hamilton [H 6] to the extent sufficient to makesure that no counterexample to the Thurston geometrization conjecture can

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occur among them. Thus, the implementation of Hamilton program would

imply the geometrization conjecture for closed three-manifolds.In this paper we carry out some details of Hamilton program. The moretechnically complicated arguments, related to the surgery, will be discussedelsewhere. We have not been able to confirm Hamilton’s hope that the so-lution that exists for all time t → ∞ necessarily has bounded normalizedcurvature; still we are able to show that the region where this does not holdis locally collapsed with curvature bounded below; by our earlier (partlyunpublished) work this is enough for topological conclusions.

Our present work has also some applications to the Hamilton-Tian conjecture concerning Kahler-Ricci flow on Kahler manifolds with positive firstChern class; these will be discussed in a separate paper.

2. The Ricci flow has also been discussed in quantum field theory, as an ap-proximation to the renormalization group (RG) flow for the two-dimensionalnonlinear σ-model, see [Gaw,§3] and references therein. While my back-ground in quantum physics is insufficient to discuss this on a technical level,I would like to speculate on the Wilsonian picture of the RG flow.

In this picture, t corresponds to the scale parameter; the larger is t, thelarger is the distance scale and the smaller is the energy scale; to computesomething on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale. In otherwords, decreasing of t should correspond to looking at our Space through

a microscope with higher resolution, where Space is now described not bysome (riemannian or any other) metric, but by an hierarchy of riemannianmetrics, connected by the Ricci flow equation. Note that we have a paradoxhere: the regions that appear to be far from each other at larger distancescale may become close at smaller distance scale; moreover, if we allow Ricciflow through singularities, the regions that are in different connected compo-nents at larger distance scale may become neighboring when viewed throughmicroscope.

Anyway, this connection between the Ricci flow and the RG flow sug-gests that Ricci flow must be gradient-like; the present work confirms thisexpectation.

3. The paper is organized as follows. In §1 we explain why Ricci flow can beregarded as a gradient flow. In §2, 3 we prove that Ricci flow, considered asa dynamical system on the space of riemannian metrics modulo diffeomor-phisms and scaling, has no nontrivial periodic orbits. The easy (and known)

3



case of metrics with negative minimum of scalar curvature is treated in §2;

the other case is dealt with in §3, using our main monotonicity formula (3.4)and the Gaussian logarithmic Sobolev inequality, due to L.Gross. In §4 weapply our monotonicity formula to prove that for a smooth solution on afinite time interval, the injectivity radius at each point is controlled by thecurvatures at nearby points. This result removes the major stumbling blockin Hamilton’s approach to geometrization. In §5 we give an interpretationof our monotonicity formula in terms of the entropy for certain canonicalensemble. In §6 we try to interpret the formal expressions , arising in thestudy of the Ricci flow, as the natural geometric quantities for a certainRiemannian manifold of potentially infinite dimension. The Bishop-Gromovrelative volume comparison theorem for this particular manifold can in turn

be interpreted as another monotonicity formula for the Ricci flow. This for-mula is rigorously proved in §7; it may be more useful than the first onein local considerations. In §8 it is applied to obtain the injectivity radiuscontrol under somewhat different assumptions than in §4. In §9 we considerone more way to localize the original monotonicity formula, this time usingthe differential Harnack inequality for the solutions of the conjugate heatequation, in the spirit of Li-Yau and Hamilton. The technique of §9 and thelogarithmic Sobolev inequality are then used in §10 to show that Ricci flowcan not quickly turn an almost euclidean region into a very curved one, nomatter what happens far away. The results of sections 1 through 10 require

no dimensional or curvature restrictions, and are not immediately related toHamilton program for geometrization of three manifolds.The work on details of this program starts in §11, where we describe

the ancient solutions with nonnegative curvature that may occur as blow-uplimits of finite time singularities ( they must satisfy a certain noncollaps-ing assumption, which, in the interpretation of §5, corresponds to havingbounded entropy). Then in §12 we describe the regions of high curvatureunder the assumption of almost nonnegative curvature, which is guaranteedto hold by the Hamilton and Ivey result, mentioned above. We also prove,under the same assumption, some results on the control of the curvaturesforward and backward in time in terms of the curvature and volume at a

given time in a given ball. Finally, in §13 we give a brief sketch of the proof of geometrization conjecture.

The subsections marked by * contain historical remarks and references.See also [Cao-C] for a relatively recent survey on the Ricci flow.

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1 Ricci flow as a gradient flow

1.1. Consider the functional F = M

(R + |∇f |2)e−f dV for a riemannianmetric gij and a function f on a closed manifold M . Its first variation canbe expressed as follows:

δ F (vij, h) =

M

e−f [−v + ∇i∇ jvij − Rijvij

−vij∇if ∇ jf + 2 < ∇f, ∇h > +(R + |∇f |2)(v/2 − h)]

=

M

e−f [−vij(Rij + ∇i∇ jf ) + (v/2 − h)(2f − |∇f |2 + R)],

where δgij = vij , δf = h, v = gij

vij . Notice that v/2 − h vanishes identicallyiff the measure dm = e−f dV is kept fixed. Therefore, the symmetric tensor−(Rij+∇i∇ jf ) is the L2 gradient of the functional F m =

M

(R + |∇f |2)dm,where now f denotes log(dV/dm). Thus given a measure m , we may considerthe gradient flow (gij)t = −2(Rij + ∇i∇ jf ) for F m. For general m this flowmay not exist even for short time; however, when it exists, it is just theRicci flow, modified by a diffeomorphism. The remarkable fact here is thatdifferent choices of m lead to the same flow, up to a diffeomorphism; that is,the choice of m is analogous to the choice of gauge.

1.2 Proposition. Suppose that the gradient flow for F m exists for t ∈ [0, T ].

Then at t = 0 we have F m

≤ n2T M dm.

Proof. We may assume M

dm = 1. The evolution equations for thegradient flow of F m are

(gij)t = −2(Rij + ∇i∇ jf ), f t = −R − f, (1.1)

and F m satisfies

F mt = 2

|Rij + ∇i∇ jf |2dm (1.2)

Modifying by an appropriate diffeomorphism, we get evolution equations

(gij)t = −2Rij , f t = −f + |∇f |2

− R, (1.3)

and retain (1.2) in the form

F t = 2

|Rij + ∇i∇ jf |2e−f dV (1.4)

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Now we compute

F t ≥ 2

n

(R + f )2e−f dV ≥ 2

n(

(R + f )e−f dV )2 =

2

nF 2,

and the proposition follows.1.3 Remark. The functional F m has a natural interpretation in terms

of Bochner-Lichnerovicz formulas. The classical formulas of Bochner (forone-forms) and Lichnerovicz (for spinors) are ∇∗∇ui = (d∗d + dd∗)ui − Riju jand ∇∗∇ψ = δ 2ψ − 1/4Rψ. Here the operators ∇∗ , d∗ are defined usingthe riemannian volume form; this volume form is also implicitly used inthe definition of the Dirac operator δ via the requirement δ ∗ = δ. A rou-tine computation shows that if we substitute dm = e−f dV for dV , we getmodified Bochner-Lichnerovicz formulas ∇∗m∇ui = (d∗md + dd∗m)ui − Rm

ij u jand ∇∗m∇ψ = (δ m)2ψ − 1/4Rmψ, where δ mψ = δψ − 1/2(∇f ) · ψ , Rm

ij =Rij+∇i∇ jf , Rm = 2f −|∇f |2+R. Note that gijRm

ij = R+f = Rm. How-ever, we do have the Bianchi identity ∇∗m

i Rmij = ∇iR

mij −Rij∇if = 1/2∇ jRm.

Now F m = M

Rmdm = M

gijRmij dm.

1.4* The Ricci flow modified by a diffeomorphism was considered byDeTurck, who observed that by an appropriate choice of diffeomorphism onecan turn the equation from weakly parabolic into strongly parabolic, thusconsiderably simplifying the proof of short time existence and uniqueness; anice version of DeTurck trick can be found in [H 4,§6].

The functional F and its first variation formula can be found in theliterature on the string theory, where it describes the low energy effectiveaction; the function f is called dilaton field; see [D,§6] for instance.

The Ricci tensor Rmij for a riemannian manifold with a smooth measure

has been used by Bakry and Emery [B-Em]. See also a very recent paper[Lott].

2 No breathers theorem I

2.1. A metric gij(t) evolving by the Ricci flow is called a breather, if for some

t1 < t2 and α > 0 the metrics αgij(t1) and gij(t2) differ only by a diffeomor-phism; the cases α = 1, α < 1, α > 1 correspond to steady, shrinking andexpanding breathers, respectively. Trivial breathers, for which the metricsgij(t1) and gij(t2) differ only by diffeomorphism and scaling for each pair of

6



t1 and t2, are called Ricci solitons. (Thus, if one considers Ricci flow as a dy-

namical system on the space of riemannian metrics modulo diffeomorphismand scaling, then breathers and solitons correspond to periodic orbits andfixed points respectively). At each time the Ricci soliton metric satisfies anequation of the form Rij + cgij + ∇ib j + ∇ jbi = 0, where c is a number andbi is a one-form; in particular, when bi =

12∇ia for some function a on M, we

get a gradient Ricci soliton. An important example of a gradient shrinkingsoliton is the Gaussian soliton, for which the metric gij is just the euclideanmetric on Rn, c = 1 and a = −|x|2/2.

In this and the next section we use the gradient interpretation of the Ricciflow to rule out nontrivial breathers (on closed M ). The argument in thesteady case is pretty straightforward; the expanding case is a little bit more

subtle, because our functional F is not scale invariant. The more difficultshrinking case is discussed in section 3.

2.2. Define λ(gij) = inf F (gij, f ), where infimum is taken over all smooth f ,satisfying

M

e−f dV = 1. Clearly, λ(gij) is just the lowest eigenvalue of theoperator −4+R. Then formula (1.4) implies that λ(gij(t)) is nondecreasingin t, and moreover, if λ(t1) = λ(t2), then for t ∈ [t1, t2] we have Rij+∇i∇ jf =0 for f which minimizes F . Thus a steady breather is necessarily a steadysoliton.

2.3. To deal with the expanding case consider a scale invariant version¯λ(gij) = λ(gij)V

2/n

(gij). The nontrivial expanding breathers will be ruledout once we prove the followingClaim λ is nondecreasing along the Ricci flow whenever it is nonpositive;

moreover, the monotonicity is strict unless we are on a gradient soliton.(Indeed, on an expanding breather we would necessarily have dV /dt > 0

for some t∈[t1, t2]. On the other hand, for every t, − ddt

logV = 1V

RdV ≥

λ(t), so λ can not be nonnegative everywhere on [t1, t2], and the claim ap-plies.)

Proof of the claim.

dλ(t)/dt ≥ 2V 2/n

|Rij + ∇i∇ jf |2e−f dV + 2

nV (2−n)/nλ

−RdV ≥

2V 2/n[ |Rij + ∇i∇ jf − 1

n(R + f )gij |2e−f dV +1n(

(R + f )2e−f dV − (

(R + f )e−f dV )2)] ≥ 0,

where f is the minimizer for F .

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2.4. The arguments above also show that there are no nontrivial (that is with

non-constant Ricci curvature) steady or expanding Ricci solitons (on closedM ). Indeed, the equality case in the chain of inequalities above requires thatR+f be constant on M ; on the other hand, the Euler-Lagrange equation forthe minimizer f is 2f −|∇f |2 + R = const. Thus, f −|∇f |2 = const = 0,because

(f − |∇f |2)e−f dV = 0. Therefore, f is constant by the maximum

principle.

2.5*. A similar, but simpler proof of the results in this section, follows im-mediately from [H 6,§2], where Hamilton checks that the minimum of RV

2

n

is nondecreasing whenever it is nonpositive, and monotonicity is strict unlessthe metric has constant Ricci curvature.

3 No breathers theorem II

3.1. In order to handle the shrinking case when λ > 0, we need to replaceour functional F by its generalization, which contains explicit insertions of the scale parameter, to be denoted by τ. Thus consider the functional

W (gij, f , τ ) =

M

[τ (|∇f |2 + R) + f − n](4πτ )−n

2 e−f dV , (3.1)

restricted to f satisfying

M

(4πτ )−n

2 e−f dV = 1, (3.2)

τ > 0. Clearly W is invariant under simultaneous scaling of τ and gij. Theevolution equations, generalizing (1.3) are

(gij)t = −2Rij , f t = −f + |∇f |2 − R + n

2τ , τ t = −1 (3.3)

The evolution equation for f can also be written as follows: ∗u = 0, whereu = (4πτ )−

n

2 e−f , and ∗ = −∂/∂t − + R is the conjugate heat operator.

Now a routine computation gives

dW /dt =

M

2τ |Rij + ∇i∇ jf − 1

2τ gij|2(4πτ )−

n

2 e−f dV . (3.4)

Therefore, if we let µ(gij, τ ) = inf W (gij , f , τ ) over smooth f satisfying (3.2),and ν (gij) = inf µ(gij, τ ) over all positive τ, then ν (gij(t)) is nondecreasing

8



along the Ricci flow. It is not hard to show that in the definition of µ there

always exists a smooth minimizer f (on a closed M ). It is also clear thatlimτ →∞ µ(gij, τ ) = +∞ whenever the first eigenvalue of −4 + R is positive.Thus, our statement that there is no shrinking breathers other than gradientsolitons, is implied by the following

Claim For an arbitrary metric gij on a closed manifold M, the function µ(gij, τ ) is negative for small τ > 0 and tends to zero as τ tends to zero.

Proof of the Claim. (sketch) Assume that τ > 0 is so small that Ricciflow starting from gij exists on [0, τ ]. Let u = (4πτ )−

n

2 e−f be the solutionof the conjugate heat equation, starting from a δ -function at t = τ , τ (t) =τ − t. Then W (gij(t), f (t), τ (t)) tends to zero as t tends to τ , and thereforeµ(gij, τ )

≤ W (gij(0), f (0), τ (0)) < 0 by (3.4).

Now let τ → 0 and assume that f τ are the minimizers, such that

W (1

2τ −1gij, f τ ,

1

2) = W (gij , f τ , τ ) = µ(gij, τ ) ≤ c < 0.

The metrics 12τ −1gij ”converge” to the euclidean metric, and if we could

extract a converging subsequence from f τ , we would get a function f on Rn,such that

Rn

(2π)−n

2 e−f dx = 1 and

Rn

[1

2|∇f |2 + f − n](2π)−

n

2 e−f dx < 0

The latter inequality contradicts the Gaussian logarithmic Sobolev inequality,due to L.Gross. (To pass to its standard form, take f = |x|2/2 − 2log φ andintegrate by parts) This argument is not hard to make rigorous; the detailsare left to the reader.

3.2 Remark. Our monotonicity formula (3.4) can in fact be used toprove a version of the logarithmic Sobolev inequality (with description of the equality cases) on shrinking Ricci solitons. Indeed, assume that a metricgij satisfies Rij − gij − ∇ib j − ∇ jbi = 0. Then under Ricci flow, gij(t) isisometric to (1 − 2t)gij(0), µ(gij(t), 12 − t) = µ(gij(0), 12), and therefore themonotonicity formula (3.4) implies that the minimizer f for µ(gij, 1

2) satisfies

Rij + ∇i∇ jf − gij = 0. Of course, this argument requires the existence of minimizer, and justification of the integration by parts; this is easy if M is closed, but can also be done with more efforts on some complete M , forinstance when M is the Gaussian soliton.

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3.3* The no breathers theorem in dimension three was proved by Ivey [I];

in fact, he also ruled out nontrivial Ricci solitons; his proof uses the almostnonnegative curvature estimate, mentioned in the introduction.Logarithmic Sobolev inequalities is a vast area of research; see [G] for a

survey and bibliography up to the year 1992; the influence of the curvaturewas discussed by Bakry-Emery [B-Em]. In the context of geometric evolutionequations, the logarithmic Sobolev inequality occurs in Ecker [E 1].

4 No local collapsing theorem I

In this section we present an application of the monotonicity formula (3.4)

to the analysis of singularities of the Ricci flow.4.1. Let gij(t) be a smooth solution to the Ricci flow (gij)t = −2Rij on [0, T ).We say that gij(t) is locally collapsing at T, if there is a sequence of timestk → T and a sequence of metric balls Bk = B( pk, rk) at times tk, such thatr2k/tk is bounded, |Rm|(gij(tk)) ≤ r−2k in Bk and r−nk V ol(Bk) → 0.

Theorem. If M is closed and T < ∞, then gij(t) is not locally collapsing at T.

Proof. Assume that there is a sequence of collapsing balls Bk = B( pk, rk)at times tk → T. Then we claim that µ(gij(tk), r2k) → −∞. Indeed onecan take f k(x) = − log φ(disttk(x, pk)r−1k ) + ck, where φ is a function of one

variable, equal 1 on [0, 1/2], decreasing on [1/2, 1], and very close to 0 on[1, ∞), and ck is a constant; clearly ck → −∞ as r−nk V ol(Bk) → 0. Therefore,applying the monotonicity formula (3.4), we get µ(gij(0), tk + r2

k) → −∞.However this is impossible, since tk + r2k is bounded.

4.2. Definition We say that a metric gij is κ-noncollapsed on the scale ρ, if every metric ball B of radius r < ρ, which satisfies |Rm|(x) ≤ r−2 for every x ∈ B, has volume at least κrn.

It is clear that a limit of κ-noncollapsed metrics on the scale ρ is alsoκ-noncollapsed on the scale ρ; it is also clear that α2gij is κ-noncollapsedon the scale αρ whenever gij is κ-noncollapsed on the scale ρ. The theorem

above essentially says that given a metric gij on a closed manifold M andT < ∞, one can find κ = κ(gij , T ) > 0, such that the solution gij(t) to theRicci flow starting at gij is κ-noncollapsed on the scale T 1/2 for all t ∈ [0, T ),provided it exists on this interval. Therefore, using the convergence theoremof Hamilton, we obtain the following

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Corollary. Let gij(t), t ∈ [0, T ) be a solution to the Ricci flow on a closed

manifold M, T < ∞. Assume that for some sequences tk → T, pk ∈ M and some constant C we have Qk = |Rm|( pk, tk) → ∞ and |Rm|(x, t) ≤ CQk,whenever t < tk. Then (a subsequence of) the scalings of gij(tk) at pk with

factors Qk converges to a complete ancient solution to the Ricci flow, which is κ-noncollapsed on all scales for some κ > 0.

5 A statistical analogy

In this section we show that the functional W , introduced in section 3, is ina sense analogous to minus entropy.

