NOTES ON PERELMAN S PAPER

Transcription

1 NOTES ON PERELAN S PAPER BRUCE KLEINER AND JOHN LOTT 1. Introduction These are informal notes on Perelman s paper The Entropy Formula for the Ricci Flow and its Geometric Applications [11]. The purpose of the notes is to add detail to some of Perelman s arguments. The notes are not meant to be complete. They are not self-contained and are meant to be read along with [11]. Selected portions of Sections 1-4 and 7-13 of [11] are covered. We include four appendices that discuss some bacground material and techniques that are used throughout the notes. In the body of the notes we will refer to Section X.Y of [11] as I.X.Y, and Section X.Y of [1] as II.X.Y. Disclaimer : We do not guarantee the accuracy of any of the statements or proofs in these notes. It is quite possible that errors, due to the authors, have crept into the exposition. Any corrections or improvements to the notes are welcome. Contents 1. Introduction 1. Derivation of the first equation of Section Basic Example for Section Argument for Section Basic Example for Section Derivation of I.(3.4) 5 7. Argument for Section Basic Example for Section Remars about the regularity of L-Geodesics and L exp 8 1. Derivation of I.(7.7) Derivation of I.(7.1) Derivation of I.(7.11) onotonicity of Ṽ Last Statement in I Argument for I Argument for I Proof of Lemma I.8.3(b) Proof of Theorem I Date: ay 5, 3. Research supported by NSF grants DS-7154 and DS

2 BRUCE KLEINER AND JOHN LOTT 19. Derivation of I.(9.1) 15. Argument for Corollary I Argument for Corollary I Argument for Corollary I Argument for Claim of I Argument for Claim 3 of I Argument for Theorem I Proof of Corollary I Argument for Corollary I Argument for I Examples for I Argument for I.(11.1) Argument for I Argument for I Corollaries of I An alternate proof of I.11.3 using I.11.4 and Corollary Argument for I Argument for I Argument for I Argument for a wea version of Corollary I Argument for a strong version of Corollary I ore properties of κ-solutions Argument for I Proof of Lemma II Proof of Theorem I Proof of Theorem I Proof of Corollary I Proof of Corollary I Proof of Theorem I Appendix A. Alexandrov spaces 4 Appendix B. Finding balls with controlled curvature 41 Appendix C. φ-almost nonnegative curvature 41 Appendix D. Convergent subsequences of Ricci flow solutions 4 References 43. Derivation of the first equation of Section 1.1 With the conventions of I, = div grad. From a standard formula, (.1) δr = v + i j v ij R ij v ij. As (.) f = g ij i f j f, we have (.3) δ f = v ij i f j f + f, h.

3 NOTES ON PERELAN S PAPER 3 As dv = det(g) dx 1... dx n, we have δ(dv ) = v dv, so (.4) δ ( e f dv ) ( v ) = h e f dv. Putting this together gives (.5) δf = e f [ v + i j v ij R ij v ij v ij i f j f + ( v )] f, h + (R + f ) h dv. As (.6) e f = ( f f) e f, we have (.7) Next, (.8) Finally, (.9) Then (.1) e f [ v] dv = e f v dv = e f i j v ij dv = δf = = = ( i j e f ) v ij dv = e f ( f f ) v dv. e f ( i f j f i j f) v ij dv. e f f, h dv = e f, h dv = = e f ( f f) h dv. i (e f j f) v ij dv ( e f ) h dv [( v ) e f h ( f f ) v ij (R ij + i j f) ( v ) ] + h (R + f ) dv ( v ) ] e [ f v ij (R ij + i j f) + h ( f f + R) dv. 3. Basic Example for Section 1 Consider R n with the standard metric, constant in time. Fix t >. Put τ = t t and (3.1) f(t, x) = x 4τ so + n ln(4πτ), (3.) e f = (4πτ) n/ e x 4τ.

