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Resolução da Conjectura de Poincaré, Notas de estudo de Mecatrônica

Grigori Yakovlevich Perelman

Tipologia: Notas de estudo

2010

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Baixe Resolução da Conjectura de Poincaré e outras Notas de estudo em PDF para Mecatrônica, somente na Docsity! ar X iv :m at h/ 02 11 15 9v 1 [ m at h. D G ] 1 1 N ov 2 00 2 The entropy formula for the Ricci flow and its geometric applications Grisha Perelman∗ February 1, 2008 Introduction 1. The Ricci flow equation, introduced by Richard Hamilton [H 1], is the evolution equation d dt gij(t) = −2Rij for a riemannian metric gij(t). In his seminal paper, Hamilton proved that this equation has a unique solution for a short time for an arbitrary (smooth) metric on a closed manifold. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric|2, so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched point- wisely as the curvature is getting large. This observation allowed him to prove the convergence results: the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator ∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelman@pdmi.ras.ru or perelman@math.sunysb.edu ; I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I’d like to thank everyone who worked to make those opportunities available to me. 1 in dimension four converge, modulo scaling, to metrics of constant positive curvature. Without assumptions on curvature the long time behavior of the metric evolving by Ricci flow may be more complicated. In particular, as t ap- proaches some finite time T, the curvatures may become arbitrarily large in some region while staying bounded in its complement. In such a case, it is useful to look at the blow up of the solution for t close to T at a point where curvature 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 a subsequence of such scalings smoothly converges (modulo diffeomorphisms) to a complete solution to the Ricci flow whenever the curvatures of the scaled metrics are uniformly bounded (on some time interval), and their injectivity radii at the origin are bounded away from zero; moreover, if the size of the scaled time interval goes to infinity, then the limit solution is ancient, that is defined on a time interval of the form (−∞, T ). In general it may be hard to analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton [H 4] proved that in dimension three, at the points where scalar curvature is large, the negative part of the curvature tensor is small compared to the scalar curvature, and therefore the blow-up limits have necessarily nonneg- ative sectional curvature. On the other hand, Hamilton [H 3] discovered a remarkable 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 and different times. These results lead Hamilton to certain conjectures on the structure of the blow-up limits in dimension three, see [H 4,§26]; the present work confirms them. The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck. In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. The exact procedure was described by Hamilton [H 5] in the case of four-manifolds, satisfying certain curvature assumptions. He also expressed the hope that a similar procedure would work in the three dimensional case, without any a priory assumptions, and that after finite number of surgeries, the Ricci flow would exist for all time t → ∞, and be nonsingular, in the sense that the normalized curvatures R̃m(x, t) = tRm(x, t) would stay bounded. The topology of such nonsingular solutions was described by Hamilton [H 6] to the extent sufficient to make sure that no counterexample to the Thurston geometrization conjecture can 2 1 Ricci flow as a gradient flow 1.1. Consider the functional F = ∫ M (R + |∇f |2)e−fdV for a riemannian metric gij and a function f on a closed manifold M . Its first variation can be expressed as follows: δF(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)(2△f − |∇f |2 +R)], where δgij = vij , δf = h, v = g ijvij . Notice that v/2− h vanishes identically iff the measure dm = e−fdV is kept fixed. Therefore, the symmetric tensor −(Rij+∇i∇jf) is the L2 gradient of the functional Fm = ∫ M (R + |∇f |2)dm, where now f denotes log(dV/dm). Thus given a measure m , we may consider the gradient flow (gij)t = −2(Rij + ∇i∇jf) for Fm. For general m this flow may not exist even for short time; however, when it exists, it is just the Ricci flow, modified by a diffeomorphism. The remarkable fact here is that different 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 Fm exists for t ∈ [0, T ]. Then at t = 0 we have Fm ≤ n 2T ∫ M dm. Proof. We may assume ∫ M dm = 1. The evolution equations for the gradient flow of Fm are (gij)t = −2(Rij + ∇i∇jf), ft = −R −△f, (1.1) and Fm satisfies Fmt = 2 ∫ |Rij + ∇i∇jf |2dm (1.2) Modifying by an appropriate diffeomorphism, we get evolution equations (gij)t = −2Rij , ft = −△f + |∇f |2 − R, (1.3) and retain (1.2) in the form Ft = 2 ∫ |Rij + ∇i∇jf |2e−fdV (1.4) 5 Now we compute Ft ≥ 2 n ∫ (R + △f)2e−fdV ≥ 2 n ( ∫ (R + △f)e−fdV )2 = 2 n F2, and the proposition follows. 