5.1 Recall that the partition function for the canonical ensemble at tem-perature β −1 is given by Z =

exp(−βE )dω(E ), where ω(E ) is a ”densityof states” measure, which does not depend on β. Then one computes theaverage energy < E >= − ∂

∂β log Z, the entropy S = β < E > +log Z, and

the fluctuation σ =< (E − < E >)2 >= ∂ 2

(∂β )2 log Z.Now fix a closed manifold M with a probability measure m, and suppose

that our system is described by a metric gij(τ ), which depends on the temper-ature τ according to equation (gij)τ = 2(Rij +∇i∇ jf ), where dm = udV, u =(4πτ )−

n

2 e−f , and the partition function is given by log Z =

(−f + n2

)dm.(We do not discuss here what assumptions on gij guarantee that the corre-sponding ”density of states” measure can be found) Then we compute

< E >= −τ 2 M

(R + |∇f |2 − n

2τ )dm,

S = − M

(τ (R + |∇f |2) + f − n)dm,

σ = 2τ 4 M

|Rij + ∇i∇ jf − 1

2τ gij |2dm

Alternatively, we could prescribe the evolution equations by replacing thet-derivatives by minus τ -derivatives in (3.3 ), and get the same formulas for

Z,< E >, S, σ, with dm replaced by udV.Clearly, σ is nonnegative; it vanishes only on a gradient shrinking soliton.

< E > is nonnegative as well, whenever the flow exists for all sufficientlysmall τ > 0 (by proposition 1.2). Furthermore, if (a) u tends to a δ -functionas τ → 0, or (b) u is a limit of a sequence of functions ui, such that each ui

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to zero (mod N −1). The heat equation and the conjugate heat equation on

M can be interpreted via Laplace equation on ˜M for functions and volumeforms respectively: u satisfies the heat equation on M iff u (the extension of

u to M constant along the SN fibres) satisfies u = 0 mod N −1; similarly, u

satisfies the conjugate heat equation on M iff u∗ = τ −N −1

2 u satisfies u∗ =0 mod N −1 on M .

6.2 Starting from g, we can also construct a metric gm on M, isometricto g (mod N −1), which corresponds to the backward m-preserving Ricci flow( given by equations (1.1) with t-derivatives replaced by minus τ -derivatives,dm = (4πτ )−

n

2 e−f dV ). To achieve this, first apply to g a (small) diffeomor-phism, mapping each point (xi, yα, τ ) into (xi, yα, τ (1 − 2f

N )); we would get a

metric gm, with components (mod N −1)

gmij = gij, gm

αβ = (1 − 2f

N )gαβ , gm

00 = g00 − 2f τ − f

τ , gm

i0 = −∇if, gmiα = gm

α0 = 0;

then apply a horizontal (that is, along the M factor) diffeomorphism to getgm satisfying (gm

ij )τ = 2(Rij + ∇i∇ jf ); the other components of gm become(mod N −1)

gmαβ = (1− 2f

N )gαβ , gm

00 = gm00−|∇f |2 =

1

τ (

N

2 − [τ (2f −|∇f |2 + R) + f −n]),

gmi0 = gm

α0 = gmiα = 0

Note that the hypersurface τ =const in the metric gm has the volume formτ N/2e−f times the canonical form on M and SN , and the scalar curvatureof this hypersurface is 1

τ (N 2 + τ (2f − |∇f |2 + R) + f ) mod N −1. Thus the

entropy S multiplied by the inverse temperature β is essentially minus thetotal scalar curvature of this hypersurface.

6.3 Now we return to the metric g and try to use its Ricci-flatness byinterpreting the Bishop-Gromov relative volume comparison theorem. Con-sider a metric ball in ( M, g) centered at some point p where τ = 0. Thenclearly the shortest geodesic between p and an arbitrary point q is alwaysorthogonal to the SN fibre. The length of such curve γ (τ ) can be computed

as τ (q)0

N

2τ + R + |γ M (τ )|2dτ

=

2Nτ (q ) + 1√

2N

τ (q)0

√ τ (R + |γ M (τ )|2)dτ + O(N −

3

2 )

13



Thus a shortest geodesic should minimize L(γ ) = τ (q)

0

√ τ (R + |γ M (τ )|2)dτ ,

an expression defined entirely in terms of M . Let L(q M ) denote the corre-sponding infimum. It follows that a metric sphere in M of radius

2Nτ (q )centered at p is O(N −1)-close to the hypersurface τ = τ (q ), and its volumecan be computed as V (SN )

M (

τ (q ) − 12N L(x) + O(N −2))N dx, so the ratio

of this volume to

2Nτ (q )N +n

is just constant times N −n

2 times

M

τ (q )−n

2 exp(− 1 2τ (q )

L(x))dx + O(N −1)

The computation suggests that this integral, which we will call the reducedvolume and denote by V (τ (q )), should be increasing as τ decreases. A rig-

orous proof of this monotonicity is given in the next section.6.4* The first geometric interpretation of Hamilton’s Harnack expres-

sions was found by Chow and Chu [C-Chu 1,2]; they construct a potentiallydegenerate riemannian metric on M ×R, which potentially satisfies the Riccisoliton equation; our construction is, in a certain sense, dual to theirs.

Our formula for the reduced volume resembles the expression in Huiskenmonotonicity formula for the mean curvature flow [Hu]; however, in our casethe monotonicity is in the opposite direction.

7 A comparison geometry approach to the

Ricci flow

7.1 In this section we consider an evolving metric (gij)τ = 2Rij on a manifoldM ; we assume that either M is closed, or gij(τ ) are complete and haveuniformly bounded curvatures. To each curve γ (τ ), 0 < τ 1 ≤ τ ≤ τ 2, weassociate its L-length

L(γ ) =

τ 2τ 1

√ τ (R(γ (τ )) + |γ (τ )|2)dτ

(of course, R(γ (τ )) and |γ (τ )|2

are computed using gij(τ ))Let X (τ ) = γ (τ ), and let Y (τ ) be any vector field along γ (τ ). Then thefirst variation formula can be derived as follows:

δ Y (L) =

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τ 2

τ 1

√ τ (< Y ,

∇R > +2 <

∇Y X, X >)dτ =

τ 2

τ 1

√ τ (< Y,

∇R > +2 <

∇X Y, X >)dτ

=

τ 2τ 1

√ τ (< Y, ∇R > +2

d

dτ < Y,X > −2 < Y, ∇X X > −4Ric(Y, X ))dτ

= 2√

τ < X, Y >τ 2τ 1

+

τ 2τ 1

√ τ < Y, ∇R − 2∇X X − 4Ric(X, ·) − 1

τ X > dτ

(7.1)Thus L-geodesics must satisfy

∇X X − 1

2∇R +

1

2τ X + 2Ric(X, ·) = 0 (7.2)

Given two points p, q and τ 2 > τ 1 > 0, we can always find an L-shortestcurve γ (τ ), τ ∈ [τ 1, τ 2] between them, and every such L-shortest curve is L-geodesic. It is easy to extend this to the case τ 1 = 0; in this case

√ τ X (τ )

has a limit as τ → 0. From now on we fix p and τ 1 = 0 and denote by L(q, τ )the L-length of the L-shortest curve γ (τ ), 0 ≤ τ ≤ τ , connecting p and q. Inthe computations below we pretend that shortest L-geodesics between p andq are unique for all pairs (q, τ ); if this is not the case, the inequalities thatwe obtain are still valid when understood in the barrier sense, or in the senseof distributions.

The first variation formula (7.1) implies that ∇L(q, τ ) = 2√

τ X (τ ), sothat

|∇L

|2 = 4τ

|X

|2 =

−4τ R + 4τ (R +

|X

|2). We can also compute

Lτ (q, τ ) =√

τ (R + |X |2)− < X, ∇L >= 2√

τR − √ τ (R + |X |2)

To evaluate R + |X |2 we compute (using (7.2))

d

dτ (R(γ (τ )) + |X (τ )|2) = Rτ + < ∇R, X > +2 < ∇X X, X > +2Ric(X, X )

= Rτ + 1

τ R + 2 < ∇R, X > −2Ric(X, X ) − 1

τ (R + |X |2)

=

−H (X )

−

1

τ

(R +

|X

|2), (7.3)

where H (X ) is the Hamilton’s expression for the trace Harnack inequality(with t = −τ ). Hence,

τ 3

2 (R + |X |2)(τ ) = −K + 1

2L(q, τ ), (7.4)

15



so |Y (τ )|2 = τ τ , and in particular, Y (0) = 0. Making a substitution into (7.7),

we get HessL(Y, Y ) ≤ τ 0

√ τ (∇Y ∇Y R + 2 < R(Y, X ), Y , X > +2∇X Ric(Y, Y ) − 4∇Y Ric(Y, X )

+2|Ric(Y, ·)|2 − 2

τ Ric(Y, Y ) +

1

2τ τ )dτ

To put this in a more convenient form, observe that

d

dτ Ric(Y (τ ), Y (τ )) = Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + 2Ric(∇X Y, Y )

= Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + 1τ

Ric(Y, Y ) − 2|Ric(Y, ·)|2,

so

HessL(Y, Y ) ≤ 1√ τ

− 2√

τ Ric(Y, Y ) − τ 0

√ τ H (X, Y )dτ , (7.9)

where

H (X, Y ) = −∇Y ∇Y R−2 < R(Y, X )Y, X > −4(∇X Ric(Y, Y )−∇Y Ric(Y, X ))

−2Ricτ (Y, Y ) + 2|Ric(Y, ·)|2 − 1

τ Ric(Y, Y )

is the Hamilton’s expression for the matrix Harnack inequality (with t = −τ ).Thus

L ≤ −2√

τR + n√

τ − 1

τ K (7.10)

A field Y (τ ) along L-geodesic γ (τ ) is called L-Jacobi, if it is the derivativeof a variation of γ among L-geodesics. For an L-Jacobi field Y with |Y (τ )| =1 we have

d

dτ |Y |2 = 2Ric(Y, Y ) + 2 < ∇X Y, Y >= 2Ric(Y, Y ) + 2 < ∇Y X, Y >

= 2Ric(Y, Y ) + 1√ τ HessL(Y, Y ) ≤

1

τ − 1√ τ τ 0

τ 1

2 H (X, Y )dτ , (7.11)

where Y is obtained by solving ODE (7.8) with initial data Y (τ ) = Y (τ ).Moreover, the equality in (7.11) holds only if Y is L-Jacobi and henceddτ

|Y |2 = 2Ric(Y, Y ) + 1√ τ

HessL(Y, Y ) = 1τ

.

17



H (X ) ≥ −R( 1τ + 1

τ 0−τ ). Therefore, whenever τ is bounded away from τ 0

(say, τ ≤ (1 − c)τ 0, c > 0), we get (using (7.6), (7.11))

|∇l|2 + R ≤ Cl

τ , (7.16)

and for L-Jacobi fields Y

d

dτ log|Y |2 ≤ 1

τ (Cl + 1) (7.17)

7.3 As the first application of the comparison inequalities above, let usgive an alternative proof of a weakened version of the no local collapsingtheorem 4.1. Namely, rather than assuming

|Rm

|(x, tk)

≤ r−2

k for x

∈ Bk,

we require |Rm|(x, t) ≤ r−2k whenever x ∈ Bk, tk − r2k ≤ t ≤ tk. Then the

proof can go as follows: let τ k(t) = tk−t, p = pk, ǫk = r−1k V ol(Bk)1

n . We claim

that V k(ǫkr2k) < 3ǫn

2

k when k is large. Indeed, using the L-exponential map wecan integrate over T pM rather than M ; the vectors in T pM of length at most12ǫ− 1

2

k give rise to L-geodesics, which can not escape from Bk in time ǫkr2k, so

their contribution to the reduced volume does not exceed 2ǫn

2

k ; on the other

hand, the contribution of the longer vectors does not exceed exp(−12ǫ− 1

2

k ) by

the jacobian comparison theorem. However, V k(tk) (that is, at t = 0) staysbounded away from zero. Indeed, since min lk(

·, tk

−12

T )

≤ n2

, we can pick a

point q k, where it is attained, and obtain a universal upper bound on lk(·, tk)by considering only curves γ with γ (tk − 1

2T ) = q k, and using the fact that allgeometric quantities in gij(t) are uniformly bounded when t ∈ [0, 1

2T ]. Since

the monotonicity of the reduced volume requires V k(tk) ≤ V k(ǫkr2k), this is acontradiction.

A similar argument shows that the statement of the corollary in 4.2 canbe strengthened by adding another property of the ancient solution, obtainedas a blow-up limit. Namely, we may claim that if, say, this solution is definedfor t ∈ (−∞, 0), then for any point p and any t0 > 0, the reduced volumefunction V (τ ), constructed using p and τ (t) = t0 − t, is bounded below by κ.

7.4* The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in anytextbook on Riemannian geometry; an even closer reference is [L-Y], wherethey use ”length”, associated to a linear parabolic equation, which is prettymuch the same as in our case.

19



8 No local collapsing theorem II

8.1 Let us first formalize the notion of local collapsing, that was used in 7.3.Definition. A solution to the Ricci flow (gij)t = −2Rij is said to be

κ-collapsed at (x0, t0) on the scale r > 0 if |Rm|(x, t) ≤ r−2 for all (x, t)satisfying dist t0(x, x0) < r and t0 − r2 ≤ t ≤ t0, and the volume of the metric ball B(x0, r2) at time t0 is less than κrn.

8.2 Theorem. For any A > 0 there exists κ = κ(A) > 0 with the fol-lowing property. If gij(t) is a smooth solution to the Ricci flow (gij)t =−2Rij , 0 ≤ t ≤ r20, which has |Rm|(x, t) ≤ r−20 for all (x, t), satisfying dist 0(x, x0) < r0, and the volume of the metric ball B(x0, r0) at time zerois at least A−1rn0 , then gij(t) can not be κ-collapsed on the scales less than r0

at a point (x, r20) with dist r20(x, x0) ≤ Ar0.Proof. By scaling we may assume r0 = 1; we may also assume dist1(x, x0) =

A. Let us apply the constructions of 7.1 choosing p = x, τ (t) = 1− t. Arguingas in 7.3, we see that if our solution is collapsed at x on the scale r ≤ 1, thenthe reduced volume V (r2) must be very small; on the other hand, V (1) cannot be small unless min l(x, 12) over x satisfying dist 1

2

(x, x0) ≤ 110 is large.

Thus all we need is to estimate l, or equivalently L, in that ball. Recall thatL satisfies the differential inequality (7.15). In order to use it efficiently ina maximum principle argument, we need first to check the following simpleassertion.

8.3 Lemma. Suppose we have a solution to the Ricci flow (gij

)t =

−2R

ij.

(a) Suppose Ric (x, t0) ≤ (n − 1)K when dist t0(x, x0) < r0. Then the distance function d(x, t) = dist t(x, x0) satisfies at t = t0 outside B (x0, r0) the differential inequality

dt − d ≥ −(n − 1)(2

3Kr0 + r−10 )

(the inequality must be understood in the barrier sense, when necessary)(b) (cf. [H 4,§17]) Suppose Ric (x, t0) ≤ (n − 1)K when dist t0(x, x0) < r0,

or dist t0(x, x1) < r0. Then

d

dtdist t(x0, x1) ≥ −2(n − 1)(2

3Kr0 + r−10 ) at t = t0

Proof of Lemma. (a) Clearly, dt(x) = γ −Ric(X, X ), where γ is the shortest

geodesic between x and x0 and X is its unit tangent vector, On the otherhand, d ≤ n−1

k=1 s′′Y k(γ ), where Y k are vector fields along γ, vanishing at

20



x0 and forming an orthonormal basis at x when complemented by X, and

s′′Y k(γ ) denotes the second variation along Y k of the length of γ. Take Y k to beparallel between x and x1, and linear between x1 and x0, where d(x1, t0) = r0.Then

d ≤n−1k=1

s′′Y k(γ ) =

d(x,t0)r0

−Ric(X, X )ds+

r00

(s2

r20(−Ric(X, X )) +

n − 1

r20)ds

=

γ

−Ric(X, X )+

r00

(Ric(X, X )(1 − s2

r20) +

n − 1

r20)ds ≤ dt+(n−1)(

2

3Kr0+r−10 )

The proof of (b) is similar.

Continuing the proof of theorem, apply the maximum principle to thefunction h(y, t) = φ(d(y, t) − A(2t − 1))(L(y, 1 − t) + 2n + 1), where d(y, t) =distt(x, x0), and φ is a function of one variable, equal 1 on (−∞, 120), andrapidly increasing to infinity on ( 1

20, 110

), in such a way that

2(φ′)2/φ − φ′′ ≥ (2A + 100n)φ′ − C (A)φ, (8.1)

for some constant C (A) < ∞. Note that L + 2n + 1 ≥ 1 for t ≥ 12

by theremark in the very end of 7.1. Clearly, min h(y, 1) ≤ h(x, 1) = 2n + 1. Onthe other hand, min h(y, 12) is achieved for some y satisfying d(y, 1

2) ≤ 1

10.

Now we compute

h = (L+2n+1)(−φ′′+(dt−d−2A)φ′)−2 < ∇φ∇L > +(Lt−L)φ (8.2)

∇h = (L + 2n + 1)∇φ + φ∇L (8.3)

At a minimum point of h we have ∇h = 0, so (8.2) becomes

h = (L + 2n +1)(−φ′′ + (dt −d − 2A)φ′ + 2(φ′)2/φ) + (Lt − L)φ (8.4)

Now since d(y, t) ≥ 120

whenever φ′ = 0, and since Ric ≤ n − 1 in B(x0, 120),we can apply our lemma (a) to get dt − d ≥ −100(n − 1) on the set whereφ′ = 0. Thus, using (8.1) and (7.15), we get

h ≥ −(L + 2n + 1)C (A)φ − 2nφ ≥ −(2n + C (A))h

This implies that min h can not decrease too fast, and we get the requiredestimate.

21



9 Differential Harnack inequality for solutions

of the conjugate heat equation9.1 Proposition. Let gij(t) be a solution to the Ricci flow (gij)t = −2Rij, 0 ≤t ≤ T, and let u = (4π(T − t))−

n

2 e−f satisfy the conjugate heat equation ∗u = −ut − u + Ru = 0. Then v = [(T − t)(2f − |∇f |2 + R) + f − n]u

satisfies

∗v = −2(T − t)|Rij + ∇i∇ jf − 1

2(T − t)gij |2 (9.1)

Proof. Routine computation.Clearly, this proposition immediately implies the monotonicity formula

(3.4); its advantage over (3.4) shows up when one has to work locally.

9.2 Corollary. Under the same assumptions, on a closed manifold M ,or whenever the application of the maximum principle can be justified, min v/uis nondecreasing in t.

9.3 Corollary. Under the same assumptions, if u tends to a δ -function as t → T, then v ≤ 0 for all t < T.

Proof. If h satisfies the ordinary heat equation ht = h with respect tothe evolving metric gij(t), then we have d

dt

hu = 0 and d

dt

hv ≥ 0. Thus we

only need to check that for everywhere positive h the limit of hv as t

→T

is nonpositive. But it is easy to see, that this limit is in fact zero.