4 4 BRUCE KLEINER AND JOHN LOTT This is the standard heat ernel when considered for τ going from to t, i.e. for t going from t to. One can chec that f solves the second equation in I.(1.3). As x (3.3) 4τ dv = (4πτ) n/, R n e f is properly normalized. Then f = x τ and f = x 4τ. Differentiating (3.3) with respect to τ gives (3.4) Rn x 4τ e x 4τ dv = (4πτ) n/ n τ, so (3.5) f e f dv = n R τ. n Then F(t) = n n =. In particular, this is nondecreasing as a function of t [, t τ (t t) ). 4. Argument for Section. The formal argument to show that λ is nondecreasing along the Ricci flow is as follows. Let g(t) be a solution of the Ricci flow. For a given t let f(t) minimize F(g(t), f), subject to the constraint that e f dv = 1. Formally we can write (4.1) where f runs over the constrained set. Then (4.) dλ dt = δf dg (g(t), f(t)), δg dt + δf δf δf (g(t), f(t)) =, δf df (g(t), f(t)), dt = δf δg (g(t), f(t)), dg dt. Solve the second equation in I.(1.3) for f(t), with initial condition f() = f(). From I.(1.4), df (g(t), f(t)). Along this trajectory, dt (4.3) At t =, (4.4) df(g(t), f(t)) dt df(g(t), f(t)) dt t= = δf dg (g(t), f(t)), δg = δf δg dt + δf δf (g(t), f(t)), df dt. dg (g(), f()), + δf df (g(), f()),. dt t= δf dt t= As f() = f(), we have δf (g(), f()) =. Thus δf dλ df(g(t), f(t)) (4.5) =. dt t= dt t= As the choice of origin was arbitrary, it follows that λ is nondecreasing in t. This argument may seem slightly dubious, since it involves solving the bacward heat equation for f in I.(1.3). However, one can rewor the argument as follows. Fix some t > on the Ricci flow trajectory. We can solve the bacward heat equation for f bacward in time, to get a solution on an interval [t ɛ, t ] for some ɛ >. Then the above calculations, done on this interval bacward from t, show that dλ(t dt ).

5 NOTES ON PERELAN S PAPER 5 In this argument we assumed that there is a unique minimizer f(t) which varies smoothly in t. To see that this is the case, write ( (4.6) F = Re f + 4 e f/ ) dv. Putting Φ = e f/, (4.7) F = ( 4 Φ + R Φ ) dv = Φ ( 4 Φ + RΦ) dv. The constraint equation becomes Φ dv = 1. Then λ(t) is the smallest eigenvalue of 4 + R and e f/ is a corresponding normalized eigenvector. As the operator is a Schrödinger operator, there is a unique normalized positive eigenvector. From eigenvalue perturbation theory, the eigenvalue depends smoothly on t. One can use eigenvalue perturbation theory to compute dλ, to give an independent proof dt that it is nonnegative. 5. Basic Example for Section 3.1 Consider R n with the standard metric, constant in time. Fix t >. Put τ = t t and (5.1) f(t, x) = x 4τ, so (5.) e f = e x 4τ. One can chec that (g(t), f(t), τ(t)) satisfies I.(3.) and I.(3.3). Now (5.3) τ( f + R) + f n = τ x 4τ + x 4τ It follows from (3.3) and (3.4) that W(t) = for all t [, t ). n = x τ n. Put σ = δτ. Then (6.1) δ ( (4πτ) n/ e f dv ) = Writing (6.) W = 6. Derivation of I.(3.4) ( v h nσ ) τ (4πτ) n/ e f dv. [ τ(r + f ) + f n ] (4πτ) n/ e f dv we can use the fact that τ R along with (.1) to obtain (6.3) δw = [ ( v ) σ(r + f ) + τ h ( f f ) τv ij (R ij + i j f) + h + [ τ(r + f ) + f n ] ( v h nσ )] (4πτ) n/ e f dv. τ