1.3 Remark. The functional Fm has a natural interpretation in terms of Bochner-Lichnerovicz formulas. The classical formulas of Bochner (for one-forms) and Lichnerovicz (for spinors) are ∇∗∇ui = (d∗d+dd∗)ui−Rijuj and ∇∗∇ψ = δ2ψ − 1/4Rψ. Here the operators ∇∗ , d∗ are defined using the riemannian volume form; this volume form is also implicitly used in the definition of the Dirac operator δ via the requirement δ∗ = δ. A rou- tine computation shows that if we substitute dm = e−fdV for dV , we get modified Bochner-Lichnerovicz formulas ∇∗m∇ui = (d∗md+ dd∗m)ui −Rmijuj and ∇∗m∇ψ = (δm)2ψ − 1/4Rmψ, where δmψ = δψ − 1/2(∇f) · ψ , Rmij = Rij+∇i∇jf , Rm = 2△f−|∇f |2+R. Note that gijRmij = R+△f 6= Rm. How- ever, we do have the Bianchi identity ∇∗mi Rmij = ∇iRmij −Rij∇if = 1/2∇jRm. Now Fm = ∫ M Rmdm = ∫ M gijRmij dm. 1.4* The Ricci flow modified by a diffeomorphism was considered by DeTurck, who observed that by an appropriate choice of diffeomorphism one can turn the equation from weakly parabolic into strongly parabolic, thus considerably simplifying the proof of short time existence and uniqueness; a nice version of DeTurck trick can be found in [H 4,§6]. The functional F and its first variation formula can be found in the literature on the string theory, where it describes the low energy effective action; 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 and expanding breathers, respectively. Trivial breathers, for which the metrics gij(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 diffeomorphism and scaling, then breathers and solitons correspond to periodic orbits and fixed points respectively). At each time the Ricci soliton metric satisfies an equation of the form Rij + cgij + ∇ibj + ∇jbi = 0, where c is a number and bi is a one-form; in particular, when bi = 1 2 ∇ia for some function a on M, we get a gradient Ricci soliton. An important example of a gradient shrinking soliton is the Gaussian soliton, for which the metric gij is just the euclidean metric on Rn, c = 1 and a = −|x|2/2. In this and the next section we use the gradient interpretation of the Ricci flow to rule out nontrivial breathers (on closed M). The argument in the steady case is pretty straightforward; the expanding case is a little bit more subtle, because our functional F is not scale invariant. The more difficult shrinking 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−fdV = 1. Clearly, λ(gij) is just the lowest eigenvalue of the operator −4△+R. Then formula (1.4) implies that λ(gij(t)) is nondecreasing in 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 steady soliton. 2.3. To deal with the expanding case consider a scale invariant version λ̄(gij) = λ(gij)V 2/n(gij). The nontrivial expanding breathers will be ruled out once we prove the following Claim λ̄ 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 λ̄ can not be nonnegative everywhere on [t1, t2], and the claim ap- plies.) Proof of the claim. dλ̄(t)/dt ≥ 2V 2/n ∫ |Rij + ∇i∇jf |2e−fdV + 2nV (2−n)/nλ ∫ −RdV ≥ 2V 2/n[ ∫ |Rij + ∇i∇jf − 1n(R + △f)gij|2e−fdV + 1 n( ∫ (R + △f)2e−fdV − ( ∫ (R + △f)e−fdV )2)] ≥ 0, where f is the minimizer for F . 7 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 almost nonnegative 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 curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, 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 times tk → T and a sequence of metric balls Bk = B(pk, rk) at times tk, such that r2k/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 one can take fk(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 + r 2 k) → −∞. However this is impossible, since tk + r 2 k 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 also κ-noncollapsed on the scale ρ; it is also clear that α2gij is κ-noncollapsed on 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 and T < ∞, one can find κ = κ(gij , T ) > 0, such that the solution gij(t) to the Ricci 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 theorem of Hamilton, we obtain the following 10 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 in a 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 ”density of states” measure, which does not depend on β. Then one computes the average energy < E >= − ∂ ∂β logZ, the entropy S = β < E > + logZ, and the fluctuation σ =< (E− < E >)2 >= ∂2 (∂β)2 logZ. 