9.4 Corollary. Under assumptions of the previous corollary, for any smooth curve γ (t) in M holds

− d

dtf (γ (t), t) ≤ 1

2(R(γ (t), t) + |γ (t)|2) − 1

2(T − t)f (γ (t), t) (9.2)

Proof. From the evolution equation f t = −f + |∇f |2 − R + n2(T −t) and

v ≤ 0 we get f t+ 12R− 1

2|∇f |2− f

2(T −t) ≥ 0. On the other hand,− ddtf (γ (t), t) =

−f t

−<

∇f, γ (t) >

≤ −f t +

12

|∇f

|2 + 1

2

|γ

|2. Summing these two inequalities,

we get (9.2).

9.5 Corollary. If under assumptions of the previous corollary, p is the point where the limit δ -function is concentrated, then f (q, t) ≤ l(q, T − t), where lis the reduced distance, defined in 7.1, using p and τ (t) = T − t.

22



Proof. Use (7.13) in the form ∗exp(−l) ≤ 0.

9.6 Remark. Ricci flow can be characterized among all other evolutionequations by the infinitesimal behavior of the fundamental solutions of theconjugate heat equation. Namely, suppose we have a riemannian metric gij(t)evolving with time according to an equation (gij)t = Aij(t). Then we havethe heat operator = ∂

∂t− and its conjugate

∗ = − ∂ ∂t

−− 12A, so that

ddt

uv =

((u)v − u(∗v)). (Here A = g ijAij) Consider the fundamental

solution u = (−4πt)−n

2 e−f for ∗, starting as δ -function at some point ( p, 0).

Then for general Aij the function (f + f t

)(q, t), where f = f − f u, is of theorder O(1) for (q, t) near ( p, 0). The Ricci flow Aij = −2Rij is characterized

by the condition (f + f t

)(q, t) = o(1); in fact, it is O(| pq |2 + |t|) in this case.9.7* Inequalities of the type of (9.2) are known as differential Harnack

inequalities; such inequality was proved by Li and Yau [L-Y] for the solutionsof linear parabolic equations on riemannian manifolds. Hamilton [H 7,8] useddifferential Harnack inequalities for the solutions of backward heat equationon a manifold to prove monotonicity formulas for certain parabolic flows. Alocal monotonicity formula for mean curvature flow making use of solutionsof backward heat equation was obtained by Ecker [E 2].

10 Pseudolocality theorem

10.1 Theorem. For every α > 0 there exist δ > 0, ǫ > 0 with the follow-ing property. Suppose we have a smooth solution to the Ricci flow (gij)t =−2Rij , 0 ≤ t ≤ (ǫr0)2, and assume that at t = 0 we have R(x) ≥ −r−20 and V ol(∂ Ω)n ≥ (1 − δ )cnV ol(Ω)n−1 for any x, Ω ⊂ B(x0, r0), where cn is the euclidean isoperimetric constant. Then we have an estimate |Rm|(x, t) ≤αt−1 + (ǫr0)−2 whenever 0 < t ≤ (ǫr0)2, d(x, t) = dist t(x, x0) < ǫr0.

Thus, under the Ricci flow, the almost singular regions (where curvatureis large) can not instantly significantly influence the almost euclidean regions.Or , using the interpretation via renormalization group flow, if a region lookstrivial (almost euclidean) on higher energy scale, then it can not suddenlybecome highly nontrivial on a slightly lower energy scale.

Proof. It is an argument by contradiction. The idea is to pick a point(x, t) not far from (x0, 0) and consider the solution u to the conjugate heatequation, starting as δ -function at (x, t), and the corresponding nonpositivefunction v as in 9.3. If the curvatures at (x, t) are not small compared to

23



t−1 and are larger than at nearby points, then one can show that

v at time

t is bounded away from zero for (small) time intervals t − t of the order of |Rm|−1(x, t). By monotonicity we conclude that

v is bounded away fromzero at t = 0. In fact, using (9.1) and an appropriate cut-off function, wecan show that at t = 0 already the integral of v over B(x0, r) is boundedaway from zero, whereas the integral of u over this ball is close to 1, where rcan be made as small as we like compared to r0. Now using the control overthe scalar curvature and isoperimetric constant in B(x0r0), we can obtain acontradiction to the logarithmic Sobolev inequality.

Now let us go into details. By scaling assume that r0 = 1. We may alsoassume that α is small, say α < 1

100n . From now on we fix α and denote byM α the set of pairs (x, t), such that

|Rm

|(x, t)

≥αt−1.

Claim 1.For any A > 0, if gij(t) solves the Ricci flow equation on 0 ≤t ≤ ǫ2, Aǫ < 1

100n , and |Rm|(x, t) > αt−1 + ǫ−2 for some (x, t), satisfying 0 ≤t ≤ ǫ2, d(x, t) < ǫ, then one can find (x, t) ∈ M α, with 0 < t ≤ ǫ2, d(x, t) <(2A + 1)ǫ, such that

|Rm|(x, t) ≤ 4|Rm|(x, t), (10.1)

whenever

(x, t) ∈ M α, 0 < t ≤ t, d(x, t) ≤ d(x, t) + A|Rm|− 1

2 (x, t) (10.2)

Proof of Claim 1. We construct (x, t) as a limit of a (finite) sequence(xk, tk), defined in the following way. Let (x1, t1) be an arbitrary point,satisfying 0 < t1 ≤ ǫ2, d(x1, t1) < ǫ, |Rm|(x1, t1) ≥ αt−1 + ǫ−2. Now if (xk, tk)is already constructed, and if it can not be taken for (x, t), because thereis some (x, t) satisfying (10.2), but not (10.1), then take any such (x, t)for (xk+1, tk+1). Clearly, the sequence, constructed in such a way, satisfies|Rm|(xk, tk) ≥ 4k−1|Rm|(x1, t1) ≥ 4k−1ǫ−2, and therefore, d(xk, tk) ≤ (2A +1)ǫ. Since the solution is smooth, the sequence is finite, and its last elementfits.

Claim 2. For (x, t), constructed above, (10.1) holds whenever

t

−

1

2

αQ−1

≤t

≤t, dist t(x, x)

≤

1

10

AQ− 1

2 , (10.3)

where Q = |Rm|(x, t).Proof of Claim 2. We only need to show that if (x, t) satisfies (10.3),

then it must satisfy (10.1) or (10.2). Since (x, t) ∈ M α, we have Q ≥ αt−1, sot− 1

2αQ−1 ≥ 12 t. Hence, if (x, t) does not satisfy (10.1), it definitely belongs to

24



M α. Now by the triangle inequality, d(x, t) ≤ d(x, t) + 110AQ− 1

2 . On the other

hand, using lemma 8.3(b) we see that, as t decreases from t to t − 1

2αQ−1

,the point x can not escape from the ball of radius d(x, t) + AQ− 1

2 centeredat x0.

Continuing the proof of the theorem, and arguing by contradiction, takesequences ǫ → 0, δ → 0 and solutions gij(t), violating the statement; byreducing ǫ, we’ll assume that

|Rm|(x, t) ≤ αt−1 + 2ǫ−2 whenever 0 ≤ t ≤ ǫ2 and d(x, t) ≤ ǫ (10.4)

Take A = 1100nǫ

→ ∞, construct (x, t), and consider solutions u = (4π(t −t))−

n

2 e−f of the conjugate heat equation, starting from δ -functions at (x, t),

and corresponding nonpositive functions v.Claim 3.As ǫ, δ → 0, one can find times t ∈ [t − 12αQ−1, t], such that the

integral B v stays bounded away from zero, where B is the ball at time t of

radius

t − t centered at x.Proof of Claim 3(sketch). The statement is invariant under scaling, so

we can try to take a limit of scalings of gij(t) at points (x, t) with factorsQ. If the injectivity radii of the scaled metrics at (x, t) are bounded awayfrom zero, then a smooth limit exists, it is complete and has |Rm|(x, t) = 1and |Rm|(x, t) ≤ 4 when t − 1

2α ≤ t ≤ t. It is not hard to show that thefundamental solutions u of the conjugate heat equation converge to such asolution on the limit manifold. But on the limit manifold, B v can not be

zero for t = t − 12α, since the evolution equation (9.1) would imply in this

case that the limit is a gradient shrinking soliton, and this is incompatiblewith |Rm|(x, t) = 1.

If the injectivity radii of the scaled metrics tend to zero, then we canchange the scaling factor, to make the scaled metrics converge to a flat man-ifold with finite injectivity radius; in this case it is not hard to choose t insuch a way that

B v → −∞.

The positive lower bound for − B

v will be denoted by β .Our next goal is to construct an appropriate cut-off function. We choose

it in the form h(y, t) = φ( d(y,t)10Aǫ

), where d(y, t) = d(y, t) + 200n√

t, and φ is

a smooth function of one variable, equal one on (−∞, 1] and decreasing tozero on [1, 2]. Clearly, h vanishes at t = 0 outside B(x0, 20Aǫ); on the otherhand, it is equal to one near (x, t).

Now h = 110Aǫ(dt−d + 100n√

t )φ′− 1

(10Aǫ)2 φ′′. Note that dt−t + 100n√ t ≥ 0

on the set where φ′ = 0 − this follows from the lemma 8.3(a) and our

25



assumption (10.4). We may also choose φ so that φ′′ ≥ −10φ, (φ′)2 ≤ 10φ.

Now we can compute ( M hu)t =

M (

h)u ≤ 1

(Aǫ)2 , so M hu |t=0≥ M hu |t=t

− t(Aǫ)2

≥ 1 − A−2. Also, by (9.1), ( M

−hv)t ≤ M

−(h)v ≤ 1(Aǫ)2

M

−hv,

so by Claim 3, − M

hv |t=0≥ β exp(− t(Aǫ)2 ) ≥ β (1 − A−2).

From now on we”ll work at t = 0 only. Let u = hu and correspondinglyf = f − logh. Then

β (1 − A−2) ≤ − M

hv =

M

[(−2f + |∇f |2 − R)t − f + n]hu

=

M

[−t|∇f |2 − f + n]u +

M

[t(|∇h|2/h − Rh) − hlogh]u

≤ M

[−t|∇f |2 − f − n]u + A−2 + 100ǫ2

( Note that M

−uh log h does not exceed the integral of u overB(x0, 20Aǫ)\B(x0, 10Aǫ), and

B(x0,10Aǫ)

u ≥ M

hu ≥ 1 − A−2,

where h = φ( d5Aǫ

))Now scaling the metric by the factor 1

2 t−1 and sending ǫ, δ to zero, weget a sequence of metric balls with radii going to infinity, and a sequenceof compactly supported nonnegative functions u = (2π)−

n

2 e−f with

u → 1and

[−1

2 |∇f |2 − f + n]u bounded away from zero by a positive constant.

We also have isoperimetric inequalities with the constants tending to the eu-clidean one. This set up is in conflict with the Gaussian logarithmic Sobolevinequality, as can be seen by using spherical symmetrization.

10.2 Corollary(from the proof) Under the same assumptions, we alsohave at time t, 0 < t ≤ (ǫr0)2, an estimate V olB(x,

√ t) ≥ c

√ tn

for x ∈B(x0, ǫr0), where c = c(n) is a universal constant.

10.3 Theorem. There exist ǫ, δ > 0 with the following property. Suppose gij(t) is a smooth solution to the Ricci flow on [0, (ǫr0)2], and assume that at t = 0 we have |Rm|(x) ≤ r−20 in B(x0, r0), and V olB(x0, r0) ≥ (1 − δ )ωnrn0 ,where ωn is the volume of the unit ball in Rn. Then the estimate |Rm|(x, t) ≤(ǫr0)−2 holds whenever 0

≤t

≤(ǫr0)2, dist t(x, x0) < ǫr0.

The proof is a slight modification of the proof of theorem 10.1, and is leftto the reader. A natural question is whether the assumption on the volumeof the ball is superfluous.

10.4 Corollary(from 8.2, 10.1, 10.2) There exist ǫ, δ > 0 and for any A > 0 there exists κ(A) > 0 with the following property. If gij(t) is a

26



Proposition.The scalings of gij(t0 − τ ) at q (τ ) with factors τ −1 converge

along a subsequence of τ → ∞ to a non-flat gradient shrinking soliton.Proof (sketch). It is not hard to deduce from (7.16) that for any ǫ > 0one can find δ > 0 such that both l(q, τ ) and τR(q, t0 − τ ) do not exceedδ −1 whenever 1

2τ ≤ τ ≤ τ and dist2t0−τ (q, q (τ )) ≤ ǫ−1τ for some τ > 0.

Therefore, taking into account the κ-noncollapsing assumption, we can takea blow-down limit, say gij(τ ), defined for τ ∈ ( 1

2, 1), (gij)τ = 2 Rij . We may

assume also that functions l tend to a locally Lipschitz function l, satisfying(7.13),(7.14) in the sense of distributions. Now, since V (τ ) is nonincreasingand bounded away from zero (because the scaled metrics are not collapsednear q (τ )) the limit function V (τ ) must be a positive constant; this constantis strictly less than limτ

→0V (τ ) = (4π)

n

2 , since gij(t) is not flat. Therefore,

on the one hand, (7.14) must become an equality, hence l is smooth, and onthe other hand, by the description of the equality case in (7.12), gij(τ ) mustbe a gradient shrinking soliton with Rij + ∇i

∇ j l − 12τ

gij = 0. If this solitonis flat, then l is uniquely determined by the equality in (7.14), and it turnsout that the value of V is exactly (4π)

n

2 , which was ruled out.

11.3 Corollary. There is only one oriented two-dimensional solution, sat-isfying the assumptions stated in 11.1, - the round sphere.

Proof. Hamilton [H 10] proved that round sphere is the only non-flatoriented nonnegatively curved gradient shrinking soliton in dimension two.

Thus, the scalings of our ancient solution must converge to a round sphere.However, Hamilton [H 10] has also shown that an almost round sphere isgetting more round under Ricci flow, therefore our ancient solution must beround.

11.4. Recall that for any non-compact complete riemannian manifold M of nonnegative Ricci curvature and a point p ∈ M, the function V olB( p, r)r−n

is nonincreasing in r > 0; therefore, one can define an asymptotic volumeratio V as the limit of this function as r → ∞.

Proposition.Under assumptions of 11.1, V = 0 for each t.Proof. Induction on dimension. In dimension two the statement is

vacuous, as we have just shown. Now let n ≥ 3, suppose that V > 0for some t = t0, and consider the asymptotic scalar curvature ratio R =lim supR(x, t0)d2(x) as d(x) → ∞. (d(x) denotes the distance, at time t0,from x to some fixed point x0) If R = ∞, then we can find a sequence of points xk and radii rk > 0, such that rk/d(xk) → 0, R(xk)r2k → ∞, and

28



that z k be the closest point to xk (at t = 0), satisfying R(z k, 0)dist20(xk, z k) = 1.

We claim that R(z, 0)/R(z k, 0) is uniformly bounded for z ∈ B(z k, 2R(z k, 0)−1

2

).Indeed, otherwise we could show, using 11.5 and relative volume comparisonin nonnegative curvature, that the balls B(z k, R(z k, 0)−

1

2 ) are collapsing onthe scale of their radii. Therefore, using the local derivative estimate, due toW.-X.Shi (see [H 4,§13]), we get a bound on Rt(z k, t) of the order of R2(z k, 0).Then we can compare 1 = R(xk, 0) ≥ cR(z k, −cR−1(z k, 0)) ≥ cR(z k, 0) forsome small c > 0, where the first inequality comes from the Harnack inequal-ity, obtained by integrating (11.1). Thus, R(z k, 0) are bounded. But nowthe existence of the sequence yk at bounded distance from xk implies, via11.5 and relative volume comparison, that balls B(xk, c) are collapsing - acontradiction.

It remains to show that the limit has bounded curvature at t = 0. If thiswas not the case, then we could find a sequence yi going to infinity, suchthat R(yi, 0) → ∞ and R(y, 0) ≤ 2R(yi, 0) for y ∈ B(yi, AiR(yi, 0)−

1

2 ), Ai →∞. Then the limit of scalings at (yi, 0) with factors R(yi, 0) satisfies theassumptions in 11.1 and splits off a line. Thus by 11.3 it must be a roundinfinite cylinder. It follows that for large i each yi is contained in a roundcylindrical ”neck” of radius ( 1

2R(yi, 0))−1

2 → 0, - something that can nothappen in an open manifold of nonnegative curvature.

11.8. Fix ǫ > 0. Let gij(t) be an ancient solution on a noncompact orientedthree-manifold M, satisfying the assumptions in 11.1. We say that a point

x0 ∈ M is the center of an ǫ-neck, if the solution gij(t) in the set (x, t) :−(ǫQ)−1 < t ≤ 0, dist20(x, x0) < (ǫQ)−1, where Q = R(x0, 0), is, after scalingwith factor Q, ǫ-close (in some fixed smooth topology) to the correspondingsubset of the evolving round cylinder, having scalar curvature one at t = 0.

Corollary (from theorem 11.7 and its proof) For any ǫ > 0 there exists C = C (ǫ, κ) > 0, such that if gij(t) satisfies the assumptions in 11.1, and M ǫ denotes the set of points in M, which are not centers of ǫ-necks, then M ǫ is compact and moreover, diam M ǫ ≤ CQ− 1

2 , and C −1Q ≤ R(x, 0) ≤ CQwhenever x ∈ M ǫ, where Q = R(x0, 0) for some x0 ∈ ∂M ǫ.

11.9 Remark. It can be shown that there exists κ0 > 0, such that if anancient solution on a noncompact three-manifold satisfies the assumptions in11.1 with some κ > 0, then it would satisfy these assumptions with κ = κ0.This follows from the arguments in 7.3, 11.2, and the statement (which isnot hard to prove) that there are no noncompact three-dimensional gradient

31



shrinking solitons, satisfying 11.1, other than the round cylinder and its Z2-

quotients.Furthermore, I believe that there is only one (up to scaling) noncom-pact three-dimensional κ-noncollapsed ancient solution with bounded posi-tive curvature - the rotationally symmetric gradient steady soliton, studiedby R.Bryant. In this direction, I have a plausible, but not quite rigorousargument, showing that any such ancient solution can be made eternal, thatis, can be extended for t ∈ (−∞, +∞); also I can prove uniqueness in theclass of gradient steady solitons.

11.10* The earlier work on ancient solutions and all that can be foundin [H 4, §16 − 22, 25, 26].

12 Almost nonnegative curvature in dimen-

sion three

12.1 Let φ be a decreasing function of one variable, tending to zero at infinity.A solution to the Ricci flow is said to have φ-almost nonnegative curvatureif it satisfies Rm(x, t) ≥ −φ(R(x, t))R(x, t) for each (x, t).