6 6 BRUCE KLEINER AND JOHN LOTT Adding a Lie derivative, we can replace I.(3.3) by the equations (6.4) Thus we want to tae (6.5) (g ij ) t = (R ij + i j f), f t = f R + n τ, τ t = 1. v ij = (R ij + i j f), h = f R + n τ, σ = 1. We see that v h nσ =. Then τ dw [ (6.6) = (R + f ) τ n dt τ ( f f ) + τ R ij f R + n ] (4πτ) n/ e f dv. τ + i j f Using the fact that (6.7) = we obtain (6.8) dw dt = = e f dv = ( f f) e f dv, [ τ R ij + i j f f R + n ] (4πτ) n/ e f dv τ τ R ij + i j f 1 τ g ij (4πτ) n/ e f dv. The existence of a minimizer f for W follows from [15] (reference indirectly from Jeff Viaclovsy). 7. Argument for Section 4.1 Remar : In the definition of noncollapsing, T could be infinite. This is why it is written that r /t stays bounded, while if T < then this is obviously the same as saying that r stays bounded. In Section 5 of the notes we showed that in the case of flat R n, taing e f (x) = e x we get W(g, f, τ) =. So putting τ = r and x e f (x) = e 4r, we have W(g, f, r ) =. In the collapsing case, the idea is to use a test function f so that (7.1) e f (x) e c e distt (x,p ) 4r, where c is determined by the normalization condition (7.) (4πr) n/ e f dv = 1. 4τ,

7 NOTES ON PERELAN S PAPER 7 The main difference between computing (7.) in and in R n comes from the difference in volumes, which means that e c 1. In particular, as, we have c r n vol(b ). Now that f is normalized correctly, the main difference between computing W(g(t ), f, r ) in, and the analogous computation for the Gaussian in R n, comes from the f term in the integrand of W. Since f c, this will drive W(g(t ), f, r ) to as, so µ(g(t ), r ). To write this out precisely, let us put Φ = e f/, so that [ (7.3) W(g, Φ, τ) = (4πτ) n/ 4τ Φ + (τr ln Φ n) Φ ] dv. For the argument it is enough to obtain small values of W for positive Φ, and by approximation, it is enough to obtain small values of W for nonnegative Φ. Tae (7.4) Φ (x) = e c / φ(dist t (x, p )/r ), where φ : [, ) [, 1] is an appropriate function such that φ() = 1 and φ(s) = for s 1. The constant c is determined by (7.5) e c = (4πr) n/ φ (dist t (x, p )/r ) dv (4πr) n/ vol(b ). Thus c. Next, (7.6) W(g(t ), Φ, r ) = (4πr ) n/ [ ] 4r Φ + (rr ln Φ n) Φ dv. Let A (s) be the mass of the distance sphere S(p, r s) around p. Put (7.7) R (s) = r A (s) 1 R d area. S(p,r s) We can compute the integral in (7.6) radially to get 1 [ (7.8) W(g(t ), Φ, r) 4(φ (s)) + (R (s) + c ln φ(s) n) φ (s) ] A (s) ds = 1. φ (s) A (s) ds Tae φ to be the radial function φ(s) which minimizes the corresponding ratio 1 (7.9) [4(φ (s)) ( ln φ(s) + n) φ (s)] A(s) ds 1 φ (s) A(s) ds for a unit ball in H n, with boundary conditions φ() = 1 and φ(1) =. (The existence of φ is shown for a similar problem on the interval in [14].) Let λ be the corresponding minimum value. The method of proof of Cheng s theorem [3, Theorem 3.4] will go through to show that 1 (7.1) [4(φ (s)) ( ln φ(s) + n) φ (s)] A (s) ds 1 λ. φ (s) A (s) ds Here we are using the lower Ricci curvature bound on B. Also, from the curvature assumption, R (s) n(n 1) for s [, 1]. Then W(g(t ), Φ, r ) λ + n(n 1) + c and W(g(t ), Φ, r ).