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 logZ = ∫ (−f + n 2 )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 the t-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 sufficiently small τ > 0 (by proposition 1.2). Furthermore, if (a) u tends to a δ-function as τ → 0, or (b) u is a limit of a sequence of functions ui, such that each ui 11 tends to a δ-function as τ → τi > 0, and τi → 0, then S is also nonnegative. In case (a) all the quantities < E >, S, σ tend to zero as τ → 0, while in case (b), which may be interesting if gij(τ) goes singular at τ = 0, the entropy S may tend to a positive limit. If the flow is defined for all sufficiently large τ (that is, we have an ancient solution to the Ricci flow, in Hamilton’s terminology), we may be interested in the behavior of the entropy S as τ → ∞. A natural question is whether we have a gradient shrinking soliton whenever S stays bounded. 5.2 Remark. Heuristically, this statistical analogy is related to the de- scription of the renormalization group flow, mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states, whereas in the former those states are suppressed by the exponential factor. 5.3* An entropy formula for the Ricci flow in dimension two was found by Chow [C]; there seems to be no relation between his formula and ours. The interplay of statistical physics and (pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond my understanding at the moment. 6 Riemannian formalism in potentially infi- nite dimensions When one is talking of the canonical ensemble, one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature (thermostat). In this section we attempt to describe such an embedding using the formalism of Rimannian geometry. 6.1 Consider the manifold M̃ = M × SN ×R+ with the following metric: g̃ij = gij , g̃αβ = τgαβ , g̃00 = N 2τ +R, g̃iα = g̃i0 = g̃α0 = 0, where i, j denote coordinate indices on the M factor, α, β denote those on the SN factor, and the coordinate τ on R+ has index 0; gij evolves with τ by the backward Ricci flow (gij)τ = 2Rij , gαβ is the metric on S N of constant curvature 1 2N . It turns out that the components of the curvature tensor of this metric coincide (modulo N−1) with the components of the matrix Harnack expression (and its traces), discovered by Hamilton [H 3]. One can also compute that all the components of the Ricci tensor are equal 12 ∫ τ2 τ1 √ τ (< Y,∇R > +2 < ∇YX,X >)dτ = ∫ τ2 τ1 √ τ(< Y,∇R > +2 < ∇XY,X >)dτ = ∫ τ2 τ1 √ τ(< Y,∇R > +2 d dτ < Y,X > −2 < Y,∇XX > −4Ric(Y,X))dτ = 2 √ τ < X, Y > ∣ ∣ τ2 τ1 + ∫ τ2 τ1 √ τ < Y,∇R− 2∇XX − 4Ric(X, ·) − 1 τ X > dτ (7.1) Thus L-geodesics must satisfy ∇XX − 1 2 ∇R + 1 2τ X + 2Ric(X, ·) = 0 (7.2) Given two points p, q and τ2 > τ1 > 0, we can always find an L-shortest curve γ(τ), τ ∈ [τ1, τ2] between them, and every such L-shortest curve is L- geodesic. It is easy to extend this to the case τ1 = 0; in this case √ τX(τ) has a limit as τ → 0. From now on we fix p and τ1 = 0 and denote by L(q, τ̄ ) the L-length of the L-shortest curve γ(τ), 0 ≤ τ ≤ τ̄ , connecting p and q. In the computations below we pretend that shortest L-geodesics between p and q are unique for all pairs (q, τ̄); if this is not the case, the inequalities that we obtain are still valid when understood in the barrier sense, or in the sense of distributions. The first variation formula (7.1) implies that ∇L(q, τ̄ ) = 2 √ τ̄X(τ̄), so that |∇L|2 = 4τ̄ |X|2 = −4τ̄R + 4τ̄(R + |X|2). We can also compute Lτ̄ (q, τ̄) = √ τ̄(R + |X|2)− < X,∇L >= 2 √ τ̄R − √ τ̄ (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 < ∇XX,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, τ̄ 3 2 (R + |X|2)(τ̄) = −K + 1 2 L(q, τ̄), (7.4) 15 where K = K(γ, τ̄) denotes the integral ∫ τ̄ 0 τ 3 2H(X)dτ, which we’ll encounter a few times below. Thus we get Lτ̄ = 2 √ τ̄R− 1 2τ̄ L+ 1 τ̄ K (7.5) |∇L|2 = −4τ̄R + 2√ τ̄ L− 4√ τ̄ K (7.6) Finally we need to estimate the second variation of L. We compute δ2Y (L) = ∫ τ̄ 0 √ τ(Y · Y · R + 2 < ∇Y ∇YX,X > +2|∇YX|2)dτ = ∫ τ̄ 0 √ τ(Y · Y · R + 