Theorem. Given ǫ > 0, κ > 0 and a function φ as above, one can find r0 > 0 with the following property. If gij(t), 0 ≤ t ≤ T is a solution tothe Ricci flow on a closed three-manifold M, which has φ-almost nonnegative

curvature and is κ-noncollapsed on scales < r0, then for any point (x0, t0)with t0 ≥ 1 and Q = R(x0, t0) ≥ r−20 , the solution in (x, t) : dist 2t0(x, x0) <(ǫQ)−1, t0 − (ǫQ)−1 ≤ t ≤ t0 is , after scaling by the factor Q, ǫ-close to the corresponding subset of some ancient solution, satisfying the assumptions in 11.1.

Proof. An argument by contradiction. Take a sequence of r0 convergingto zero, and consider the solutions gij(t), such that the conclusion does nothold for some (x0, t0); moreover, by tampering with the condition t0 ≥ 1 alittle bit, choose among all such (x0, t0), in the solution under consideration,the one with nearly the smallest curvature Q. (More precisely, we can choose(x0, t0) in such a way that the conclusion of the theorem holds for all (x, t),

satisfying R(x, t) > 2Q, t0 − HQ−1 ≤ t ≤ t0, where H → ∞ as r0 → 0)Our goal is to show that the sequence of blow-ups of such solutions at suchpoints with factors Q would converge, along some subsequence of r0 → 0, toan ancient solution, satisfying 11.1.

32



Claim 1. For each (x, t) with t0 − HQ−1 ≤ t ≤ t0 we have R(x, t) ≤ 4 Q

whenever t − c Q−1

≤ t ≤ t and dist t(x, x) ≤ c Q−1

2

, where Q = Q + R(x, t)and c = c(κ) > 0 is a small constant.Proof of Claim 1. Use the fact ( following from the choice of (x0, t0) and

the description of the ancient solutions) that for each (x, t) with R(x, t) >2Q and t0 − HQ−1 ≤ t ≤ t0 we have the estimates |Rt(x, t)| ≤ CR2(x, t),

|∇R|(x, t) ≤ CR3

2 (x, t).Claim 2. There exists c = c(κ) > 0 and for any A > 0 there exist

D = D(A) < ∞, ρ0 = ρ0(A) > 0, with the following property. Suppose that r0 < ρ0, and let γ be a shortest geodesic with endpoints x, x in gij(t), for some t ∈ [t0 − HQ−1, t0], such that R(y, t) > 2Q for each y ∈ γ. Let z ∈ γ satisfy

cR(z, t) > R(x, t) = Q. Then dist t(x, z )≥

AQ− 1

2 whenever R(x, t)≥

D Q.Proof of Claim 2. Note that from the choice of (x0, t0) and the description

of the ancient solutions it follows that an appropriate parabolic (backward intime) neighborhood of a point y ∈ γ at t = t is ǫ-close to the evolving roundcylinder, provided c−1 Q ≤ R(y, t) ≤ cR(x, t) for an appropriate c = c(κ).Now assume that the conclusion of the claim does not hold, take r0 to zero,R(x, t) - to infinity, and consider the scalings around (x, t) with factors Q.We can imagine two possibilities for the behavior of the curvature along γ inthe scaled metric: either it stays bounded at bounded distances from x, ornot. In the first case we can take a limit (for a subsequence) of the scaledmetrics along γ and get a nonnegatively curved almost cylindrical metric,

with γ going to infinity. Clearly, in this case the curvature at any point of the limit does not exceed c−1; therefore, the point z must have escaped toinfinity, and the conclusion of the claim stands.

In the second case, we can also take a limit along γ ; it is a smooth non-negatively curved manifold near x and has cylindrical shape where curvatureis large; the radius of the cylinder goes to zero as we approach the (first)singular point, which is located at finite distance from x; the region beyondthe first singular point will be ignored. Thus, at t = t we have a metric,which is a smooth metric of nonnegative curvature away from a single sin-gular point o. Since the metric is cylindrical at points close to o, and theradius of the cylinder is at most ǫ times the distance from o, the curvature

at o is nonnegative in Aleksandrov sense. Thus, the metric near o must becone-like. In other words, the scalings of our metric at points xi → o withfactors R(xi, t) converge to a piece of nonnegatively curved non-flat metriccone. Moreover, using claim 1, we see that we actually have the convergence

33



of the solutions to the Ricci flow on some time interval, and not just met-

rics at t = t. Therefore, we get a contradiction with the strong maximumprinciple of Hamilton [H 2].Now continue the proof of theorem, and recall that we are considering

scalings at (x0, t0) with factor Q. It follows from claim 2 that at t = t0the curvature of the scaled metric is bounded at bounded distances fromx0. This allows us to extract a smooth limit at t = t0 (of course, we usethe κ-noncollapsing assumption here). The limit has bounded nonnegativecurvature (if the curvatures were unbounded, we would have a sequence of cylindrical necks with radii going to zero in a complete manifold of nonneg-ative curvature). Therefore, by claim 1, we have a limit not only at t = t0,but also in some interval of times smaller than t0.

We want to show that the limit actually exists for all t < t0. Assume thatthis is not the case, and let t′ be the smallest value of time, such that the blow-up limit can be taken on (t′, t0]. From the differential Harnack inequality of Hamilton [H 3] we have an estimate Rt(x, t) ≥ −R(x, t)(t−t′)−1, therefore, if Q denotes the maximum of scalar curvature at t = t0, then R(x, t) ≤ Q t0−t′

t−t′ .Hence by lemma 8.3(b) distt(x, y) ≤ distt0(x, y) + C for all t, where C =

10n(t0 − t′)

Q.The next step is needed only if our limit is noncompact. In this case

there exists D > 0, such that for any y satisfying d = distt0(x0, y) > D, onecan find x satisfying distt0(x, y) = d, distt0(x, x0) > 3

2d. We claim that the

scalar curvature R(y, t) is uniformly bounded for all such y and all t ∈ (t′, t0].Indeed, if R(y, t) is large, then the neighborhood of (y, t) is like in an ancientsolution; therefore, (long) shortest geodesics γ and γ 0, connecting at time tthe point y to x and x0 respectively, make the angle close to 0 or π at y ; theformer case is ruled out by the assumptions on distances, if D > 10C ; in thelatter case, x and x0 are separated at time t by a small neighborhood of y,with diameter of order R(y, t)−

1

2 , hence the same must be true at time t0,which is impossible if R(y, t) is too large.

Thus we have a uniform bound on curvature outside a certain compactset, which has uniformly bounded diameter for all t ∈ (t′, t0]. Then claim 2gives a uniform bound on curvature everywhere. Hence, by claim 1, we canextend our blow-up limit past t′ - a contradiction.

12.2 Theorem. Given a function φ as above, for any A > 0 there exists K = K (A) < ∞ with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which

34



satisfies the assumptions of theorem 8.2 with r0 = 1. Then R(x, 1) ≤ K

whenever dist 1(x, x0) < A.Proof. In the first step of the proof we check the followingClaim. There exists K = K (A) < ∞, such that a point (x, 1) satisfies

the conclusion of the previous theorem 12.1 (for some fixed small ǫ > 0),whenever R(x, 1) > K and dist 1(x, x0) < A.

The proof of this statement essentially repeats the proof of the previoustheorem (the κ-noncollapsing assumption is ensured by theorem 8.2). Theonly difference is in the beginning. So let us argue by contradiction, andsuppose we have a sequence of solutions and points x with dist1(x, x0) < Aand R(x, 1) → ∞, which do not satisfy the conclusion of 12.1. Then anargument, similar to the one proving claims 1,2 in 10.1, delivers points ( x, t)

with 12 ≤ t ≤ 1, distt(x, x0) < 2A, with Q = R(x, t) → ∞, and such that(x, t) satisfies the conclusion of 12.1 whenever R(x, t) > 2Q, t − DQ−1 ≤ t ≤t, distt(x, x) < DQ− 1

2 , where D → ∞. (There is a little subtlety here in theapplication of lemma 8.3(b); nevertheless, it works, since we need to apply itonly when the endpoint other than x0 either satisfies the conclusion of 12.1,or has scalar curvature at most 2Q) After such (x, t) are found, the proof of 12.1 applies.

Now, having checked the claim, we can prove the theorem by applying theclaim 2 of the previous theorem to the appropriate segment of the shortestgeodesic, connecting x and x0.

12.3 Theorem. For any w > 0 there exist τ = τ (w) > 0, K = K (w) <∞, ρ = ρ(w) > 0 with the following property. Suppose we have a solution gij(t) to the Ricci flow, defined on M × [0, T ), where M is a closed three-manifold, and a point (x0, t0), such that the ball B(x0, r0) at t = t0 has volume ≥ wrn

0 , and sectional curvatures ≥ −r−20 at each point. Suppose that gij(t) is φ-almost nonnegatively curved for some function φ as above.Then we have an estimate R(x, t) < Kr−20 whenever t0 ≥ 4τ r20, t ∈ [t0 −τr20, t0], dist t(x, x0) ≤ 1

4r0, provided that φ(r−20 ) < ρ.Proof. If we knew that sectional curvatures are ≥ −r−20 for all t, then we

could just apply corollary 11.6(b) (with the remark after its proof) and takeτ (w) = τ 0(w)/2, K (w) = C (w) + 2B(w)/τ 0(w). Now fix these values of τ, K,

consider a φ-almost nonnegatively curved solution gij(t), a point (x0, t0) anda radius r0 > 0, such that the assumptions of the theorem do hold whereasthe conclusion does not. We may assume that any other point (x′, t′) andradius r′ > 0 with that property has either t′ > t0 or t′ < t0 − 2τ r20, or

35



2r′ > r0. Our goal is to show that φ(r−20 ) is bounded away from zero.

Let τ ′ > 0 be the largest time interval such that Rm(x, t) ≥ −r−2

0 when-ever t ∈ [t0 − τ ′r20, t0], distt(x, x0) ≤ r0. If τ ′ ≥ 2τ, we are done by corollary11.6(b). Otherwise, by elementary Aleksandrov space theory, we can find attime t′ = t0 − τ ′r20 a ball B(x′, r′) ⊂ B (x0, r0) with V olB(x′, r′) ≥ 1

2ωn(r′)n,

and with radius r ′ ≥ cr0 for some small constant c = c(w) > 0. By the choiceof (x0, t0) and r0, the conclusion of our theorem holds for (x′, t′), r′. Thus wehave an estimate R(x, t) ≤ K (r′)−2 whenever t ∈ [t′−τ (r′)2, t′], distt(x, x′) ≤14r′. Now we can apply the previous theorem (or rather its scaled version) andget an estimate on R(x, t) whenever t ∈ [t′ − 1

2τ (r′)2, t′], distt(x′, x) ≤ 10r0.

Therefore, if r0 > 0 is small enough, we have Rm(x, t) ≥ −r−20 for those(x, t), which is a contradiction to the choice of τ ′.

12.4 Corollary (from 12.2 and 12.3) Given a function φ as above, for any w > 0 one can find ρ > 0 such that if gij(t) is a φ-almost nonnegatively curved solution to the Ricci flow, defined on M × [0, T ), where M is a closed three-manifold, and if B(x0, r0) is a metric ball at time t0 ≥ 1, with r0 <ρ, and such that min Rm(x, t0) over x ∈ B(x0, r0) is equal to −r−20 , then V olB(x0, r0) ≤ wrn

0 .

13 The global picture of the Ricci flow in di-

mension three

13.1 Let gij(t) be a smooth solution to the Ricci flow on M × [1, ∞), whereM is a closed oriented three-manifold. Then, according to [H 6, theorem 4.1],the normalized curvatures Rm(x, t) = tRm(x, t) satisfy an estimate of theform Rm(x, t) ≥ −φ( R(x, t)) R(x, t), where φ behaves at infinity as 1

log . This

estimate allows us to apply the results 12.3,12.4, and obtain the followingTheorem. For any w > 0 there exist K = K (w) < ∞, ρ = ρ(w) > 0,

such that for sufficiently large times t the manifold M admits a thick-thin decomposition M = M thick

M thin with the following properties. (a) For

every x ∈ M thick we have an estimate | Rm| ≤ K in the ball B(x, ρ(w)√

t).and the volume of this ball is at least 1

10w(ρ(w)

√ t)n. (b) For every y

∈M thin

there exists r = r(y), 0 < r < ρ(w)√ t, such that for all points in the ball B(y, r) we have Rm ≥ −r−2, and the volume of this ball is < wrn.

Now the arguments in [H 6] show that either M thick is empty for larget, or , for an appropriate sequence of t → 0 and w → 0, it converges to

36



a (possibly, disconnected) complete hyperbolic manifold of finite volume,

whose cusps (if there are any) are incompressible in M . On the other hand,collapsing with lower curvature bound in dimension three is understood wellenough to claim that, for sufficiently small w > 0, M thin is homeomorphicto a graph manifold.

The natural questions that remain open are whether the normalized cur-vatures must stay bounded as t → ∞, and whether reducible manifoldsand manifolds with finite fundamental group can have metrics which evolvesmoothly by the Ricci flow on the infinite time interval.

13.2 Now suppose that gij(t) is defined on M × [1, T ), T < ∞, and goessingular as t → T. Then using 12.1 we see that, as t → T, either the curvaturegoes to infinity everywhere, and then M is a quotient of either S3 or S2

×R,

or the region of high curvature in gij(t) is the union of several necks andcapped necks, which in the limit turn into horns (the horns most likely havefinite diameter, but at the moment I don’t have a proof of that). Then atthe time T we can replace the tips of the horns by smooth caps and continuerunning the Ricci flow until the solution goes singular for the next time, e.t.c.It turns out that those tips can be chosen in such a way that the need for thesurgery will arise only finite number of times on every finite time interval.The proof of this is in the same spirit, as our proof of 12.1; it is technicallyquite complicated, but requires no essentially new ideas. It is likely that bypassing to the limit in this construction one would get a canonically defined

Ricci flow through singularities, but at the moment I don’t have a proof of that. (The positive answer to the conjecture in 11.9 on the uniqueness of ancient solutions would help here)

Moreover, it can be shown, using an argument based on 12.2, that everymaximal horn at any time T, when the solution goes singular, has volumeat least cT n; this easily implies that the solution is smooth (if nonempty)from some finite time on. Thus the topology of the original manifold canbe reconstructed as a connected sum of manifolds, admitting a thick-thindecomposition as in 13.1, and quotients of S3 and S2 × R.

13.3* Another differential-geometric approach to the geometrization conjecture is being developed by Anderson [A]; he studies the elliptic equations,

arising as Euler-Lagrange equations for certain functionals of the riemannianmetric, perturbing the total scalar curvature functional, and one can observecertain parallelism between his work and that of Hamilton, especially takinginto account that, as we have shown in 1.1, Ricci flow is the gradient flow fora functional, that closely resembles the total scalar curvature.

37



a r X i v : m a t h / 0

3 0 3 1 0 9 v 1 [ m a t h . D

G ] 1 0 M a r 2 0 0 3 Ricci flow with surgery on three-manifolds

Grisha Perelman∗

February 1, 2008

This is a technical paper, which is a continuation of [I]. Here we verify mostof the assertions, made in [I, §13]; the exceptions are (1) the statement that a3-manifold which collapses with local lower bound for sectional curvature is agraph manifold - this is deferred to a separate paper, as the proof has nothing todo with the Ricci flow, and (2) the claim about the lower bound for the volumesof the maximal horns and the smoothness of the solution from some time on,which turned out to be unjustified, and, on the other hand, irrelevant for theother conclusions.

The Ricci flow with surgery was considered by Hamilton [H 5,§4,5]; unfortu-nately, his argument, as written, contains an unjustified statement (RMAX = Γ,on page 62, lines 7-10 from the bottom), which I was unable to fix. Our approachis somewhat different, and is aimed at eventually constructing a canonical Ricciflow, defined on a largest possible subset of space-time, - a goal, that has notbeen achieved yet in the present work. For this reason, we consider two scalebounds: the cutoff radius h, which is the radius of the necks, where the surg-eries are performed, and the much larger radius r, such that the solution on the

scales less than r has standard geometry. The point is to make h arbitrarilysmall while keeping r bounded away from zero.

Notation and terminology

B(x,t,r) denotes the open metric ball of radius r, with respect to the metricat time t, centered at x.

P (x,t,r, t) denotes a parabolic neighborhood, that is the set of all points(x′, t′) with x′ ∈ B(x,t,r) and t′ ∈ [t, t + t] or t′ ∈ [t + t, t], depending onthe sign of t.

A ball B(x,t,ǫ−1r) is called an ǫ-neck, if, after scaling the metric with factorr−2, it is ǫ-close to the standard neck S2 × I, with the product metric, where S2

has constant scalar curvature one, and I has length 2ǫ−1; here ǫ-close refers toC N topology, with N > ǫ−1.

A parabolic neighborhood P (x,t,ǫ−1r, r2) is called a strong ǫ-neck, if, afterscaling with factor r−2, it is ǫ-close to the evolving standard neck, which at each

∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg

191011, Russia. Email: [email protected] or [email protected]

1

http://arxiv.org/abs/math/0303109v1









































time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′)−1.A metric on S2

×I, such that each point is contained in some ǫ-neck, is

called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature staysbounded on both ends, stays bounded on one end and tends to infinity on theother, and tends to infinity on both ends, respectively.

A metric on B3 or RP3 \ B3, such that each point outside some compactsubset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if thescalar curvature stays bounded or tends to infinity on the end, respectively.

We denote by ǫ a fixed small positive constant. In contrast, δ denotes apositive quantity, which is supposed to be as small as needed in each particularargument.

1 Ancient solutions with bounded entropy

1.1 In this section we review some of the results, proved or quoted in [I,§11],correcting a few inaccuracies. We consider smooth solutions gij(t) to the Ricciflow on oriented 3-manifold M , defined for −∞ < t ≤ 0, such that for each tthe metric gij(t) is a complete non-flat metric of bounded nonnegative sectionalcurvature, κ-noncollapsed on all scales for some fixed κ > 0; such solutions willbe called ancient κ-solutions for short. By Theorem I.11.7, the set of all suchsolutions with fixed κ is compact modulo scaling, that is from any sequenceof such solutions (M α, gαij(t)) and points (xα, 0) with R(xα, 0) = 1, we canextract a smoothly (pointed) convergent subsequence, and the limit (M, gij(t))belongs to the same class of solutions. (The assumption in I.11.7. that M α

be noncompact was clearly redundant, as it was not used in the proof. Notealso that M need not have the same topology as M α.) Moreover, accordingto Proposition I.11.2, the scalings of any ancient κ-solution gij(t) with factors

(−t)−1 about appropriate points converge along a subsequence of t → −∞ toa non-flat gradient shrinking soliton, which will be called an asymptotic solitonof the ancient solution. If the sectional curvature of this asymptotic soliton isnot strictly positive, then by Hamilton’s strong maximum principle it admitslocal metric splitting, and it is easy to see that in this case the soliton is eitherthe round infinite cylinder, or its Z2 quotient, containing one-sided projectiveplane. If the curvature is strictly positive and the soliton is compact, then ithas to be a metric quotient of the round 3-sphere, by [H 1]. The noncompactcase is ruled out below.