8 8 BRUCE KLEINER AND JOHN LOTT Thus µ(g(t ), r ). From an argument as in Section 4 of the notes, for any t, µ(g(t), t t) is nondecreasing in t. Hence µ(g(), t +r ) µ(g(t ), r ), so µ(g(), t +r ). Since T is finite, t and r are bounded, which gives the contradiction. If is flat R n and p is the origin then (8.1) γ(τ) = 8. Basic Example for Section 7 ( τ τ ) 1/ q, (8.) L(q, τ) = 1 τ 1/ q, (8.3) l(q, τ) = q 4τ and (8.4) L(q, τ) = q. 9. Remars about the regularity of L-Geodesics and L exp We are assuming that (, g( )) is a Ricci flow, where the curvature operator of is uniformly bounded on a τ-interval [τ 1, τ ], and each tau slice (, g(τ)) is complete for τ [τ 1, τ ]. Thus by [9, Theorem 13.1], for every τ < τ there is a constant D = D(τ ) such that (9.1) R(x, τ) < D for all x, τ τ. aing the change of variable s = τ in the formula for L-length, we get s ( 1 (9.) L(γ) = dγ ) + s R(γ(s)) ds. 4 ds The Euler-Lagrange equation becomes (9.3) ˆX ˆX s R + 4s Ric( ˆX, ) =, s 1 where ˆX := dγ ds = sx. Putting s 1 = τ 1, it follows from standard existence theory for ODE s that for each p and v T p, there is a unique solution γ(s) to (9.3), defined on an interval [s 1, s 1 + ɛ), with γ(s 1 ) = p and 1 dγ γ (s 1 ) = lim τ τ τ1 dτ = v. If γ(s) is defined for s [s 1, s ] then d ds ˆX = d ds ˆX, ˆX = 4s Ric( ˆX, ˆX) + ˆX ˆX, ˆX C ˆX by (9.3) and (9.1). Since the metrics g(τ) are uniformly comparable for τ [τ 1, τ ], we conclude (by a continuity argument) that the L-geodesic γ v with 1 γ v(s 1 ) = v is defined on the whole interval [s 1, s ), where s = τ. In particular, for each s = τ [s 1, s ) and

9 NOTES ON PERELAN S PAPER 9 every p, we get a globally defined and smooth L-exponential map L exp τ : T p which taes each v T p to γ v ( τ). We now fix p, tae τ 1 =, and let L(q, τ) be the minimizer function as in I.7.1. We can imitate the traditional Riemannian geometry proof that geodesics minimize for a short time. By the implicit function theorem, there is an r = r(p) > (which varies continuously with p) such that for every q with d(q, p) r at τ =, and every τ r, there is a unique L-geodesic γ (q, τ) : [, τ], starting at p and ending at q, which remains within the ball B(p, r) (in the τ = slice (, g())), and γ (q, τ) varies smoothly with (q, τ). Thus, the L-length of γ (q, τ) varies smoothly with (q, τ), and defines a function ˆL(q, τ) near (p, ). We claim that ˆL = L near (p, ). For if α : [, τ] is a smooth curve then d dτ ˆL(α(τ), τ) = τx, dα dτ + τ(r X ) τ ( R + dα ) = d dτ dτ ( Llength(α ) [,τ] ) ; compare [11, p. 15, b1-b8]. Thus γ (q, τ) minimizes when (q, τ) is close to (p, ). We can now deduce that for all (q, τ), there is an L-geodesic γ : [, τ] which has infimal L-length among all piecewise smooth curves starting at p and ending at q (with domain [, τ]). This can be done by imitating the usual broen geodesic argument. Another technical issue is the justification of the change of variables from to T p in the proof of monotonicity of reduced volume, before I.(7.13). Fix p and τ >, and let L exp τ : T p be the map which taes v T p to γ v (τ), where γ v : [, τ] is the unique L-geodesic with τ dγv v as τ. Let B be the set of points which are dτ either endpoints of more than one minimizing L-geodesic, or which are the endpoint of a minimizing geodesic γ v : [, τ] where v T p is a critical point of L exp τ. Let G be the complement of B, and let Ω p T p be the corresponding set of initial conditions. Then Ω p is an open set, and L exp maps it diffeomorphically onto G. We claim that B has measure zero. By Sard s theorem, to prove this it suffices to prove that the set B of points q B which are regular values of L exp τ, has measure zero. Pic q B, and distinct points v 1, v T p such that γ vi : [, τ] are both minimizing geodesics ending at q. Then L exp τ is a local diffeomorphism near each v i. The first variation formula and the implicit function theorem then show that there are neighborhoods U i of v i, and a smooth hypersurface H passing through q, such that if we have points w i U i with q := L exp τ (w 1 ) = L exp τ (w ) and L length(γ w1 ) = L length(γ w ), then q lies on H. Thus B is contained in a countable union of hypersurfaces, and hence has measure zero. Therefore one may compute the integral of any integrable function on by pulling it bac to Ω p T p and using the change of variables formula. 1. Derivation of I.(7.7) From the general equation for the Levi-Civita connection in terms of the metric [3, (1.9)], if g(τ) is a 1-parameter family of metrics, with ġ = dg and = d, then dτ dτ (1.1) X Y, Z = ( X ġ)(y, Z) + ( Y ġ)(z, X) ( Z ġ)(x, Y ).