2 < ∇X∇Y Y,X > +2 < R(Y,X), Y,X > +2|∇XY |2)dτ Now d dτ < ∇Y Y,X >=< ∇X∇Y Y,X > + < ∇Y Y,∇XX > +2Y ·Ric(Y,X)−X·Ric(Y, Y ), so, if Y (0) = 0 then δ2Y (L) = 2 < ∇Y Y,X > √ τ̄+ ∫ τ̄ 0 √ τ(∇Y ∇YR + 2 < R(Y,X), Y,X > +2|∇XY |2 + 2∇XRic(Y, Y ) − 4∇Y Ric(Y,X))dτ, (7.7) where we discarded the scalar product of −2∇Y Y with the left hand side of (7.2). Now fix the value of Y at τ = τ̄ , assuming |Y (τ̄ )| = 1, and construct Y on [0, τ̄ ] by solving the ODE ∇XY = −Ric(Y, ·) + 1 2τ Y (7.8) We compute d dτ < Y, Y >= 2Ric(Y, Y ) + 2 < ∇XY, Y >= 1 τ < Y, Y >, 16 so |Y (τ)|2 = τ τ̄ , and in particular, Y (0) = 0. Making a substitution into (7.7), we get HessL(Y, Y ) ≤ ∫ τ̄ 0 √ τ (∇Y ∇YR + 2 < R(Y,X), Y,X > +2∇XRic(Y, Y ) − 4∇Y Ric(Y,X) +2|Ric(Y, ·)|2 − 2 τ Ric(Y, Y ) + 1 2τ τ̄ )dτ To put this in a more convenient form, observe that d dτ Ric(Y (τ), Y (τ)) = Ricτ (Y, Y ) + ∇XRic(Y, Y ) + 2Ric(∇XY, Y ) = Ricτ (Y, Y ) + ∇XRic(Y, Y ) + 1 τ Ric(Y, Y ) − 2|Ric(Y, ·)|2, so HessL(Y, Y ) ≤ 1√ τ̄ − 2 √ τ̄Ric(Y, Y ) − ∫ τ̄ 0 √ τH(X, Y )dτ, (7.9) where H(X, Y ) = −∇Y ∇YR−2 < R(Y,X)Y,X > −4(∇XRic(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 derivative of a variation of γ among L-geodesics. For an L-Jacobi field Y with |Y (τ̄)| = 1 we have d dτ |Y |2 = 2Ric(Y, Y ) + 2 < ∇XY, Y >= 2Ric(Y, Y ) + 2 < ∇YX, Y > = 2Ric(Y, Y ) + 1√ τ̄ HessL(Y, Y ) ≤ 1 τ̄ − 1√ τ̄ ∫ τ̄ 0 τ 1 2H(X, Ỹ )dτ, (7.11) where Ỹ is obtained by solving ODE (7.8) with initial data Ỹ (τ̄) = Y (τ̄ ). Moreover, the equality in (7.11) holds only if Ỹ is L-Jacobi and hence d dτ |Y |2 = 2Ric(Y, Y ) + 1√ τ̄ HessL(Y, Y ) = 1 τ̄ . 17 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 distt0(x, x0) < r and t0 − r2 ≤ t ≤ t0, and the volume of the metric ball B(x0, r 2) at time t0 is less than κr n. 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 dist0(x, x0) < r0, and the volume of the metric ball B(x0, r0) at time zero is at least A−1rn0 , then gij(t) can not be κ-collapsed on the scales less than r0 at a point (x, r20) with distr2 0 (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. Arguing as in 7.3, we see that if our solution is collapsed at x on the scale r ≤ 1, then the reduced volume Ṽ (r2) must be very small; on the other hand, Ṽ (1) can not be small unless min l(x, 1 2 ) over x satisfying dist 1 2 (x, x0) ≤ 110 is large. Thus all we need is to estimate l, or equivalently L̄, in that ball. Recall that L̄ satisfies the differential inequality (7.15). In order to use it efficiently in a maximum principle argument, we need first to check the following simple assertion. 8.3 Lemma. Suppose we have a solution to the Ricci flow (gij)t = −2Rij . (a) Suppose Ric(x, t0) ≤ (n − 1)K when distt0(x, x0) < r0. Then the distance function d(x, t) = distt(x, x0) satisfies at t = t0 outside B(x0, r0) the differential inequality dt −△d ≥ −(n− 1)( 2 3 Kr0 + r −1 0 ) (the inequality must be understood in the barrier sense, when necessary) (b) (cf. [H 4,§17]) Suppose Ric(x, t0) ≤ (n− 1)K when distt0(x, x0) < r0, or distt0(x, x1) < r0. Then d dt distt(x0, x1) ≥ −2(n− 1)( 2 3 Kr0 + r −1 0 ) 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 other hand, △d ≤ ∑n−1 k=1 s ′′ Yk (γ), where Yk are vector fields along γ, vanishing at 20 x0 and forming an orthonormal basis at x when complemented by X, and s′′Yk(γ) denotes the second variation along Yk of the length of γ. Take Yk to be parallel between x and x1, and linear between x1 and x0, where d(x1, t0) = r0. Then △d ≤ n−1 ∑ k=1 s′′Yk(γ) = ∫ d(x,t0) r0 −Ric(X,X)ds+ ∫ r0 0 ( s2 r20 (−Ric(X,X)) + n− 1 r20 )ds = ∫ γ −Ric(X,X)+ ∫ r0 0 (Ric(X,X)(1 − s 2 r20 ) + n− 1 r20 )ds ≤ dt+(n−1)( 2 3 Kr0+r −1 0 ) The proof of (b) is similar. Continuing the proof of theorem, apply the maximum principle to the function h(y, t) = φ(d(y, t)−A(2t− 1))(L̄(y, 1− t)+ 2n+1), where d(y, t) = distt(x, x0), and φ is a function of one variable, equal 1 on (−∞, 120), and rapidly increasing to infinity on ( 1 20 , 1 10 ), 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 L̄ + 2n + 1 ≥ 1 for t ≥ 1 2 by the remark in the very end of 7.1. Clearly, min h(y, 1) ≤ h(x, 1) = 2n + 1. On the other hand, min h(y, 1 2 ) is achieved for some y satisfying d(y, 1 2 ) ≤ 1 10 . Now we compute 2h = (L̄+2n+1)(−φ′′+(dt−△d−2A)φ′)−2 < ∇φ∇L̄ > +(L̄t−△L̄)φ (8.2) ∇h = (L̄+ 2n+ 1)∇φ+ φ∇L̄ (8.3) At a minimum point of h we have ∇h = 0, so (8.2) becomes 2h = (L̄+ 2n+ 1)(−φ′′ + (dt −△d− 2A)φ′ + 2(φ′)2/φ) + (L̄t −△L̄)φ (8.4) Now since d(y, t) ≥ 1 20 whenever φ′ 6= 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 where φ′ 6= 0. Thus, using (8.1) and (7.15), we get 2h ≥ −(L̄+ 2n+ 1)C(A)φ− 2nφ ≥ −(2n+ C(A))h This implies that min h can not decrease too fast, and we get the required estimate. 