1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curva-ture.

Proof. A gradient shrinking soliton gij(t),−∞

< t < 0, satisfies the equation

∇i∇jf + Rij + 1

2tgij = 0 (1.1)

Differentiating and switching the order of differentiation, we get

∇iR = 2Rij∇jf (1.2)

2



Fix some t < 0, say t = −1, and consider a long shortest geodesic γ (s), 0 ≤s

≤ s; let x = γ (0), x = γ (s), X (s) = γ (s). Since the curvature is bounded and

positive, it is clear from the second variation formula that s0 Ric(X, X )ds ≤

const. Therefore, s0 |Ric(X, ·)|2ds ≤ const, and

s0 |Ric(X, Y )|ds ≤ const(

√ s +

1) for any unit vector field Y along γ, orthogonal to X. Thus by integrating(1.1) we get X · f (γ (s)) ≥ s

2 +const, |Y · f (γ (s))| ≤ const(√

s + 1). We concludethat at large distances from x0 the function f has no critical points, and itsgradient makes small angle with the gradient of the distance function from x0.

Now from (1.2) we see that R is increasing along the gradient curves of f,in particular, R = lim sup R > 0. If we take a limit of our soliton about points(xα, −1) where R(xα) → R, then we get an ancient κ-solution, which splitsoff a line, and it follows from I.11.3, that this solution is the shrinking roundinfinite cylinder with scalar curvature R at time t = −1. Now comparing theevolution equations for the scalar curvature on a round cylinder and for the

asymptotic scalar curvature on a shrinking soliton we conclude that ¯R = 1.Hence, R(x) < 1 when the distance from x to x0 is large enough, and R(x) → 1

when this distance tends to infinity.Now let us check that the level surfaces of f, sufficiently distant from x0, are

convex. Indeed, if Y is a unit tangent vector to such a surface, then ∇Y ∇Y f =12 − Ric(Y, Y ) ≥ 1

2 − R

2 > 0. Therefore, the area of the level surfaces grows as f

increases, and is converging to the area of the round sphere of scalar curvatureone. On the other hand, the intrinsic scalar curvature of a level surface turnsout to be less than one. Indeed, denoting by X the unit normal vector, thisintrinsic curvature can be computed as

R − 2Ric(X, X ) + 2det(Hessf )

|∇f |2 ≤ R − 2Ric(X, X ) + (1 − R + Ric(X, X ))2

2|∇f |2 < 1

when R is close to one and |∇f | is large. Thus we get a contradiction to theGauss-Bonnet formula.

1.3 Now, having listed all the asymptotic solitons, we can classify the ancientκ-solutions. If such a solution has a compact asymptotic soliton, then it is itself a metric quotient of the round 3-sphere, because the positive curvature pinchingcan only improve in time [H 1]. If the asymptotic soliton contains the one-sidedprojective plane, then the solution has a Z2 cover, whose asymptotic soliton isthe round infinite cylinder. Finally, if the asymptotic soliton is the cylinder,thenthe solution can be either noncompact (the round cylinder itself, or the Bryantsoliton, for instance), or compact. The latter possibility, which was overlookedin the first paragraph of [I.11.7], is illustrated by the example below, which alsogives the negative answer to the question in the very end of [I.5.1].

1.4 Example. Consider a solution to the Ricci flow, starting from a metricon S3 that looks like a long round cylinder S2 × I (say, with radius one andlength L >> 1), with two spherical caps, smoothly attached to its boundarycomponents. By [H 1] we know that the flow shrinks such a metric to a pointin time, comparable to one (because both the lower bound for scalar curvatureand the upper bound for sectional curvature are comparable to one) , and after

3



normalization, the flow converges to the round 3-sphere. Scale the initial metricand choose the time parameter in such a way that the flow starts at time t0 =

t0(L) < 0, goes singular at t = 0, and at t = −1 has the ratio of the maximalsectional curvature to the minimal one equal to 1 + ǫ. The argument in [I.7.3]shows that our solutions are κ-noncollapsed for some κ > 0 independent of L.We also claim that t0(L) → −∞ as L → ∞. Indeed, the Harnack inequality of

Hamilton [H 3] implies that Rt ≥ Rt0−t , hence R ≤ 2(−1−t0)t−t0

for t ≤ −1,

and then the distance change estimate ddt

distt(x, y) ≥ −const

Rmax(t)from [H 2,§17] implies that the diameter of gij(t0) does not exceed −const · t0,which is less than L

√ −t0 unless t0 is large enough. Thus, a subsequence of our solutions with L → ∞ converges to an ancient κ-solution on S3, whoseasymptotic soliton can not be anything but the cylinder.

1.5 The important conclusion from the classification above and the proof of Proposition I.11.2 is that there exists κ0 > 0, such that every ancient κ-solution

is either κ0-solution, or a metric quotient of the round sphere. Therefore, thecompactness theorem I.11.7 implies the existence of a universal constant η, suchthat at each point of every ancient κ-solution we have estimates

|∇R| < ηR32 , |Rt| < ηR2 (1.3)

Moreover, for every sufficiently small ǫ > 0 one can find C 1,2 = C 1,2(ǫ), suchthat for each point (x, t) in every ancient κ-solution there is a radius r, 0 < r <

C 1R(x, t)−12 , and a neighborhood B, B(x,t,r) ⊂ B ⊂ B(x,t, 2r), which falls

into one of the four categories:(a) B is a strong ǫ-neck (more precisely, the slice of a strong ǫ-neck at its

maximal time), or(b) B is an ǫ-cap, or

(c) B is a closed manifold, diffeomorphic to S3

or RP3

, or(d) B is a closed manifold of constant positive sectional curvature;furthermore, the scalar curvature in B at time t is between C −1

2 R(x, t) and

C 2R(x, t), its volume in cases (a),(b),(c) is greater than C −12 R(x, t)−

32 , and in

case (c) the sectional curvature in B at time t is greater than C −12 R(x, t).

2 The standard solution

Consider a rotationally symmetric metric on R3 with nonnegative sectional cur-vature, which splits at infinity as the metric product of a ray and the round2-sphere of scalar curvature one. At this point we make some choice for themetric on the cap, and will refer to it as the standard cap; unfortunately, the

most obvious choice, the round hemisphere, does not fit, because the metric onR3 would not be smooth enough, however we can make our choice as close to

it as we like. Take such a metric on R3 as the initial data for a solution gij(t)to the Ricci flow on some time interval [0, T ), which has bounded curvature foreach t ∈ [0, T ).

Claim 1. The solution is rotationally symmetric for all t.

4



Indeed, if ui is a vector field evolving by uit = ui + Rijuj , then vij = ∇iuj

evolves by (vij)t =

vij + 2Rikjlvkl −

Rikvkj −

Rkjvik. Therefore, if ui was aKilling field at time zero, it would stay Killing by the maximum principle. It isalso clear that the center of the cap, that is the unique maximum point for theBusemann function, and the unique point, where all the Killing fields vanish,retains these properties, and the gradient of the distance function from thispoint stays orthogonal to all the Killing fields. Thus, the rotational symmetryis preserved.

Claim 2. The solution converges at infinity to the standard solution on the round infinite cylinder of scalar curvature one. In particular, T ≤ 1.

Claim 3. The solution is unique.Indeed, using Claim 1, we can reduce the linearized Ricci flow equation to

the system of two equations on (−∞, +∞) of the following type

f t = f ′′ + a1f ′ + b1g′ + c1f + d1g, gt = a2f ′ + b2g′ + c2f + d2g,

where the coefficients and their derivatives are bounded, and the unknowns f, gand their derivatives tend to zero at infinity by Claim 2. So we get uniqueness

by looking at the integrals A−A

(f 2 + g2) as A → ∞.Claim 4. The solution can be extended to the time interval [0, 1).Indeed, we can obtain our solution as a limit of the solutions on S3, starting

from the round cylinder S2 × I of length L and scalar curvature one, with twocaps attached; the limit is taken about the center p of one of the caps, L → ∞.Assume that our solution goes singular at some time T < 1. Take T 1 < T very close to T, T − T 1 << 1 − T. By Claim 2, given δ > 0, we can findL, D < ∞, depending on δ and T 1, such that for any point x at distance Dfrom p at time zero, in the solution with L ≥ L, the ball B (x, T 1, 1) is δ -close

to the corresponding ball in the round cylinder of scalar curvature (1 − T 1)

−1

.We can also find r = r(δ, T ), independent of T 1, such that the ball B(x, T 1, r)is δ -close to the corresponding euclidean ball. Now we can apply TheoremI.10.1 and get a uniform estimate on the curvature at x as t → T , providedthat T − T 1 < ǫ2r(δ, T )2. Therefore, the t → T limit of our limit solutionon the capped infinite cylinder will be smooth near x. Thus, this limit willbe a positively curved space with a conical point. However, this leads to acontradiction via a blow-up argument; see the end of the proof of the Claim 2in I.12.1.

The solution constructed above will be called the standard solution.Claim 5. The standard solution satisfies the conclusions of 1.5 , for an

appropriate choice of ǫ, η,C 1(ǫ), C 2(ǫ), except that the ǫ-neck neighborhood need not be strong; more precisely, we claim that if (x, t) has neither an ǫ-cap neigh-

borhood as in 1.5(b), nor a strong ǫ-neck neighborhood as in 1.5(a), then x is not in B( p, 0, ǫ−1), t < 3/4, and there is an ǫ-neck B(x,t,ǫ−1r), such that the solution in P (x,t,ǫ−1r, −t) is, after scaling with factor r−2, ǫ-close to the appropriate piece of the evolving round infinite cylinder.

Moreover, we have an estimate Rmin(t) ≥ const · (1 − t)−1.

5



Indeed, the statements follow from compactness and Claim 2 on compactsubintervals of [0, 1), and from the same arguments as for ancient solutions,

when t is close to one.

3 The structure of solutions at the first singular

time

Consider a smooth solution gij(t) to the Ricci flow on M × [0, T ), where M isa closed oriented 3-manifold, T < ∞. Assume that curvature of gij(t) does notstay bounded as t → T. Recall that we have a pinching estimate Rm ≥ −φ(R)Rfor some function φ decreasing to zero at infinity [H 4,§4], and that the solution isκ-noncollapsed on the scales ≤ r for some κ > 0, r > 0 [I, §4].Then by TheoremI.12.1 and the conclusions of 1.5 we can find r = r(ǫ) > 0, such that each point(x, t) with R(x, t)

≥ r−2 satisfies the estimates (1.3) and has a neighborhood,

which is either an ǫ-neck, or an ǫ-cap, or a closed positively curved manifold.In the latter case the solution becomes extinct at time T, so we don’t need toconsider it any more.

If this case does not occur, then let Ω denote the set of all points in M,where curvature stays bounded as t → T . The estimates (1.3) imply that Ω isopen and that R(x, t) → ∞ as t → T for each x ∈ M \Ω. If Ω is empty, thenthe solution becomes extinct at time T and it is entirely covered by ǫ-necks andcaps shortly before that time, so it is easy to see that M is diffeomorphic toeither S3, or RP3, or S2 × S

1, or RP3 ♯ RP3.Otherwise, if Ω is not empty, we may (using the local derivative estimates

due to W.-X.Shi, see [H 2,§13]) consider a smooth metric gij on Ω, which is thelimit of gij(t) as t → T. Let Ωρ for some ρ < r denotes the set of points x ∈ Ω,

where the scalar curvature R(x) ≤ ρ−2

. We claim that Ωρ is compact. Indeed, if R(x) ≤ ρ−2, then we can estimate the scalar curvature R(x, t) on [T −η−1ρ2, T )using (1.3), and for earlier times by compactness, so x is contained in Ω with aball of definite size, depending on ρ.

Now take any ǫ-neck in (Ω, gij) and consider a point x on one of its boundarycomponents. If x ∈ Ω\Ωρ, then there is either an ǫ-cap or an ǫ-neck, adjacentto the initial ǫ-neck. In the latter case we can take a point on the boundary of the second ǫ-neck and continue. This procedure can either terminate when wereach a point in Ωρ or an ǫ-cap, or go on indefinitely, producing an ǫ-horn. Thesame procedure can be repeated for the other boundary component of the initialǫ-neck. Therefore, taking into account that Ω has no compact components, weconclude that each ǫ-neck of (Ω, gij) is contained in a subset of Ω of one of thefollowing types:

(a) An ǫ-tube with boundary components in Ωρ, or(b) An ǫ-cap with boundary in Ωρ, or(c) An ǫ-horn with boundary in Ωρ, or(d) A capped ǫ-horn, or(e) A double ǫ-horn.

6



Clearly, each ǫ-cap, disjoint from Ωρ, is also contained in one of the subsetsabove. It is also clear that there is a definite lower bound (depending on ρ) for

the volume of subsets of types (a),(b),(c), so there can be only finite number of them. Thus we can conclude that there is only a finite number of componentsof Ω, containing points of Ωρ, and every such component has a finite numberof ends, each being an ǫ-horn. On the other hand, every component of Ω,containing no points of Ωρ, is either a capped ǫ-horn, or a double ǫ-horn.

Now, by looking at our solution for times t just before T, it is easy to seethat the topology of M can be reconstructed as follows: take the componentsΩj , 1 ≤ j ≤ i of Ω which contain points of Ωρ, truncate their ǫ-horns, and glue tothe boundary components of truncated Ωj a collection of tubes S2 × I and capsB3 or RP3\B3. Thus, M is diffeomorphic to a connected sum of Ωj , 1 ≤ j ≤ i,

with a finite number of S2 × S1 (which correspond to gluing a tube to two

boundary components of the same Ωj), and a finite number of RP3; here Ωjdenotes Ωj with each ǫ-horn one point compactified.

4 Ricci flow with cutoff

4.1 Suppose we are given a collection of smooth solutions gij(t) to the Ricci flow,defined on M k× [t−k , t+k ), which go singular as t → t+k . Let (Ωk, gkij) be the limits

of the corresponding solutions as t → t+k , as in the previous section. Suppose

also that for each k we have t−k = t+k−1, and (Ωk−1, gk−1ij ) and (M k, gkij(t−k ))

contain compact (possibly disconnected) three-dimensional submanifolds withsmooth boundary, which are isometric. Then we can identify these isometricsubmanifolds and talk about the solution to the Ricci flow with surgery on theunion of all [t−k , t+k ).

Fix a small number ǫ > 0 which is admissible in sections 1,2. In this sectionwe consider only solutions to the Ricci flow with surgery, which satisfy thefollowing a priori assumptions:

(pinching) There exists a function φ, decreasing to zero at infinity, such thatRm ≥ −φ(R)R,

(canonical neighborhood) There exists r > 0, such that every point wherescalar curvature is at least r−2 has a neighborhood, satisfying the conclusionsof 1.5. (In particular, this means that if in case (a) the neighborhood in ques-tion is B (x0, t0, ǫ−1r0), then the solution is required to be defined in the wholeP (x0, t0, ǫ−1r0, −r20); however, this does not rule out a surgery in the time in-terval (t0 − r20, t0), that occurs sufficiently far from x0.)

Recall that from the pinching estimate of Ivey and Hamilton, and TheoremI.12.1, we know that the a priori assumptions above hold for a smooth solution

on any finite time interval. For Ricci flow with surgery they will be justified inthe next section.4.2 Claim 1. Suppose we have a solution to the Ricci flow with surgery, sat-

isfying the canonical neighborhood assumption, and let Q = R(x0, t0)+r−2. Then

we have estimate R(x, t) ≤ 8Q for those (x, t) ∈ P (x0, t0, 12η−1Q−12 , − 1

8η−1Q−1), for which the solution is defined.

7



Indeed, this follows from estimates (1.3).Claim 2. For any A <

∞ one can find Q = Q(A) <

∞ and ξ = ξ (A) > 0

with the following property. Suppose we have a solution to the Ricci flow with surgery, satisfying the pinching and the canonical neighborhood assump-tions. Let γ be a shortest geodesic in gij(t0) with endpoints x0 and x, such that R(y, t0) > r−2 for each y ∈ γ, and Q0 = R(x0, t0) is so large that φ(Q0) < ξ. Finally, let z ∈ γ be any point satisfying R(z, t0) > 10C 2R(x0, t0).

Then distt0(x0, z) ≥ AQ−

12

0 whenever R(x, t0) > QQ0.The proof is exactly the same as for Claim 2 in Theorem I.12.1; in the very

end of it, when we get a piece of a non-flat metric cone as a blow-up limit,we get a contradiction to the canonical neighborhood assumption, because thecanonical neighborhoods of types other than (a) are not close to a piece of metric cone, and type (a) is ruled out by the strong maximum principle, sincethe ǫ-neck in question is strong.

4.3 Suppose we have a solution to the Ricci flow with surgery, satisfying oura priori assumptions, defined on [0, T ), and going singular at time T . Choose asmall δ > 0 and let ρ = δr. As in the previous section, consider the limit (Ω, gij)of our solution as t → T, and the corresponding compact set Ωρ.

Lemma. There exists a radius h, 0 < h < δρ, depending only on δ, ρ and the pinching function φ, such that for each point x with h(x) = R−

12 (x) ≤ h in an ǫ-

horn of (Ω, gij) with boundary in Ωρ, the neighborhood P (x , T , δ −1h(x), −h2(x))is a strong δ -neck.

Proof. An argument by contradiction. Assuming the contrary, take a se-quence of solutions with limit metrics (Ωα, gαij) and points xα with h(xα) → 0.Since xα lies deeply inside an ǫ-horn, its canonical neighborhood is a strongǫ-neck. Now Claim 2 gives the curvature estimate that allows us to take a limitof appropriate scalings of the metrics gαij on [T − h2(xα), T ] about xα, for a sub-

sequence of α → ∞. By shifting the time parameter we may assume that thelimit is defined on [−1, 0]. Clearly, for each time in this interval, the limit is acomplete manifold with nonnegative sectional curvature; moreover, since xα wascontained in an ǫ-horn with boundary in Ωαρ , and h(xα)/ρ → 0, this manifoldhas two ends. Thus, by Toponogov, it admits a metric splitting S2 × R. Thisimplies that the canonical neighborhood of the point (xα, T − h2(xα)) is also of type (a), that is a strong ǫ-neck, and we can repeat the procedure to get thelimit, defined on [−2, 0], and so on. This argument works for the limit in anyfinite time interval [−A, 0], because h(xα)/ρ → 0. Therefore, we can constructa limit on [−∞, 0]; hence it is the round cylinder, and we get a contradiction.