10 1 BRUCE KLEINER AND JOHN LOTT In our case ġ = Ric and so (1.) d dτ Y Y, X = X Y Y, X + Y Y, X X + Ric( Y Y, X) + Y Y, X = X Y Y, X + Y Y, X X + Ric( Y Y, X) + ( Y Ric)(Y, X) ( X Ric)(Y, Y ). On the other hand, (1.3) Y Ric(Y, X) X Ric(Y, Y ) = ( Y Ric)(Y, X) + Ric( Y Y, X) + Ric(Y, Y X) ( X Ric)(Y, Y ) Ric( X Y, Y ) = Ric( Y Y, X) + ( Y Ric)(Y, X) ( X Ric)(Y, Y ) Ric([X, Y ], Y ). We are assuming that the variation field Y satisfies [X, Y ] = (this was used in I.(7.1)). Hence one obtains the formula (1.4) d dτ Y Y, X = X Y Y, X + Y Y, X X + Y Ric(Y, X) X Ric(Y, Y ). Next, using I.(7.), (1.5) τ 1/ Y Y, X = It follows that = = = τ τ τ τ d dτ ( τ 1/ Y Y, X ) dτ τ 1/ [ 1 τ Y Y, X + d dτ Y Y, X τ 1/ [ 1 τ Y Y, X + X Y Y, X + Y Y, X X + 4 Ric( Y Y, X) + 4( Y Ric)(Y, X) ( X Ric)(Y, Y )] dτ τ 1/ [ X Y Y, X + ( Y Y )R + ] dτ 4( Y Ric)(Y, X) ( X Ric)(Y, Y )] dτ. (1.6) δ Y L τ 1/ Y Y, X = = τ τ 1/ [Y Y R ( Y Y )R + R(Y, X)Y, X + X Y 4( Y Ric)(Y, X) + ( X Ric)(Y, Y ) ] dτ τ τ [ 1/ Hess(Y, Y )R + R(Y, X)Y, X + X Y 4( Y Ric)(Y, X) + ( X Ric)(Y, Y )] dτ.

11 NOTES ON PERELAN S PAPER Derivation of I.(7.1) If {Y i } is an orthonormal basis at γ(τ), let Y i (τ) be the corresponding vector field as in I.(7.8). Then as in the computation below I.(7.8), it follows that Y i (τ), Y j (τ) = τ δ τ ij. Putting Y i (τ) = ( ) τ 1/ ei (τ), substituting into I.(7.9) and summing over i gives τ (11.1) L n τ τr 1 τ where (11.) H(X, e i ) = R + Ric(X, X) i τ τ 3/ i H(X, e i ) dτ, 4( R, X i ei Ric(e i, X)) i Ric τ (e i, e i ) + Ric 1 τ R. Tracing the second Bianchi identity gives (11.3) ei Ric(e i, X) = 1 R, X. i From a nown identity, (11.4) R τ = R Ric. On the other hand, (11.5) R τ = (g ij R ij ) τ = Ric + Ric τ (e i, e i ), i so (11.6) Ric τ (e i, e i ) = R. i Putting all this together gives i H(X, e i) = H(X), where H(X) is defined in I.(7.3). Thus (11.7) L n τr 1 τ τ τ = n τ τr 1 τ K. τ 3/ H(X) dτ The Hessian of L is given by 1. Derivation of I.(7.11) (1.1) Hess L (Y, Y ) = δ Y L δ Y Y L = δ Y L τ X, Y Y, which is given by the right-hand-side of (1.6). As with ordinary geodesics, the equation for an L-Jacobi field will differ from the integrand in (1.6) by a total derivative, namely (1.) d dτ ( τ 1/ Y, X Y ).