21 9 Differential Harnack inequality for solutions of the conjugate heat equation 9.1 Proposition. Let gij(t) be a solution to the Ricci flow (gij)t = −2Rij , 0 ≤ t ≤ T, and let u = (4π(T − t))−n2 e−f satisfy the conjugate heat equation 2 ∗u = −ut −△u+Ru = 0. Then v = [(T − t)(2△f − |∇f |2 +R) + f − n]u satisfies 2 ∗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/u is 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 to the 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 dt f(γ(t), t) ≤ 1 2 (R(γ(t), t) + |γ̇(t)|2) − 1 2(T − t)f(γ(t), t) (9.2) Proof. From the evolution equation ft = −△f + |∇f |2 − R + n2(T−t) and v ≤ 0 we get ft+ 12R− 12 |∇f |2− f 2(T−t) ≥ 0. On the other hand,− ddtf(γ(t), t) = −ft− < ∇f, γ̇(t) >≤ −ft + 12 |∇f |2 + 12 |γ̇|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 l is the reduced distance, defined in 7.1, using p and τ(t) = T − t. 22 Mα. Now by the triangle inequality, d(x, t̄) ≤ d(x̄, t̄)+ 110AQ− 1 2 . On the other hand, using lemma 8.3(b) we see that, as t decreases from t̄ to t̄ − 1 2 αQ−1, the point x can not escape from the ball of radius d(x̄, t̄) + AQ− 1 2 centered at x0. Continuing the proof of the theorem, and arguing by contradiction, take sequences ǫ → 0, δ → 0 and solutions gij(t), violating the statement; by reducing 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 = 1 100nǫ → ∞, construct (x̄, t̄), and consider solutions u = (4π(t̄ − t))− n 2 e−f of the conjugate heat equation, starting from δ-functions at (x̄, t̄), and corresponding nonpositive functions v. Claim 3.As ǫ, δ → 0, one can find times t̃ ∈ [t̄− 1 2 αQ−1, t̄], such that the integral ∫ B v stays bounded away from zero, where B is the ball at time t̃ of radius √ t̄− t̃ centered at x̄. 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 (x̄, t̄) with factors Q. If the injectivity radii of the scaled metrics at (x̄, t̄) are bounded away from zero, then a smooth limit exists, it is complete and has |Rm|(x̄, t̄) = 1 and |Rm|(x, t) ≤ 4 when t̄ − 1 2 α ≤ t ≤ t̄. It is not hard to show that the fundamental solutions u of the conjugate heat equation converge to such a solution on the limit manifold. But on the limit manifold, ∫ B v can not be zero for t̃ = t̄ − 1 2 α, since the evolution equation (9.1) would imply in this case that the limit is a gradient shrinking soliton, and this is incompatible with |Rm|(x̄, t̄) = 1. If the injectivity radii of the scaled metrics tend to zero, then we can change 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 t̃ in such 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( d̃(y,t) 10Aǫ ), where d̃(y, t) = d(y, t) + 200n √ t, and φ is a smooth function of one variable, equal one on (−∞, 1] and decreasing to zero on [1, 2]. Clearly, h vanishes at t = 0 outside B(x0, 20Aǫ); on the other hand, it is equal to one near (x̄, t̄). Now 2h = 1 10Aǫ (dt−△d+ 100n√t )φ ′− 1 (10Aǫ)2 φ′′. Note that dt−△t+ 100n√t ≥ 0 on the set where φ′ 6= 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 (2h)u ≤ 1 (Aǫ)2 , so ∫ M hu |t=0≥ ∫ M hu |t=t̄ − t̄ (Aǫ)2 ≥ 1 − A−2. Also, by (9.1), ( ∫ M −hv)t ≤ ∫ M −(2h)v ≤ 1 (Aǫ)2 ∫ M −hv, so by Claim 3, − ∫ M hv |t=0≥ βexp(− t̄(Aǫ)2 ) ≥ β(1 −A−2). From now on we”ll work at t = 0 only. Let ũ = hu and correspondingly f̃ = f − logh. Then β(1 − A−2) ≤ − ∫ M hv = ∫ M [(−2△f + |∇f |2 − R)t̄− f + n]hu = ∫ M [−t̄|∇f̃ |2 − f̃ + n]ũ+ ∫ M [t̄(|∇h|2/h− Rh) − hlogh]u ≤ ∫ M [−t̄|∇f̃ |2 − f̃ − n]ũ+ A−2 + 100ǫ2 ( Note that ∫ M −uh log h does not exceed the integral of u over B(x0, 20Aǫ)\B(x0, 10Aǫ), and ∫ B(x0,10Aǫ) u ≥ ∫ M h̄u ≥ 1 −A−2, where h̄ = φ( d̃ 5Aǫ )) Now scaling the metric by the factor 1 2 t̄−1 and sending ǫ, δ to zero, we get a sequence of metric balls with radii going to infinity, and a sequence of compactly supported nonnegative functions u = (2π)− n 2 e−f with ∫ u → 1 and ∫ [−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 Sobolev inequality, as can be seen by using spherical symmetrization. 