4.4 Now we can specialize our surgery and define the Ricci flow with δ -cutoff.Fix δ > 0, compute ρ = δ r and determine h from the lemma above. Given asmooth metric gij on a closed manifold, run the Ricci flow until it goes singular

at some time t+; form the limit (Ω, gij). If Ωρ is empty, the procedure stopshere, and we say that the solution became extinct. Otherwise we remove thecomponents of Ω which contain no points of Ωρ, and in every ǫ-horn of each of the remaining components we find a δ -neck of radius h, cut it along the middletwo-sphere, remove the horn-shaped end, and glue in an almost standard cap

8



scalar curvature on this subset does not exceed 2Qh−2. Now if for each point inB( p, T 0, A1h) the solution is defined on [t1, t2], then we can repeat the procedure,

defining A2 etc. Continuing this way, we eventually define AN , and it wouldremain to choose δ so small, and correspondingly A0 so large, that AN > A.

Now assume that for some k, 0 ≤ k < N, and for some x ∈ B( p, T 0, Akh) thesolution is defined on [t0, tk] but not on [tk, tk+1]. Then we can find a surgerytime t+ ∈ [tk, tk+1], such that the solution on B( p, T 0, Akh) is defined on [t0, t+),but for some points of this ball it is not defined past t+. Clearly, the A−1

k+1-closeness assertion holds on P ( p, T 0, Ak+1h, t+ − T 0). On the other hand, thesolution on B( p, T 0, Akh) is at least ǫ-close to the standard one for all t ∈ [tk, t+),hence no point of this set can be the center of a δ -neck neighborhood at timet+. However, the surgery is always done along the middle two-sphere of such aneck. It follows that for each point of B ( p, T 0, Akh) the solution terminates att+.

4.6 Corollary. For any l < ∞

one can find A = A(l) < ∞

and θ =θ(l), 0 < θ < 1, with the following property. Suppose we are in the situation of the lemma above, with δ < δ (A, θ). Consider smooth curves γ in the set B( p, T 0, Ah), parametrized by t ∈ [T 0, T γ ], such that γ (T 0) ∈ B( p, T 0,Ah/2)and either T γ = T 1 < T , or T γ < T 1 and γ (T γ ) ∈ ∂B( p, T 0, Ah). Then T γT 0

(R(γ (t), t) + |γ (t)|2)dt > l.Proof. Indeed, if T γ = T 1, then on the standard solution we would have T γT 0

R(γ (t), t)dt ≥ const θ0 (1 − t)−1dt = −const · (log(1 − θ))−1, so by choosing

θ sufficiently close to one we can handle this case. Then we can choose A so largethat on the standard solution distt( p, ∂B( p, 0, A)) ≥ 3A/4 for each t ∈ [0, θ].

Now if γ (T γ ) ∈ ∂B ( p, T 0, Ah) then T γT 0

|γ (t)|2dt ≥ A2/100, so by taking A largeenough, we can handle this case as well.

4.7 Corollary. For any Q <

∞ there exists θ = θ(Q), 0 < θ < 1 with

the following property. Suppose we are in the situation of the lemma above,with δ < δ (A, θ), A > ǫ−1. Suppose that for some point x ∈ B( p, T 0, Ah) the solution is defined at x (at least) on [T 0, T x], T x ≤ T, and satisfies Q−1R(x, t) ≤R(x, T x) ≤ Q(T x − T 0)−1 for all t ∈ [T 0, T x]. Then T x ≤ T 0 + θh2.

Proof. Indeed, if T x > T 0 + θh2, then by lemma R(x, T 0 + θh2) ≥const · (1 − θ)−1h−2, whence R(x, T x) ≥ const · Q−1(1 − θ)−1h−2, and T x− T 0 ≤const · Q2(1 − θ)h2 < θh2 if θ is close enough to one.

5 Justification of the a priori assumption

5.1 Let us call a riemannian manifold (M, gij) normalized if M is a closedoriented 3-manifold, the sectional curvatures of gij do not exceed one in absolute

value, and the volume of every metric ball of radius one is at least half the volumeof the euclidean unit ball. For smooth Ricci flow with normalized initial datawe have, by [H 4, 4.1], at any time t > 0 the pinching estimate

Rm ≥ −φ(R(t + 1))R, (5.1)

10



L(γ, τ ) = t0t0−τ

√ t0 − t(R(γ (t), t) + |γ (t)|2)dt. We can also define L+(γ, τ ) by

replacing in the previous formula R with R+ = max(R, 0). ThenL+

≤ L+4T

√ T

because R ≥ −6 by the maximum principle and normalization. Now suppose wecould show that every barely admissible curve with endpoints (x0, t0) and (x, t),where t ∈ [2i−1ǫ, T ), has L+ > 2ǫ−2T

√ T ; then we could argue that either there

exists a point (x, t), t ∈ [2i−1ǫ, 2iǫ], such that R(x, t) ≤ r−2i and L+ ≤ ǫ−2T

√ T ,

in which case we can take this point in place of (x, ǫ) in the argument of theprevious paragraph, and obtain (using Claim 1 in 4.2) an estimate for κ interms of ri, κi, T, or for any γ , defined on [2i−1ǫ, t0], γ (t0) = x0, we have L+ ≥min(ǫ−2T

√ T , 23(2i−1ǫ)

32 r−2i ) > ǫ−2T

√ T , which is in contradiction with the

assumed bound for barely admissible curves and the bound min l(x, t0−2i−1ǫ) ≤3/2, valid in the smooth case. Thus, to conclude the proof it is sufficient to checkthe following assertion.

5.3 Lemma. For any L < ∞ one can find δ = δ (L, r0) > 0 with the

following property. Suppose that in the situation of the previous lemma we have a curve γ, parametrized by t ∈ [T 0, t0], 2i−1ǫ ≤ T 0 < t0, such that γ (t0) = x0,T 0 is a surgery time, and γ (T 0) ∈ B( p, T 0, ǫ−1h), where p corresponds to the center of the cap, and h is the radius of the δ -neck. Then we have an estimate t0T 0

√ t0 − t(R+(γ (t), t) + |γ (t)|2)dt ≥ L.

Proof. It is clear that if we take t = ǫr40L−2, then either γ satisfies ourestimate, or γ stays in P (x0, t0, r0, −t) for t ∈ [t0 − t, t0]. In the latter

case our estimate follows from Corollary 4.6, for l = L(t)−12 , since clearly

T γ < t0 − t when δ is small enough.5.4 Proof of proposition. Assume the contrary, and let the sequences rα, δ αβ

be such that rα → 0 as α → ∞, δ αβ → 0 as β → ∞ with fixed α, and let(M αβ , gαβij ) be normalized initial data for solutions to the Ricci flow with δ (t)-

cutoff, δ (t) < δ αβ on [2i−1ǫ, 2i+1ǫ], which satisfy the statement on [0, 2iǫ], but vi-olate the canonical neighborhood assumption with parameter rα on [2iǫ, 2i+1ǫ].Slightly abusing notation, we’ll drop the indices α, β when we consider an indi-vidual solution.

Let t be the first time when the assumption is violated at some point x;clearly such time exists, because it is an open condition. Then by lemma 5.2 wehave uniform κ-noncollapsing on [0, t]. Claims 1,2 in 4.2 are also valid on [0, t];moreover, since h << r, it follows from Claim 1 that the solution is defined onthe whole parabolic neighborhood indicated there in case R(x0, t0) ≤ r−2.

Scale our solution about (x, t) with factor R(x, t) ≥ r−2 and take a limit forsubsequences of α, β → ∞. At time t, which we’ll shift to zero in the limit, thecurvature bounds at finite distances from x for the scaled metric are ensuredby Claim 2 in 4.2. Thus, we get a smooth complete limit of nonnegative sec-

tional curvature, at time zero. Moreover, the curvature of the limit is uniformlybounded, since otherwise it would contain ǫ-necks of arbitrarily small radius.Let Q0 denote the curvature bound. Then, if there was no surgery, we could,

using Claim 1 in 4.2, take a limit on the time interval [−ǫη−1Q−10 , 0]. To prevent

this, there must exist surgery times T 0 ∈ [ t − ǫη−1Q−10 R−1(x, t), t] and points

x with dist2T 0(x, x)R−1(x, t) uniformly bounded as α, β → ∞, such that the

12



solution at x is defined on [T 0, t], but not before T 0. Using Claim 2 from 4.2 attime T 0, we see that R(x, t)h2(T 0) must be bounded away from zero. Therefore,

in this case we can apply Corollary 4.7, Lemma 4.5 and Claim 5 in section 2 toshow that the point (x, t) in fact has a canonical neighborhood, contradicting itschoice. (It is not excluded that the strong ǫ-neck neighborhood extends to timesbefore T 0, where it is a part of the strong δ -neck that existed before surgery.)

Thus we have a limit on a certain time interval. Let Q1 be the curvaturebound for this limit. Then we either can construct a limit on the time interval[−ǫη−1(Q−1

0 + Q−11 ), 0], or there is a surgery, and we get a contradiction as

before. We can continue this procedure indefinitely, and the final part of theproof of Theorem I.12.1 shows that the bounds Qk can not go to infinity whilethe limit is defined on a bounded time interval. Thus we get a limit on (−∞, 0],which is κ-noncollapsed by Lemma 5.2, and this means that (x, t) has a canonicalneighborhood by the results of section 1 - a contradiction.

6 Long time behavior I

6.1 Let us summarize what we have achieved so far. We have shown the exis-tence of decreasing (piecewise constant) positive functions r(t) and δ (t) (whichwe may assume converging to zero at infinity), such that if (M, gij) is a normal-ized manifold, and 0 < δ (t) < δ (t), then there exists a solution to the Ricci flowwith δ (t)-cutoff on the time interval [0, +∞], starting from (M, gij) and sat-isfying on each subinterval [0, t] the canonical neighborhood assumption withparameter r(t), as well as the pinching estimate (5.1).

In particular, if the initial data has positive scalar curvature, say R ≥ a > 0,then the solution becomes extinct in time at most 3

2a , and it follows that M inthis case is diffeomorphic to a connected sum of several copies of S2

×S1 and

metric quotients of round S3. ( The topological description of 3-manifolds withpositive scalar curvature modulo quotients of homotopy spheres was obtained bySchoen-Yau and Gromov-Lawson more than 20 years ago, see [G-L] for instance;in particular, it is well known and easy to check that every manifold that canbe decomposed in a connected sum above admits a metric of positive scalarcurvature.) Moreover, if the scalar curvature is only nonnegative, then by thestrong maximum principle it instantly becomes positive unless the metric is(Ricci-)flat; thus in this case, we need to add to our list the flat manifolds.

However, if the scalar curvature is negative somewhere, then we need to workmore in order to understand the long tome behavior of the solution. To achievethis we need first to prove versions of Theorems I.12.2 and I.12.3 for solutionswith cutoff.

6.2 Correction to Theorem I.12.2. Unfortunately, the statement of Theorem I.12.2 was incorrect. The assertion I had in mind is as follows:

Given a function φ as above, for any A < ∞ there exist K = K (A) < ∞and ρ = ρ(A) > 0 with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which satisfies the assumptions of theorem 8.2 for some x0, r0 with φ(r−2

0 ) < ρ. Then

13



R(x, r20) ≤ Kr−20 whenever distr2

0(x, x0) < Ar0.

It is this assertion that was used in the proof of Theorem I.12.3 and Corollary

I.12.4.6.3 Proposition. For any A < ∞ one can find κ = κ(A) > 0, K 1 =

K 1(A) < ∞, K 2 = K 2(A) < ∞, r = r(A) > 0, such that for any t0 < ∞ there exists δ = δ A(t0) > 0, decreasing in t0, with the following property. Suppose we have a solution to the Ricci flow with δ (t)-cutoff on time interval [0, T ], δ (t) <δ (t) on [0, T ], δ (t) < δ on [t0/2, t0], with normalized initial data; assume that the solution is defined in the whole parabolic neighborhood P (x0, t0, r0, −r20), 2r20 <t0, and satisfies |Rm| ≤ r−2

0 there, and that the volume of the ball B(x0, t0, r0)is at least A−1r30. Then

(a) The solution is κ-noncollapsed on the scales less than r0 in the ball B(x0, t0, Ar0).

(b) Every point x ∈ B(x0, t0, Ar0) with R(x, t0) ≥ K 1r−20 has a canonical

neighborhood as in 4.1.(c) If r0 ≤ r√ t0 then R ≤ K 2r−2

0 in B(x0, t0, Ar0).Proof. (a) This is an analog of Theorem I.8.2. Clearly we have κ-noncollapsing

on the scales less than r(t0), so we may assume r(t0) ≤ r0 ≤

t0/2 , and studythe scales ρ, r(t0) ≤ ρ ≤ r0. In particular, for fixed t0 we are interested in thescales, uniformly equivalent to one.

So assume that x ∈ B (x0, t0, Ar0) and the solution is defined in the wholeP (x, t0, ρ, −ρ2) and satisfies |Rm| ≤ ρ−2 there. An inspection of the proof of I.8.2 shows that in order to make the argument work it suffices to check thatfor any barely admissible curve γ , parametrized by t ∈ [tγ , t0], t0 − r20 ≤ tγ ≤ t0,such that γ (t0) = x, we have an estimate

2 t0 −tγ t0

tγ

√ t0

−t(R(γ (t), t) +

|γ (t)

|2)dt

≥C (A)r20 (6.1)

for a certain function C (A) that can be made explicit. Now we would like toconclude the proof by using Lemma 5.3. However, unlike the situation in Lemma5.2, here Lemma 5.3 provides the estimate we need only if t0 − tγ is boundedaway from zero, and otherwise we only get an estimate ρ2 in place of C (A)r20 .Therefore we have to return to the proof of I.8.2.

Recall that in that proof we scaled the solution to make r0 = 1 and workedon the time interval [1/2, 1]. The maximum principle for the evolution equationof the scalar curvature implies that on this time interval we have R ≥ −3. Weconsidered a function of the form h(y, t) = φ(d(y, t))L(y, τ ), where φ is a certain

cutoff function, τ = 1 − t, d(y, t) = distt(x0, y) − A(2t − 1), L(y, τ ) = L(y, τ ) + 7,and L was defined in [I,(7.15)]. Now we redefine L, taking L(y, τ ) = L(y, τ ) +

2√ τ . Clearly, L > 0 because R ≥ −3 and 2√ τ > 4τ 2

for 0 < τ ≤ 1/2. Then thecomputations and estimates of I.8.2 yield

h ≥ −C (A)h − (6 + 1√

τ )φ

14



Now denoting by h0(τ ) the minimum of h(y, 1 − t), we can estimate

ddτ

(log( h0(τ )√ τ

)) ≤ C (A) + 6√ τ + 12τ − 4τ 2

√ τ − 1

2τ ≤ C (A) + 50√

τ , (6.2)

whenceh0(τ ) ≤ √

τ exp(C (A)τ + 100√

τ ), (6.3)

because the left hand side of (6.2) tends to zero as τ → 0 + .Now we can return to our proof, replace the right hand side of (6.1) by the

right hand side of (6.3) times r20 , with τ = r−20 (t0 − tγ ), and apply Lemma 5.3.

(b) Assume the contrary, take a sequence K α1 → ∞ and consider the solu-tions violating the statement. Clearly, K α1 (rα0 )−2 < (r(tα0 ))−2, whence tα0 → ∞;

When K 1 is large enough, we can, arguing as in the proof of Claim 1 in[I.10.1], find a point (x, t), x ∈ B(x0, t, 2Ar0), t ∈ [t0 − r20/2, t0], such that Q =

R(x,¯t) > K 1r

−2

0 , (x,¯t) does not satisfy the canonical neighborhood assumption,but each point (x, t) ∈ P with R(x, t) ≥ 4 Q does, where P is the set of all (x, t)

satisfying t − 14K 1 Q−1 ≤ t ≤ t, distt(x0, x) ≤ distt(x0, x) + K

12

1 Q−

12 . (Note

that P is not a parabolic neighborhood.) Clearly we can use (a) with slightlydifferent parameters to ensure κ-noncollapsing in P .

Now we apply the argument from 5.4. First, by Claim 2 in 4.2, for anyA < ∞ we have an estimate R ≤ Q( A) Q in B(x, t, A Q−

12 ) when K 1 is large

enough; therefore we can take a limit as α → ∞ of scalings with factor Q about(x, t), shifting the time t to zero; the limit at time zero would be a smoothcomplete nonnegatively curved manifold. Next we observe that this limit hascurvature uniformly bounded, say, by Q0, and therefore, for each fixed A and forsufficiently large K 1, the parabolic neighborhood P (x, t, A Q−

12 , −ǫη−1Q−1

0 Q−1)

is contained in P . (Here we use the estimate of distance change, given by Lemma

I.8.3(a).) Thus we can take a limit on the interval [−ǫη−

1Q−

10 , 0]. (The possibil-ity of surgeries is ruled out as in 5.4) Then we repeat the procedure indefinitely,getting an ancient κ-solution in the limit, which means a contradiction.

(c) If x ∈ B(x0, t0, Ar0) has very large curvature, then on the shortestgeodesic γ at time t0, that connects x0 and x, we can find a point y, suchthat R(y, t0) = K 1(A)r−2

0 and the curvature is larger at all points of the seg-ment of γ between x and y. Then our statement follows from Claim 2 in 4.2,applied to this segment.

From now on we redefine the function δ (t) to be min(δ (t), δ 2t(2t)), so thatthe proposition above always holds for A = t0.

6.4 Proposition. There exist τ > 0, r > 0, K < ∞ with the following property. Suppose we have a solution to the Ricci flow with δ (t)-cutoff on the time interval [0, t0], with normalized initial data. Let r0, t0 satisfy 2C 1h

≤r0

≤r√ t0, where h is the maximal cutoff radius for surgeries in [t0/2, t0], and assume that the ball B(x0, t0, r0) has sectional curvatures at least −r−2

0 at each point,and the volume of any subball B(x, t0, r) ⊂ B(x0, t0, r0) with any radius r > 0 is at least (1 − ǫ) times the volume of the euclidean ball of the same radius. Then the solution is defined in P (x0, t0, r0/4, −τ r20) and satisfies R < Kr−2

0 there.

15



Proof. Let us first consider the case r0 ≤ r(t0). Then clearly R(x0, t0) ≤C 21r−2

0 , since an ǫ-neck of radius r can not contain an almost euclidean ball of

radius ≥ r. Thus we can take K = 2C 21 , τ = ǫη−1C −21 in this case, and sincer0 ≥ 2C 1h, the surgeries do not interfere in P (x0, t0, r0/4, −τ r20).

In order to handle the other case r(t0) < r0 ≤ r√

t0 we need a couple of lemmas.