12 1 BRUCE KLEINER AND JOHN LOTT Then if Y is an L-Jacobi field, (1.3) Hess L (Y, Y ) = τ 1/ Y, X Y = τ 1/ Y, Y X. Fixing p, we can write (13.1) Ṽ (τ) = 13. onotonicity of Ṽ T p τ n/ e l(x,τ) J(X, τ) dx, where the integration is up to the cut locus, l(x, τ) is computed along the L-exponential of X and similarly for J(X, τ). The monotonicity of Ṽ follows from that of τ n/ e l(x,τ) J(X, τ). We have 14. Last Statement in I.7.1 (14.1) R τ + n R R, so (14.) R R τ + n R R. If r(τ) = min x R(x, τ) then (14.3) dr 1 + dτ n, so r 1 (τ) τ is monotonically nondecreasing. n Suppose that for some τ [, τ ], we have r(τ) < n. Then (τ τ) (14.4) r 1 (τ) τ n > τ n. As (14.5) r 1 (τ) τ n r 1 (τ ) τ n, it follows that r 1 (τ ) >. As r 1 (τ) <, r 1 must vanish in [τ, τ ], which is a contradiction. Put (15.1) P abc = a R bc b R ac, 15. Argument for I.7. ab = R ab 1 a b R + R acbc R cd R ac R bc + R ab t. Given a -form U and a 1-form W, put (15.) Z(U, W ) = ab W a W b + P abc U ab W c + R abcd U ab U cd. Hamilton s Harnac inequality says that with a nonnegative curvature operator, if Z(t ) then Z(t) for all t t [9, Theorem 14.1]. Taing W a = Y a and U ab = (Y a X b X a Y b )/, and using the fact that (15.3) Ric τ (Y, Y ) = ( R ab )Y a Y b R acbd R cd Y a Y b + R ac R bc Y a Y b,

13 NOTES ON PERELAN S PAPER 13 one gets that H(X, Y ) = Z(U, W ). The choice of time parameter τ is arbitrary up to a constant. Defining by H(X, Y ) = (τ) Ric(Y,Y ), it follows that for all a, we can equally well apply the Harnac inequality τ to (τ) Ric(Y,Y ). Now suppose that for some τ (, τ τ+a ), we have ( ) 1 (15.4) H(X, Y ) < Ric(Y, Y ) τ + 1. τ τ Then (τ) Ric(Y,Y ) τ τ <. We would get a contradiction with the Harnac inequality unless for all τ (τ, τ ), we have (τ ) Ric(Y,Y ) τ τ <. However, as Ric(Y, Y ), if Ric(Y, Y ) then this implies that (τ ) is unbounded below as τ τ, which is a contradiction. We are interested in generic Y and if Ric = then g is flat, in which case the first inequality at the bottom of p. 18 of I is trivially true. The inequality at the top of p. 19 of I should read (15.5) H(X) n R but this will not affect the argument. From I.(7.6), (15.6) 4τ l = 4τR + 4l 4 τ τ 4τR + 4l + 4n τ τ = 4τR + 4l + 4n τ τ 4τR + 4l + 4n c τ ( 1 τ + 1 τ τ τ ), τ 3/ H(X) d τ τ 3/ R ( 1 τ + 1 τ τ τ 1/ R τ τ τ d τ τ 1/ R d τ 4τR + 4l + 8nl c, where the last line uses the first equation in I.7.1. Equation I.(7.16) follows, and the proof of I.(7.17) is similar. 16. Argument for I.7.3 The proof is an argument by contradiction, assuming initially that ɛ. Given an L-geodesic γ(τ) with velocity vector X(τ) = dγ, its initial vector is X dτ = lim τ τ X(τ). If X is small compared to ɛ 1/, we want to show that γ does not escape from B in time ɛ r. (We will not worry about constants.) We have d (16.1) dτ X(τ), X(τ) = Ric(X, X) + X, XX = Ric(X, X) + X, R 1 X 4 Ric(X, ) τ ) d τ = X τ Ric(X, X) + X, R,