10.2 Corollary(from the proof) Under the same assumptions, we also have at time t, 0 < t ≤ (ǫr0)2, an estimate V olB(x, √ t) ≥ c √ t n 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 R n. Then the estimate |Rm|(x, t) ≤ (ǫr0) −2 holds whenever 0 ≤ t ≤ (ǫr0)2, distt(x, x0) < ǫr0. The proof is a slight modification of the proof of theorem 10.1, and is left to the reader. A natural question is whether the assumption on the volume of 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 smooth solution to the Ricci flow on [0, (ǫr0) 2], such that at t = 0 we have R(x) ≥ −r−20 , V ol(∂Ω)n ≥ (1 − δ)cnV ol(Ω)n−1 for any x,Ω ⊂ B(x0, r0), and (x, t) satisfies A−1(ǫr0) 2 ≤ t ≤ (ǫr0)2, distt(x, x0) ≤ Ar0, then gij(t) can not be κ-collapsed at (x, t) on the scales less than √ t. 10.5 Remark. It is straightforward to get from 10.1 a version of the Cheeger diffeo finiteness theorem for manifolds, satisfying our assumptions on scalar curvature and isoperimetric constant on each ball of some fixed radius r0 > 0. In particular, these assumptions are satisfied (for some controllably smaller r0), if we assume a lower bound for Ric and an almost euclidean lower bound for the volume of the balls of radius r0. (this follows from the Levy- Gromov isoperimetric inequality); thus we get one of the results of Cheeger and Colding [Ch-Co] under somewhat weaker assumptions. 10.6* Our pseudolocality theorem is similar in some respect to the results of Ecker-Huisken [E-Hu] on the mean curvature flow. 11 Ancient solutions with nonnegative cur- vature operator and bounded entropy 11.1. In this section we consider smooth solutions to the Ricci flow (gij)t = −2Rij ,−∞ < t ≤ 0, such that for each t the metric gij(t) is a complete non-flat metric of bounded curvature and nonnegative curvature operator. Hamilton discovered a remarkable differential Harnack inequality for such solutions; we need only its trace version Rt + 2 < X,∇R > +2Ric(X,X) ≥ 0 (11.1) and its corollary, Rt ≥ 0. In particular, the scalar curvature at some time t0 ≤ 0 controls the curvatures for all t ≤ t0. We impose one more requirement on the solutions; namely, we fix some κ > 0 and require that gij(t) be κ-noncollapsed on all scales (the definitions 4.2 and 8.1 are essentially equivalent in this case). It is not hard to show that this requirement is equivalent to a uniform bound on the entropy S, defined as in 5.1 using an arbitrary fundamental solution to the conjugate heat equation. 11.2. Pick an arbitrary point (p, t0) and define Ṽ (τ), l(q, τ) as in 7.1, for τ(t) = t0 − t. Recall that for each τ > 0 we can find q = q(τ), such that l(q, τ) ≤ n 2 . 27 (b) If, rather than assuming a lower bound on volume for all t, we assume it only for t = 0, then the same conclusion holds with −τ0r20 in place of t0, provided that −t0 ≥ τ0r20. Proof. By scaling assume r0 = 1. (a) Arguing by contradiction, consider a sequence of B,C → ∞, of solutions gij(t) and points (x, t), such that distt(x, x0) ≤ 14 and R(x, t) > C + B(t − t0)−1. Then, arguing as in the proof of claims 1,2 in 10.1, we can find a point (x̄, t̄), satisfying distt̄(x̄, x0) < 1 3 , Q = R(x̄, t̄) > C + B(t̄ − t0)−1, and such that R(x′, t′) ≤ 2Q whenever t̄ − AQ−1 ≤ t′ ≤ t̄, distt̄(x′, x̄) < AQ− 1 2 , where A tends to infinity with B,C. Applying the previous corollary at (x̄, t̄) and using the relative volume comparison, we get a contradiction with the assumption involving w. (b) Let B(w), C(w) be good for (a). We claim that B = B(5−nw), C = C(5−nw) are good for (b) , for an appropriate τ0(w) > 0. Indeed, let gij(t) be a solution with nonnegative curvature operator, such that V olB(x0, 1) ≥ w at t = 0, and let [−τ, 0] be the maximal time interval, where the assumption of (a) still holds, with 5−nw in place of w and with −τ in place of t0. Then at time t = −τ we must have V olB(x0, 1) ≤ 5−nw. On the other hand, from lemma 8.3 (b) we see that the ball B(x0, 1 4 ) at time t = −τ contains the ball B(x0, 1 4 − 10(n − 1)(τ √ C + 2 √ Bτ )) at time t = 0, and the volume of the former is at least as large as the volume of the latter. Thus, it is enough to choose τ0 = τ0(w) in such a way that the radius of the latter ball is > 1 5 . Clearly, the proof also works if instead of assuming that curvature op- erator is nonnegative, we assumed that it is bounded below by −r−20 in the (time-dependent) metric ball of radius r0, centered at x0. 