6.5 Lemma. There exist τ 0 > 0 and K 0 < ∞, such that if we have a smooth solution to the Ricci flow in P (x0, 0, 1, −τ ), τ ≤ τ 0, having sectional curvatures at least −1, and the volume of the ball B(x0, 0, 1) is at least (1 − ǫ) times the volume of the euclidean unit ball, then

(a) R ≤ K 0τ −1 in P (x0, 0, 1/4, −τ/2), and (b) the ball B(x0, 1/4, −τ ) has volume at least 1

10 times the volume of the

euclidean ball of the same radius.The proof can be extracted from the proof of Lemma I.11.6.6.6 Lemma. For any w > 0 there exists θ0 = θ0(w) > 0, such that if

B(x, 1) is a metric ball of volume at least w, compactly contained in a manifold without boundary with sectional curvatures at least −1, then there exists a ball B(y, θ0) ⊂ B(x, 1), such that every subball B(z, r) ⊂ B(y, θ0) of any radius rhas volume at least (1 − ǫ) times the volume of the euclidean ball of the same radius.

This is an elementary fact from the theory of Aleksandrov spaces.6.7 Now we continue the proof of the proposition. We claim that one can

take τ = min(τ 0/2, ǫη−1C −21 ), K = max(2K 0τ −1, 2C 21). Indeed, assume the con-

trary, and take a sequence of rα → 0 and solutions, violating our assertion forthe chosen τ, K. Let tα0 be the first time when it is violated, and let B(xα0 , tα0 , rα0 )be the counterexample with the smallest radius. Clearly rα0 > r(tα0 ) and(rα0 )2(tα0 )−1 → 0 as α → ∞.

Consider any ball B(x1, t0, r) ⊂ B (x0, t0, r0), r < r0. Clearly we can applyour proposition to this ball and get the solution in P (x1, t0, r/4, −τ r2) with thecurvature bound R < Kr−2. Now if r20t−1

0 is small enough, then we can applyproposition 6.3(c) to get an estimate R(x, t) ≤ K ′(A)r−2 for (x, t) satisfyingt ∈ [t0 − τ r2/2, t0], distt(x, x1) < Ar, for some function K ′(A) that can bemade explicit. Let us choose A = 100r0r−1; then we get the solution witha curvature estimate in P (x0, t0, r0, −t), where t = K ′(A)−1r2. Now thepinching estimate implies Rm ≥ −r−2

0 on this set, if r20t−10 is small enough

while rr−10 is bounded away from zero. Thus we can use lemma 6.5(b) to

estimate the volume of the ball B(x0, t0−t, r0/4) by at least 110 of the volume

of the euclidean ball of the same radius, and then by lemma 6.6 we can find asubball B(x2, t0−t, θ0( 110)r0/4), satisfying the assumptions of our proposition.Therefore, if we put r = θ0( 110)r0/4, then we can repeat our procedure as many

times as we like, until we reach the time t0 − τ 0r20 , when the lemma 6.5(b) stops

working. But once we reach this time, we can apply lemma 6.5(a) and get therequired curvature estimate, which is a contradiction.

6.8 Corollary. For any w > 0 one can find τ = τ (w) > 0, K = K (w) <∞, r = r(w) > 0, θ = θ(w) > 0 with the following property. Suppose we have a solution to the Ricci flow with δ (t)-cutoff on the time interval [0, t0], with

16



normalized initial data. Let t0, r0 satisfy θ−1(w)h ≤ r0 ≤ r√

t0, and assume that the ball B(x0, t0, r0) has sectional curvatures at least

−r20 at each point,

and volume at least wr30 . Then the solution is defined in P (x0, t0, r0/4, −τ r20)and satisfies R < Kr−2

0 there.Indeed, we can apply proposition 6.4 to a smaller ball, provided by lemma

6.6, and then use proposition 6.3(c).

7 Long time behavior II

In this section we adapt the arguments of Hamilton [H 4] to a more generalsetting. Hamilton considered smooth Ricci flow with bounded normalized cur-vature; we drop both these assumptions. In the end of [I,13.2] I claimed that thevolumes of the maximal horns can be effectively bounded below, which wouldimply that the solution must be smooth from some time on; however, the argu-

ment I had in mind seems to be faulty. On the other hand, as we’ll see below,the presence of surgeries does not lead to any substantial problems.

From now on we assume that our initial manifold does not admit a metricwith nonnegative scalar curvature, and that once we get a component withnonnegative scalar curvature, it is immediately removed.

7.1 (cf. [H 4,§2,7]) Recall that for a solution to the smooth Ricci flow thescalar curvature satisfies the evolution equation

d

dtR = R + 2|Ric|2 = R + 2|Ric|2 +

2

3R2, (7.1)

where Ric is the trace-free part of Ric. Then Rmin(t) satisfies ddt

Rmin ≥ 23

R2min,

whence

Rmin(t) ≥ −3

2

1

t + 1/4 (7.2)

for a solution with normalized initial data. The evolution equation for thevolume is d

dtV = −

RdV, in particular

d

dtV ≤ −RminV, (7.3)

whence by (7.2) the function V (t)(t+1/4)−32 is non-increasing in t. Let V denote

its limit as t → ∞.Now the scale invariant quantity R = RminV

23 satisfies

d

dtR(t) ≥ 2

3RV −1

(Rmin − R)dV, (7.4)

which is nonnegative whenever Rmin ≤ 0, which we have assumed from thebeginning of the section. Let R denote the limit of R(t) as t → ∞.

Assume for a moment that V > 0. Then it follows from (7.2) and (7.3) that

Rmin(t) is asymptotic to − 32t ; in other words, RV −

23 = − 3

2 . Now the inequality(7.4) implies that whenever we have a sequence of parabolic neighborhoods

17



P (xα, tα, r√

tα, −r2tα), for tα → ∞ and some fixed small r > 0, such that thescalings of our solution with factor tα smoothly converge to some limit solution,

defined in an abstract parabolic neighborhood P (x, 1, r, −r2), then the scalarcurvature of this limit solution is independent of the space variables and equals− 3

2t at time t ∈ [1 − r2, 1]; moreover, the strong maximum principle for (7.1)implies that the sectional curvature of the limit at time t is constant and equals− 1

4t . This conclusion is also valid without the a priori assumption that V > 0,since otherwise it is vacuous.

Clearly the inequalities and conclusions above hold for the solutions to theRicci flow with δ (t)-cutoff, defined in the previous sections. From now on weassume that we are given such a solution, so the estimates below may dependon it.

7.2 Lemma. (a) Given w > 0, r > 0, ξ > 0 one can find T = T (w,r,ξ ) <∞, such that if the ball B(x0, t0, r

√ t0) at some time t0 ≥ T has volume at

least wr3 and sectional curvature at least −

r−2t−1

0

, then curvature at x0 at time t = t0 satisfies

|2tRij + gij| < ξ. (7.5)

(b) Given in addition A < ∞ and allowing T to depend on A, we can ensure (7.5) for all points in B (x0, t0, Ar

√ t0).

(c) The same is true for P (x0, t0, Ar√

t0, Ar2t0).Proof. (a) If T is large enough then we can apply corollary 6.8 to the ball

B(x0, t0, r0) for r0 = min(r, r(w))√

t0; then use the conclusion of 7.1.(b) The curvature control in P (x0, t0, r0/4, −τr20), provided by corollary 6.8,

allows us to apply proposition 6.3 (a),(b) to a controllably smaller neighborhoodP (x0, t0, r′

0, −(r′

0)2). Thus by 6.3(b) we know that each point in B(x0, t0, Ar√

t0)with scalar curvature at least Q = K ′1(A)r−2

0 has a canonical neighborhood.This implies that for T large enough such points do not exist, since if there

was a point with R larger than Q, there would be a point having a canonicalneighborhood with R = Q in the same ball, and that contradicts the alreadyproved assertion (a). Therefore we have curvature control in the ball in question,and applying 6.3(a) we also get volume control there, so our assertion has beenreduced to (a).

(c) If ξ is small enough, then the solution in the ball B(x0, t0, Ar√

t0) wouldstay almost homothetic to itself on the time interval [t0, t0 + Ar2t0] until (7.5)is violated at some (first) time t′ in this interval. However, if T is large enough,then this violation could not happen, because we can apply the already provedassertion (b) at time t′ for somewhat larger A.

7.3 Let ρ(x, t) denote the radius ρ of the ball B(x,t,ρ) where inf Rm =−ρ−2. It follows from corollary 6.8, proposition 6.3(c), and the pinching estimate(5.1) that for any w > 0 we can find ρ = ρ(w) > 0, such that if ρ(x, t) < ρ

√ t,

thenV ol B(x,t,ρ(x, t)) < wρ3(x, t), (7.6)

provided that t is large enough (depending on w).Let M −(w, t) denote the thin part of M, that is the set of x ∈ M where (7.6)

holds at time t, and let M +(w, t) be its complement. Then for t large enough

18



(depending on w) every point of M + satisfies the assumptions of lemma 7.2.Assume first that for some w > 0 the set M +(w, t) is not empty for a

sequence of t → ∞. Then the arguments of Hamilton [H 4,§8-12] work in oursituation. In particular, if we take a sequence of points xα ∈ M +(w, tα), tα →∞, then the scalings of gαij about xα with factors (tα)−1 converge, along asubsequence of α → ∞, to a complete hyperbolic manifold of finite volume.The limits may be different for different choices of (xα, tα). If none of the limitsis closed, and H 1 is such a limit with the least number of cusps, then, by anargument in [H 4,§8-10], based on hyperbolic rigidity, for all sufficiently smallw′, 0 < w′ < w(H 1), there exists a standard truncation H 1(w′) of H 1, such that,for t large enough, M +(w′/2, t) contains an almost isometric copy of H 1(w′),which in turn contains a component of M +(w′, t); moreover, this embeddedcopy of H 1(w′) moves by isotopy as t increases to infinity. If for some w > 0 thecomplement M +(w, t) \ H 1(w) is not empty for a sequence of t → ∞, then wecan repeat the argument and get another complete hyperbolic manifold H 2, etc.,until we find a finite collection of H j , 1 ≤ j ≤ i, such that for each sufficientlysmall w > 0 the embeddings of H j(w) cover M +(w, t) for all sufficiently large t.

Furthermore, the boundary tori of H j(w) are incompressible in M. This isproved [H 4,§11,12] by a minimal surface argument, using a result of Meeksand Yau. This argument does not use the uniform bound on the normalizedcurvature, and goes through even in the presence of surgeries, because the areaof the least area disk in question can only decrease when we make a surgery.

7.4 Let us redefine the thin part in case the thick one isn’t empty, M −(w, t) =M \(H 1(w)∪...∪H i(w)). Then, for sufficiently small w > 0 and sufficiently larget, M −(w, t) is diffeomorphic to a graph manifold, as implied by the followinggeneral result on collapsing with local lower curvature bound, applied to themetrics t−1gij(t).

Theorem. Suppose (M α

, gα

ij) is a sequence of compact oriented riemannian 3-manifolds, closed or with convex boundary, and wα → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα(x), 0 < ρ < 1, not

exceeding the diameter of the manifold, such that the ball B(x, ρ) in the metric gαij has volume at most wαρ3 and sectional curvatures at least −ρ−2;

(2) each component of the boundary of M α has diameter at most wα, and has a (topologically trivial) collar of length one, where the sectional curvatures are between −1/4 − ǫ and −1/4 + ǫ;

(3) For every w′ > 0 there exist r = r(w′) > 0 and K m = K m(w′) <∞, m = 0, 1, 2..., such that if α is large enough, 0 < r ≤ r, and the ball B(x, r)in gαij has volume at least w′r3 and sectional curvatures at least −r2, then the curvature and its m-th order covariant derivatives at x, m = 1, 2..., are bounded by K 0r−2 and K mr−m−2 respectively.

Then M α

for sufficiently large α are diffeomorphic to graph manifolds.Indeed, there is only one exceptional case, not covered by the theorem above,

namely, when M = M −(w, t), and ρ(x, t), for some x ∈ M, is much larger thanthe diameter d(t) of the manifold, whereas the ratio V (t)/d3(t) is boundedaway from zero. In this case, since by the observation after formula (7.3) the

19



volume V (t) can not grow faster than const · t32 , the diameter does not grow

faster than const

·

√ t, hence if we scale our metrics gij(t) to keep the diameter

equal to one, the scaled metrics would satisfy the assumption (3) of the theoremabove and have the minimum of sectional curvatures tending to zero. Thus wecan take a limit and get a smooth solution to the Ricci flow with nonnegativesectional curvature, but not strictly positive scalar curvature. Therefore, in thisexceptional case M is diffeomorphic to a flat manifold.

The proof of the theorem above will be given in a separate paper; it hasnothing to do with the Ricci flow; its main tool is the critical point theory fordistance functions and maps, see [P,§2] and references therein. The assump-tion (3) is in fact redundant; however, it allows to simplify the proof quite abit, by avoiding 3-dimensional Aleksandrov spaces, and in particular, the non-elementary Stability Theorem.

Summarizing, we have shown that for large t every component of the solutionis either diffeomorphic to a graph manifold, or to a closed hyperbolic manifold,or can be split by a finite collection of disjoint incompressible tori into parts, eachbeing diffeomorphic to either a graph manifold or to a complete noncompacthyperbolic manifold of finite volume. The topology of graph manifolds is wellunderstood [W]; in particular, every graph manifold can be decomposed in aconnected sum of irreducible graph manifolds, and each irreducible one can inturn be split by a finite collection of disjoint incompressible tori into Seifertfibered manifolds.

8 On the first eigenvalue of the operator −4+R

8.1 Recall from [I,§1,2] that Ricci flow is the gradient flow for the first eigenvalue

λ of the operator −

4

+ R; moreover, ddt

λ(t) ≥

2

3

λ2(t) and λ(t)V 23 (t) is non-

decreasing whenever it is nonpositive. We would like to extend these inequalitiesto the case of Ricci flow with δ (t)-cutoff. Recall that we immediately removecomponents with nonnegative scalar curvature.

Lemma. Given any positive continuous function ξ (t) one can chose δ (t)in such a way that for any solution to the Ricci flow with δ (t)-cutoff, with normalized initial data, and any surgery time T 0, after which there is at least one component, where the scalar curvature is not strictly positive, we have an estimate λ+(T 0)−λ−(T 0) ≥ ξ (T 0)(V +(T 0)−V −(T 0)), where V −, V + and λ−, λ+

are the volumes and the first eigenvalues of −4+R before and after the surgery respectively.

Proof. Consider the minimizer a for the functional

(4|∇a|

2

+ Ra2

) (8.1)

under normalization

a2 = 1, for the metric after the surgery on a componentwhere scalar curvature is not strictly positive. Clearly is satisfies the equation

4a = Ra − λ−a (8.2)

20



Observe that since the metric contains an ǫ-neck of radius about r(T 0), we canestimate λ−(T 0) from above by about r(T 0)−2.

Let M cap denote the cap, added by the surgery. It is attached to a longtube, consisting of ǫ-necks of various radii. Let us restrict our attention toa maximal subtube, on which the scalar curvature at each point is at least2λ−(T 0). Choose any ǫ-neck in this subtube, say, with radius r0, and considerthe distance function with range [0, 2ǫ−1r0], whose level sets M z are almostround two-spheres; let M +z ⊃ M cap be the part of M, chopped off by M z. Then

M z

−4aaz =

M +z

(4|∇a|2 + Ra2 − λ−a2) > r−20 /2

M +z

a2

On the other hand,

| M z2aaz

−( M z

a2)z| ≤

const

· M zǫr−1

0 a2

These two inequalities easily imply that

M +

0

a2 ≥ exp(ǫ−1/10)

M +

ǫ−1r0

a2

Now the chosen subtube contains at least about −ǫ−1log(λ−(T 0)h2(T 0)) disjointǫ-necks, where h denotes the cutoff radius, as before. Since h tends to zerowith δ, whereas r(T 0), that occurs in the bound for λ−, is independent of δ,we can ensure that the number of necks is greater then log h, and therefore, M cap

a2 < h6, say. Then standard estimates for the equation (8.2) show that

|∇a

|2 and Ra2 are bounded by const

·h on M cap, which makes it possible to

extend a to the metric before surgery in such a way that the functional (8.1) ispreserved up to const · h4. However, the loss of volume in the surgery is at leasth3, so it suffices to take δ so small that h is much smaller than ξ .

8.2 The arguments above lead to the following result(a) If (M, gij) has λ > 0, then, for an appropriate choice of the cutoff pa-

rameter, the solution becomes extinct in finite time. Thus, if M admits a metric with λ > 0 then it is diffeomorphic to a connected sum of a finite collection of S2 × S

1 and metric quotients of the round S3. Conversely, every such connected sum admits a metric with R > 0, hence with λ > 0.

(b) Suppose M does not admit any metric with λ > 0, and let λ denote the

supremum of λV 23 over all metrics on this manifold. Then λ = 0 implies that

M is a graph manifold. Conversely, a graph manifold can not have λ < 0.(c) Suppose λ < 0 and let V = (

−2

3

λ)32 . Then V is the minimum of V, such

that M can be decomposed in connected sum of a finite collection of S2 × S1,metric quotients of the round S3, and some other components, the union of which will be denoted by M ′, and there exists a (possibly disconnected) complete hyperbolic manifold, with sectional curvature −1/4 and volume V , which can be embedded in M ′ in such a way that the complement (if not empty) is a graph

21



manifold. Moreover, if such a hyperbolic manifold has volume V , then its cusps (if any) are incompressible in M ′.

For the proof one needs in addition easily verifiable statements that one canput metrics on connected sums preserving the lower bound for scalar curva-ture [G-L], that one can put metrics on graph manifolds with scalar curvaturebounded below and volume tending to zero [C-G], and that one can close a com-pressible cusp, preserving the lower bound for scalar curvature and reducing thevolume, cf. [A,5.2]. Notice that using these results we can avoid the hyperbolicrigidity and minimal surface arguments, quoted in 7.3, which, however, havethe advantage of not requiring any a priori topological information about thecomplement of the hyperbolic piece.

The results above are exact analogs of the conjectures for the Sigma constant,formulated by Anderson [A], at least in the nonpositive case.

References

[I] G.Perelman The entropy formula for the Ricci flow and its geometricapplications. arXiv:math.DG/0211159 v1

[A] M.T.Anderson Scalar curvature and geometrization conjecture for three-manifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997),49-82.

[C-G] J.Cheeger, M.Gromov Collapsing Riemannian manifolds while keepingtheir curvature bounded I. Jour. Diff. Geom. 23 (1986), 309-346.

[G-L] M.Gromov, H.B.Lawson Positive scalar curvature and the Dirac oper-ator on complete Riemannian manifolds. Publ. Math. IHES 58 (1983), 83-196.

[H 1] R.S.Hamilton Three-manifolds with positive Ricci curvature. Jour.Diff. Geom. 17 (1982), 255-306.

[H 2] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys inDiff. Geom. 2 (1995), 7-136.

[H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff.Geom. 37 (1993), 225-243.

[H 4] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds.Commun. Anal. Geom. 7 (1999), 695-729.

[H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Com-mun. Anal. Geom. 5 (1997), 1-92.