14 14 BRUCE KLEINER AND JOHN LOTT so d ( (16.) ) τ X = τ Ric(X, X) + τ X, R. dτ Letting C denote a generic n-dependent constant, for x B(p, r /) and t [t r /, t ], the fact that g satisfies the Ricci flow gives an estimate R (x, t) Cr 3 [9, Theorem 13.1]. Then d ( (16.3) ) τ X C τ X d(τ/r ) + C (τ/r) 1/ (τ X ) 1/. Equivalently, (16.4) from which (16.5) Thus (16.6) r 1 τ d ( ) τ X C τ X + C (τ/r d(τ/r ) ) 1/, τ X e Cτ/r X + C X(τ) dτ X τ/r τ/r u 1/ e Cu du + C s e C(τ/r s) ds. τ/r u u 1/ s e C(u s) ds du. Taing τ = ɛ r and X c ɛ 1/, with c small and ɛ very small, it follows that (16.7) ɛ r X(τ) dτ c r. From the Ricci flow equation g τ = Ric, it follows that the metrics g(τ) between τ = and τ = ɛ r are ecɛ -close to each other. Then for ɛ small, the length of γ, as measured with the metric at time t, will be at most 3 c r. As τ, τ n/ e l(x,τ) J(X, τ) e X /4. Then from monotonicity, (16.8) T p B(,c ɛ 1/ ) for large. τ n/ e l(x,τ) J(X, τ) d n X T p B(,c ɛ 1/ ) 17. Proof of Lemma I.8.3(b) e X /4 d n X C e c 4ɛ If γ is a minimal geodesic from x to x 1 then for any vector field V along γ that vanishes at the endpoints, d(x,x 1 ) ( X ) (17.1) V + R(V, X)V, X ds. Tae V to be linear from s = to s = r, parallel from s = r to s = d(x, x 1 ) r and linear from s = d(x, x 1 ) r to s = d(x, x 1 ). Adding over an orthonormal frame of parallel vectors gives ( ) (17.) Ric(X, X) (n 1) 3 K r + r 1. γ

15 NOTES ON PERELAN S PAPER 15 Corollary [9, Theorem 17.] If Ric K globally, with K >, then for all x, x 1, (17.4) d dt dist t(x, x 1 ) const.(n) K 1/. Proof. Put r = K 1/. If dist t (x, x 1 ) > r then the corollary follows from Lemma I.8.3(b). If dist t (x, x 1 ) r then we have the estimate (17.5) Ric(X, X) K dist t (x, x 1 ) K 1/. γ 18. Proof of Theorem I To get I.(8.1), it suffices to tae φ(t) = for t near. e (A+1n)( 1 1 t) 1 1 To see that L + n for t 1 : From the last remar at the end of I.7.1, n (18.1) R(, τ) (1 τ). Then for τ [, 1 ], (18.) L(q, τ) Hence τ τ n (1 τ) dτ n (18.3) L(q, τ) = τ 1/ L(q, τ) 4n 3 τ n 3. τ 19. Derivation of I.(9.1) The right-hand-side of I.(9.1) should be multiplied by u. To derive it, we first claim that d (19.1) = R ij i j. dt To see this, for f 1, f Cc (), we have (19.) f 1 f dv = df 1, df dv, so (19.3) so τ dτ = n 3 τ 3/. d f 1 dt f dv f 1 f R dv = Ric(df 1, df ) dv + df 1, df R dv, d (19.4) dt f R f = i (R ij j f ) i (R i f ). Then (19.1) follows from the traced second Bianchi identity. Next, one can chec that u = is equivalent to (19.5) ( t + ) f = n 1 T t + f R.