11.7. From now on we restrict our attention to oriented manifolds of dimen- sion three. Under the assumptions in 11.1, the solutions on closed manifolds must be quotients of the round S3 or S2 × R - this is proved in the same way as in two dimensions, since the gradient shrinking solitons are known from the work of Hamilton [H 1,10]. The noncompact solutions are described below. Theorem.The set of non-compact ancient solutions , satisfying the as- sumptions of 11.1, is compact modulo scaling. That is , from any sequence of such solutions and points (xk, 0) with R(xk, 0) = 1, we can extract a smoothly converging subsequence, and the limit satisfies the same conditions. Proof. To ensure a converging subsequence it is enough to show that whenever R(yk, 0) → ∞, the distances at t = 0 between xk and yk go to in- finity as well. Assume the contrary. Define a sequence zk by the requirement 30 that zk be the closest point to xk (at t = 0), satisfyingR(zk, 0)dist 2 0(xk, zk) = 1. We claim thatR(z, 0)/R(zk, 0) is uniformly bounded for z ∈ B(zk, 2R(zk, 0)− 1 2 ). Indeed, otherwise we could show, using 11.5 and relative volume comparison in nonnegative curvature, that the balls B(zk, R(zk, 0) − 1 2 ) are collapsing on the scale of their radii. Therefore, using the local derivative estimate, due to W.-X.Shi (see [H 4,§13]), we get a bound on Rt(zk, t) of the order of R2(zk, 0). Then we can compare 1 = R(xk, 0) ≥ cR(zk,−cR−1(zk, 0)) ≥ cR(zk, 0) for some small c > 0, where the first inequality comes from the Harnack inequal- ity, obtained by integrating (11.1). Thus, R(zk, 0) are bounded. But now the existence of the sequence yk at bounded distance from xk implies, via 11.5 and relative volume comparison, that balls B(xk, c) are collapsing - a contradiction. It remains to show that the limit has bounded curvature at t = 0. If this was not the case, then we could find a sequence yi going to infinity, such that 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 the assumptions in 11.1 and splits off a line. Thus by 11.3 it must be a round infinite cylinder. It follows that for large i each yi is contained in a round cylindrical ”neck” of radius (1 2 R(yi, 0)) − 1 2 → 0, - something that can not happen in an open manifold of nonnegative curvature. 11.8. Fix ǫ > 0. Let gij(t) be an ancient solution on a noncompact oriented three-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 scaling with factor Q, ǫ-close (in some fixed smooth topology) to the corresponding subset 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 Mǫ denotes the set of points in M, which are not centers of ǫ-necks, then Mǫ is compact and moreover, diamMǫ ≤ CQ− 1 2 , and C−1Q ≤ R(x, 0) ≤ CQ whenever x ∈Mǫ, where Q = R(x0, 0) for some x0 ∈ ∂Mǫ. 11.9 Remark. It can be shown that there exists κ0 > 0, such that if an ancient solution on a noncompact three-manifold satisfies the assumptions in 11.1 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 is not 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, studied by R.Bryant. In this direction, I have a plausible, but not quite rigorous argument, showing that any such ancient solution can be made eternal, that is, can be extended for t ∈ (−∞,+∞); also I can prove uniqueness in the class of gradient steady solitons. 11.10* The earlier work on ancient solutions and all that can be found in [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 curvature if 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 to the 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) : dist2t0(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 converging to zero, and consider the solutions gij(t), such that the conclusion does not hold for some (x0, t0); moreover, by tampering with the condition t0 ≥ 1 a little 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 such points with factors Q would converge, along some subsequence of r0 → 0, to an ancient solution, satisfying 11.1. 32 satisfies the assumptions of theorem 8.2 with r0 = 1. Then R(x, 1) ≤ K whenever dist1(x, x0) < A. Proof. In the first step of the proof we check the following Claim. 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 dist1(x, x0) < A. The proof of this statement essentially repeats the proof of the previous theorem (the κ-noncollapsing assumption is ensured by theorem 8.2). The only difference is in the beginning. So let us argue by contradiction, and suppose we have a sequence of solutions and points x with dist1(x, x0) < A and R(x, 1) → ∞, which do not satisfy the conclusion of 12.1. Then an argument, similar to the one proving claims 1,2 in 10.1, delivers points (x̄, t̄) with 1 2 ≤ t̄ ≤ 1, distt̄(x̄, x0) < 2A, with Q = R(x̄, t̄) → ∞, and such that (x, t) satisfies the conclusion of 12.1 whenever R(x, t) > 2Q, t̄−DQ−1 ≤ t ≤ t̄, distt̄(x̄, x) < DQ − 1 2 , where D → ∞. (There is a little subtlety here in the application of lemma 8.3(b); nevertheless, it works, since we need to apply it only when the endpoint