G.Perelman Spaces with curvature bounded below. Proceedings of ICM-1994, 517-525.

F.Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I,II. In-vent. Math. 3 (1967), 308-333 and 4 (1967), 87-117.

22

http://arxiv.org/abs/math/0211159




a r X i v : m a t h / 0

3 0 7 2 4 5 v 1

[ m a t h . D

G ] 1 7 J u l 2 0 0 3 Finite extinction time for the solutions to the

Ricci flow on certain three-manifolds

Grisha Perelman∗

February 1, 2008

In our previous paper we constructed complete solutions to the Ricci flowwith surgery for arbitrary initial riemannian metric on a (closed, oriented)

three-manifold [P,6.1], and used the behavior of such solutions to classify three-manifolds into three types [P,8.2]. In particular, the first type consisted of thosemanifolds, whose prime factors are diffeomorphic copies of spherical space formsand S2 × S

1; they were characterized by the property that they admit metrics,that give rise to solutions to the Ricci flow with surgery, which become extinctin finite time. While this classification was sufficient to answer topological ques-tions, an analytical question of significant independent interest remained open,namely, whether the solution becomes extinct in finite time for every initialmetric on a manifold of this type.

In this note we prove that this is indeed the case. Our argument (in con- junction with [P,§1-5]) also gives a direct proof of the so called ”elliptizationconjecture”. It turns out that it does not require any substantially new ideas:we use only a version of the least area disk argument from [H,§11] and a regu-larization of the curve shortening flow from [A-G].

1 Finite time extinction

1.1 Theorem. Let M be a closed oriented three-manifold, whose prime decom-position contains no aspherical factors. Then for any initial metric on M the solution to the Ricci flow with surgery becomes extinct in finite time.

Proof for irreducible M . Let ΛM denote the space of all contractible loops inC 1(S1 → M ). Given a riemannian metric g on M and c ∈ ΛM, define A(c, g) tobe the infimum of the areas of all lipschitz maps from D

2 to M, whose restrictionto ∂ D2 = S

1 is c. For a family Γ ⊂ ΛM let A(Γ, g) be the supremum of A(c, g)over all c ∈ Γ. Finally, for a nontrivial homotopy class α ∈ π∗(ΛM, M ) let

A(α, g) be the infimum of A(Γ, g) over all Γ ∈ α. Since M is not aspherical, itfollows from a classical (and elementary) result of Serre that such a nontrivialhomotopy class exists.

∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg

191023, Russia. Email: [email protected] or [email protected]

1










































1.2 Lemma. (cf. [H,§11]) If gt is a smooth solution to the Ricci flow, then for any α the rate of change of the function At = A(α, gt) satisfies the estimate

d

dtAt ≤ −2π −

1

2RtminAt

(in the sense of the lim sup of the forward difference quotients), where Rtmin

denotes the minimum of the scalar curvature of the metric gt.

A rigorous proof of this lemma will be given in §3, but the idea is simple andcan be explained here. Let us assume that at time t the value At is attainedby the family Γ, such that the loops c ∈ Γ where A(c, gt) is close to At areembedded and sufficiently smooth. For each such c consider the minimal diskDc with boundary c and with area A(c, gt). Now let the metric evolve by theRicci flow and let the curves c evolve by the curve shortening flow (which movesevery point of the curve in the direction of its curvature vector at this point)

with the same time parameter. Then the rate of change of the area of Dc canbe computed as Dc

(−Tr(RicT)) +

c

(−kg)

where RicT is the Ricci tensor of M restricted to the tangent plane of Dc, andkg is the geodesic curvature of c with respect to Dc (cf. [A-G, Lemma 3.2]). Inthree dimensions the first integrand equals −1

2R − (K − det II), where K is the

intrinsic curvature of Dc and det II, the determinant of the second fundamentalform, is nonpositive, because Dc is minimal. Thus, the rate of change of thearea of Dc can be estimated from above by

Dc

(−1

2R − K ) +

c

(−kg) =

Dc

(−1

2R) − 2π

by the Gauss-Bonnet theorem, and the statement of the lemma follows.The problem with this argument is that if Γ contains curves, which are not

immersed (for instance, a curve could pass an arc once in one direction and thenmake an about turn and pass the same arc in the opposite direction), then itis not clear how to define curve shortening flow so that it would be continuousboth in the time parameter and in the family parameter. In §3 we’ll explain howto circumvent this difficulty, essentially by adding one dimension to the ambientmanifold. This regularization of the curve shortening flow has been worked outby Altschuler and Grayson [A-G] (who were interested in approximating thesingular curve shortening flow on the plane and obtained for that case moreprecise results than what we need).

1.3 Now consider the solution to the Ricci flow with surgery. Since M is

assumed irreducible, the surgeries are topologically trivial, that is one of thecomponents of the post-surgery manifold is diffeomorphic to the pre-surgerymanifold, and all the others are spheres. Moreover, by the construction of the surgery [P,4.4], the diffeomorphism from the pre-surgery manifold to thepost-surgery one can be chosen to be distance non-increasing ( more precisely,(1 + ξ )-lipschitz, where ξ > 0 can be made as small as we like). It follows that

2



the conclusion of the lemma above holds for the solutions to the Ricci flow withsurgery as well.

Now recall that the evolution equation for the scalar curvature

d

dtR = R + 2|Ric|2 = R +

2

3R2 + 2|Ric|2

implies the estimate Rtmin ≥ − 3

2

1

t+const. It follows that At = At

t+const satisfies

ddt

At ≤ − 2πt+const

, which implies finite extinction time since the right hand side

is non-integrable at infinity whereas At can not become negative.1.4 Remark. The finite time extinction result for irreducible non-aspherical

manifolds already implies (in conjuction with the work in [P,§1-5] and theKneser finiteness theorem) the so called ”elliptization conjecture”, claiming thata closed manifold with finite fundamental group is diffeomorphic to a sphericalspace form. The analysis of the long time behavior in [P,§6-8] is not needed inthis case; moreover the argument in [P,§5] can be slightly simplified, replacingthe sequences rj , κj, δ j by single values r,κ, δ, since we already have an upperbound on the extinction time in terms of the initial metric.

In fact, we can even avoid the use of the Kneser theorem. Indeed, if we startfrom an initial metric on a homotopy sphere (not assumed irreducible), then ateach surgery time we have (almost) distance non-increasing homotopy equiva-lences from the pre-surgery manifold to each of the post-surgery components,and this is enough to keep track of the nontrivial relative homotopy class of theloop space.

1.5 Proof of theorem 1.1 for general M . The Kneser theorem implies that oursolution undergoes only finitely many topologically nontrivial surgeries, so fromsome time T on all the surgeries are trivial. Moreover, by the Milnor uniqueness

theorem, each component at time T satisfies the assumption of the theorem.Since we already know from 1.4 that there can not be any simply connectedprime factors, it follows that every such component is either irreducible, or hasnontrivial π2; in either case the proof in 1.1-1.3 works.

2 Preliminaries on the curve shortening flow

In this section we rather closely follow [A-G].2.1 Let M be a closed n-dimensional manifold, n ≥ 3, and let gt be a smooth

family of riemannian metrics on M evolving by the Ricci flow on a finite timeinterval [t0, t1]. It is known [B] that gt for t > t0 are real analytic. Let ct be asolution to the curve shortening flow in (M, gt), that is ct satisfies the equationd

dt

ct(x) = H t(x), where x is the parameter on S1, and H t is the curvature

vector field of ct with respect to gt. It is known [G-H] that for any smoothlyimmersed initial curve c the solution ct exists on some time interval [t0, t′1), eachct for t > t0 is an analytic immersed curve, and either t ′1 = t1, or the curvature

kt = gt(H t, H t)1

2 is unbounded when t → t′1.

3



Denote by X t the tangent vector field to ct, and let S t = gt(X t, X t)−1

2 X t

be the unit tangent vector field; then H = ∇S S (from now on we drop the

superscript t except where this omission can cause confusion). We compute

d

dtg(X, X ) = −2Ric(X, X ) − 2g(X, X )k2, (1)

which implies[H, S ] = (k2 + Ric(S, S ))S (2)

Now we can compute

d

dtk2 = (k2)′′ − 2g((∇S H )⊥, (∇S H )⊥) + 2k4 + ..., (3)

where primes denote differentiation with respect to the arclength parameter s,

and where dots stand for the terms containing the curvature tensor of g , whichcan be estimated in absolute value by const · (k2 + k). Thus the curvature k

satisfiesd

dtk ≤ k′′ + k3 + const · (k + 1) (4)

Now it follows from (1) and (4) that the length L and the total curvatureΘ =

kds satisfy

d

dtL ≤

(const − k2)ds, (5)

d

dtΘ ≤

const · (k + 1)ds (6)

In particular, both quantities can grow at most exponentially in t (they would

be non-increasing in a flat manifold).2.2 In general the curvature of ct may concentrate near certain points, cre-

ating singularities. However, if we know that this does not happen at sometime t∗, then we can estimate the curvature and higher derivatives at timesshortly thereafter. More precisely, there exist constants ǫ, C 1, C 2,... (which maydepend on the curvatures of the ambient space and their derivatives, but areindependent of ct), such that if at time t∗ for some r > 0 the length of ct is atleast r and the total curvature of each arc of length r does not exceed ǫ, thenfor every t ∈ (t∗, t∗ + ǫr2) the curvature k and higher derivatives satisfy theestimates k2 = g(H, H ) ≤ C 0(t − t∗)−1, g(∇S H, ∇S H ) ≤ C 1(t − t∗)−2,...

This can be proved by adapting the arguments of Ecker and Huisken [E-Hu];see also [A-G,§4].

2.3 Now suppose that our manifold (M, gt) is a metric product ( M , gt)×S1λ,

where the second factor is the circle of constant length λ; let U denote the unittangent vector field to this factor. Then u = g(S, U ) satisfies the evolutionequation

d

dtu = u′′ + (k2 + Ric(S, S ))u (7)

4



Assume that u was strictly positive everywhere at time t0 (in this case thecurve is called a ramp). Then it will remain positive and bounded away from

zero as long as the solution exists. Now combining (4) and (7) we can estimatethe right hand side of the evolution equation for the ratio k

u and conclude that

this ratio, and hence the curvature k, stays bounded (see [A-G,§2]). It followsthat ct is defined on the whole interval [t0, t1].

2.4 Assume now that we have two ramp solutions ct1, ct2, each winding oncearound the S1λ factor. Let µt be the infimum of the areas of the annuli withboundary ct1 ∪ ct2. Then

d

dtµt ≤ (2n − 1)|Rmt|µt, (8)

where |Rmt| denotes a bound on the absolute value of sectional curvaturesof gt. Indeed, the curves ct1 and ct2, being ramps, are embedded and without

substantial loss of generality we may assume them to be disjoint. In this case theresults of Morrey [M] and Hildebrandt [Hi] yield an analytic minimal annulus A,

immersed, except at most finitely many branch points, with prescribed boundaryand with area µ. The rate of change of the area of A can be computed as

A

(−Tr(RicT )) +

∂A

(−kg) ≤

A

(−Tr(RicT ) + K )

≤

A

(−Tr(RicT ) + RmT ) ≤ (2n − 1)|Rm|µ,

where the first inequality comes from the Gauss-Bonnet theorem, with possiblecontribution of the branch points, and the second one is due to the fact that aminimal surface has nonpositive extrinsic curvature with respect to any normal

vector.2.5 The estimate (8) implies that µt can grow at most exponentially; in

particular, if ct1 and ct2 were very close at time t0, then they would be close forall t ∈ [t0, t1] in the sense of minimal annulus area. In general this does notimply that the lengths of the curves are also close. However, an elementaryargument shows that if ǫ > 0 is small then, given any r > 0, one can find µ,

depending only on r and on upper bound for sectional curvatures of the ambientspace, such that if the length of ct1 is at least r, each arc of ct1 with length r hastotal curvature at most ǫ, and µt ≤ µ, then L(ct2) ≥ (1 − 100ǫ)L(ct1).

3 Proof of lemma 1.2

3.1 In this section we prove the following statementLet M be a closed three-manifold, and let (M, gt) be a smooth solution to the

Ricci flow on a finite time interval [t0, t1]. Suppose that Γ ⊂ ΛM is a compact family. Then for any ξ > 0 one can construct a continuous deformation Γt, t ∈[t0, t1], Γt0 = Γ, such that for each curve c ∈ Γ either the value A(ct1 , gt1) is bounded from above by ξ plus the value at t = t1 of the solution to the ODE

5



ddt

w(t) = −2π − 1

2Rtminw(t) with the initial data w(t0) = A(ct0 , gt0), or

L(ct1) ≤ ξ ; moreover, if c was a constant map, then all ct are constant maps.

It is clear that our statement implies lemma 1.2, because a family consistingof very short loops can not represent a nontrivial relative homotopy class.

3.2 As a first step of the proof of the statement we can replace Γ by afamily, which consists of piecewise geodesic loops with some large fixed numberof vertices and with each segment reparametrized in some standard way to makethe parametrizations of the whole curves twice continuously differentiable.

Now consider the manifold M λ = M × S1λ, 0 < λ < 1, and for each c ∈ Γ

consider the smooth embedded closed curve cλ such that p1cλ(x) = c(x) and p2cλ(x) = λx mod λ, where p1 and p2 are projections of M λ to the first andsecond factor respectively, and x is the parameter of the curve c on the standardcircle of length one. Using 2.3 we can construct a solution ctλ, t ∈ [t0, t1] tothe curve shortening flow with initial data cλ. The required deformation willbe obtained as Γt = p1Γt

λ

(where Γt

λ

denotes the family consisting of ct

λ

) forcertain sufficiently small λ > 0. We’ll verify that an appropriate λ can be foundfor each individual curve c, or for any finite number of them, and then showthat if our λ works for all elements of a µ-net in Γ, for sufficiently small µ > 0,

then it works for all elements of Γ.

3.3 In the following estimates we shall denote by C large constants thatmay depend on metrics gt, family Γ and ξ, but are independent of λ, µ and aparticular curve c.

The first step in 3.2 implies that the lengths and total curvatures of cλ areuniformly bounded, so by 2.1 the same is true for all ctλ. It follows that the areaswept by ctλ, t ∈ [t′, t′′] ⊂ [t0, t1] is bounded above by C (t′′ − t′), and therefore

we have the estimates A( p1ctλ, gt) ≤ C, A( p1ct′′

λ , gt′′

) − A( p1ct′

λ , gt′

) ≤ C (t′′ − t′).

3.4 It follows from (5) that t1

t0 k2dsdt ≤ C for any ctλ. Fix some large

constant B, to be chosen later. Then there is a subset I B(cλ) ⊂ [t0, t1] of measure at least t1 − t0 − CB−1 where

k2ds ≤ B, hence

kds ≤ ǫ on any arc

of length ≤ ǫ2B−1. Assuming that ctλ are at least that long, we can apply 2.2and construct another subset J B(cλ) ⊂ [t0, t1] of measure at least t1−t0−CB−1,

consisting of finitely many intervals of measure at least C −1B−2 each, such thatfor any t ∈ J B(cλ) we have pointwise estimates on ctλ for curvature and higherderivatives, of the form k ≤ CB,...

Now fix c,B, and consider any sequence of λ → 0. Assume again that thelengths of ctλ are bounded below by ǫ2B−1, at least for t ∈ [t0, t2], where t2 =t1 − B−1. Then an elementary argument shows that we can find a subsequenceΛc and a subset J B(c) ⊂ [t0, t2] of measure at least t1 − t0 − CB−1, consistingof finitely many intervals, such that J B(c) ⊂ J B(cλ) for all λ ∈ Λc. It followsthat on every interval of J B(c) the curve shortening flows ctλ smoothly converge

(as λ → 0 in some subsequence of Λc ) to a curve shortening flow in M.Let wc(t) be the solution of the ODE d

dtwc(t) = −2π − 1

2Rtminwc(t) with

initial data wc(t0) = A(c, gt0). Then for sufficiently small λ ∈ Λc we haveA( p1ctλ, gt) ≤ wc(t) + 1

2ξ provided that B > Cξ −1. Indeed, on the intervals of

J B(c) we can estimate the change of A for the limit flow using the minimal

6



disk argument as in 1.2, and this implies the corresponding estimate for p1ctλ if λ ∈ Λc is small enough, whereas for the intervals of the complement of J B(c)

we can use the estimate in 3.3.On the other hand, if our assumption on the lower bound for lengths does

not hold, then it follows from (5) that L(ct2λ ) ≤ CB−1 ≤ 1

2ξ.

3.5 Now apply the previous argument to all elements of some finite µ-netΓ ⊂ Γ for small µ > 0 to be determined later. We get a λ > 0 such that for eachc ∈ Γ either A( p1ct1λ , gt1) ≤ wc(t1) + 1

2ξ or L(ct2λ ) ≤ 1

2ξ. Now for any curve c ∈ Γ

pick a curve c ∈ Γ, µ-close to c, and apply the result of 2.4. It follows that if A( p1ct1λ , gt1) ≤ wc(t1) + 1

2ξ and µ ≤ C −1ξ, then A( p1ct1λ , gt1) ≤ wc(t1) + ξ. On

the other hand, if L(ct2λ ) ≤ 1

2ξ, then we can conclude that L(ct1λ ) ≤ ξ provided

that µ > 0 is small enough in comparison with ξ and B−1. Indeed, if L(ct1λ ) > ξ,

then L(ctλ) > 3

4ξ for all t ∈ [t2, t1]; on the other hand, using (5) we can find a

t ∈ [t2, t1], such that k2ds ≤ CB for ctλ; hence, applying 2.5, we get L(ctλ) > 2

3ξ

for this t, which is incompatible with L(ct2λ ) ≤

1

2ξ. The proof of the statement3.1 is complete.

References

[A-G] S.Altschuler, M.Grayson Shortening space curves and flow throughsingularities. Jour. Diff. Geom. 35 (1992), 283-298.

[B] S.Bando Real analyticity of solutions of Hamilton’s equation. Math.Zeit. 195 (1987), 93-97.

[E-Hu] K.Ecker, G.Huisken Interior estimates for hypersurfaces moving bymean curvature. Invent. Math. 105 (1991), 547-569.

[G-H] M.Gage, R.S.Hamilton The heat equation shrinking convex plane

curves. Jour. Diff. Geom. 23 (1986), 69-96.[H] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds.Commun. Anal. Geom. 7 (1999), 695-729.

[Hi] S.Hildebrandt Boundary behavior of minimal surfaces. Arch. Rat.Mech. Anal. 35 (1969), 47-82.

[M] C.B.Morrey The problem of Plateau on a riemannian manifold. Ann.Math. 49 (1948), 807-851.

[P] G.Perelman Ricci flow with surgery on three-manifolds.arXiv:math.DG/0303109 v1



perelman ricci flows y conj poincaré

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