other than x0 either satisfies the conclusion of 12.1, or has scalar curvature at most 2Q) After such (x̄, t̄) are found, the proof of 12.1 applies. Now, having checked the claim, we can prove the theorem by applying the claim 2 of the previous theorem to the appropriate segment of the shortest geodesic, 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 ≥ wrn0 , 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 ≥ 4τr20, t ∈ [t0 − τr20, t0], distt(x, x0) ≤ 14r0, provided that φ(r −2 0 ) < 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 take τ(w) = τ0(w)/2, K(w) = C(w) + 2B(w)/τ0(w). Now fix these values of τ,K, consider a φ-almost nonnegatively curved solution gij(t), a point (x0, t0) and a radius r0 > 0, such that the assumptions of the theorem do hold whereas the conclusion does not. We may assume that any other point (x′, t′) and radius r′ > 0 with that property has either t′ > t0 or t ′ < t0 − 2τr20, or 35 2r′ > r0. Our goal is to show that φ(r −2 0 ) is bounded away from zero. Let τ ′ > 0 be the largest time interval such that Rm(x, t) ≥ −r−20 when- ever t ∈ [t0 − τ ′r20, t0], distt(x, x0) ≤ r0. If τ ′ ≥ 2τ, we are done by corollary 11.6(b). Otherwise, by elementary Aleksandrov space theory, we can find at time t′ = t0 − τ ′r20 a ball B(x′, r′) ⊂ B(x0, r0) with V olB(x′, r′) ≥ 12ωn(r′)n, and with radius r′ ≥ cr0 for some small constant c = c(w) > 0. By the choice of (x0, t0) and r0, the conclusion of our theorem holds for (x ′, t′), r′. Thus we have an estimate R(x, t) ≤ K(r′)−2 whenever t ∈ [t′−τ(r′)2, t′], distt(x, x′) ≤ 1 4 r′. Now we can apply the previous theorem (or rather its scaled version) and get an estimate on R(x, t) whenever t ∈ [t′ − 1 2 τ(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 minRm(x, t0) over x ∈ B(x0, r0) is equal to −r−20 , then V olB(x0, r0) ≤ wrn0 . 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,∞), where M is a closed oriented three-manifold. Then, according to [H 6, theorem 4.1], the normalized curvatures R̃m(x, t) = tRm(x, t) satisfy an estimate of the form R̃m(x, t) ≥ −φ(R̃(x, t))R̃(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 following Theorem. 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 = Mthick ⋃ Mthin with the following properties. (a) For every x ∈ Mthick we have an estimate |R̃m| ≤ K in the ball B(x, ρ(w) √ t). and the volume of this ball is at least 1 10 w(ρ(w) √ t)n. (b) For every y ∈Mthin 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 Mthick is empty for large t, 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 well enough to claim that, for sufficiently small w > 0, Mthin is homeomorphic to a graph manifold. The natural questions that remain open are whether the normalized cur- vatures must stay bounded as t → ∞, and whether reducible manifolds and manifolds with finite fundamental group can have metrics which evolve smoothly by the Ricci flow on the infinite time interval. 13.2 Now suppose that gij(t) is defined on M × [1, T ), T < ∞, and goes singular as t→ T. Then using 12.1 we see that, as t→ T, either the curvature goes 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 and capped necks, which in the limit turn into horns (the horns most likely have finite diameter, but at the moment I don’t have a proof of that). Then at the time T we can replace the tips of the horns by smooth caps and continue running 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 the surgery 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 technically quite complicated, but requires no essentially new ideas. It is likely that by passing 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 every maximal horn at any time T, when the solution goes singular, has volume at 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 can be reconstructed as a connected sum of manifolds, admitting a thick-thin decomposition as in 13.1, and quotients of S3 and S2 × R. 13.3* Another differential-geometric approach to the geometrization con- jecture is being developed by Anderson [A]; he studies the elliptic equations, arising as Euler-Lagrange equations for certain functionals of the riemannian metric, perturbing the total scalar curvature functional, and one can observe certain parallelism between his work and that of Hamilton, especially taking into account that, as we have shown in 1.1, Ricci flow is the gradient flow for a functional, that closely resembles the total scalar curvature. 37
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