16 16 BRUCE KLEINER AND JOHN LOTT Then one obtains (19.6) u 1 v = ( t + ) [ (T t) ( f f + R) + f ] [ (T t) ( f f + R) + f ], u 1 u = f f + R (T t) ( t + ) ( f f + R) ( t + ) f + (T t) ( f f + R), f + f. Now (19.7) ( t + ) ( f f + R) = ( t )f + ( t + ) f ( t + ) f + ( t + ) R = 4 R ij i j f + ( f R) Ric(df, df) f t, f f + R + Ric + R = 4 R ij i j f + f Ric(df, df) Hence the term in u 1 v proportionate to (T t) 1 is (19.8) n The term proportionate to (T t) is ( f + f R), f f + Ric. 1 T t. (19.9) f f + R f + R + f = ( f + R). The term proportionate to (T t) is (T t) times (19.1) 4 R ij i j f f + Ric(df, df) + ( f + f R), f + f Ric + ( f f + R), f = 4 R ij i j f f + Ric(df, df) + f, f Ric = 4 R ij i j f Hess(f) Ric. Putting this together gives (19.11) v = (T t) R ij + i j f 1 (T t) g ij u.. Argument for Corollary I.9. The statement of the corollary should have max v/u instead of min v/u. To prove it, we have (.1) ( t + ) v u = v u u v u, (u v v u). u u3 Then using the fact that v, at a maximum point p of v/u we have ( v ) (.) t (p). u

17 We have d (1.1) dt hv dv = NOTES ON PERELAN S PAPER Argument for Corollary I.9.3 (( t )h v + h( t + R)v) dv = h v dv. As t T, the computation of hv approaches the flat-space calculation, which one finds to be zero.. Argument for Corollary I.9.5 The statement at the top of page 3 of I should be ( (4πτ) n/ e l). From this and the fact that ( (4πτ) n/ e f) =, the argument of the proof of Corollary I.9. gives that max e f l is nondecreasing in t, so max(f l) is nondecreasing in t. As t T one obtains the flat-space result, namely that f l vanishes. Thus f(t) l(t t) for all t [, T ). To give an alternative proof, putting τ = T t, Corollary I.9.4 says that (.1) or (.) For small τ, d dτ f(γ(τ), τ) 1 d dτ ( R(γ(τ), τ) + γ(t) ) 1 τ f(γ(τ), τ), ( τ 1/ f(γ(τ), τ) ) 1 τ 1/ ( R(γ(τ), τ) + γ(t) ). (.3) f(γ(τ), τ) d(p, γ(τ)) /4τ = O(τ ). Then integration gives τ 1/ f 1 L, or f l. 3. Argument for Claim of I.1.1 We claim first that if (x, t) satisfies t 1 αq 1 t t and d(x, t) d(x, t) + AQ 1/ then Rm (x, t) 4Q. To see this, if (x, t) α then it is true by Claim 1. If (x, t) / α then Rm (x, t) < αt 1. As (x, t) α, we now that Q αt 1. Then t 1 t and so Rm (x, t) < αt 1 Q. Thus we have a uniform curvature bound on the time-t distance ball B(x, d(x, t) + AQ 1/ ), provided that t 1 αq 1 t t. Suppose that (x, t) satisfies I.(1.3) but does not satisfy I.(1.1). As stated in I, x lies in the time-t distance ball B(x, d(x, t)+ 1 1 AQ 1/ ). Applying Lemma I.8.3(b) with r = 1 AQ 1/ and the above curvature bound, we obtain (3.1) dist t (x, x) dist t (x, x) 1 ( ) αq 1 (n 1) 3 4Q(1 AQ 1/ ) + A 1 Q 1/. Assuming that A 1 (we ll tae A later) and using the fact that α < 1 1n, it follows that d(x, t) d(x, t) + 1 AQ 1/. This shows that it is self-consistent to use the curvature bound in the above application of Lemma I.8.3(b).

18 18 BRUCE KLEINER AND JOHN LOTT 4. Argument for Claim 3 of I.1.1 If the injectivity radii of the scaled metrics are bounded away from zero : From Corollary I.9.3, v. If B v = at time t then v vanishes on B at time t. Let h be a solution to the heat equation on [ t, t) with h(x, t) a nonnegative nonzero function supported in B. As in the proof of Corollary I.9.3, hv dv is nondecreasing in t and vanishes for t = t and t t. Thus hv dv vanishes for all t [ t, t). However, for t ( t, t), h is strictly positive and v is nonpositive. Thus v vanishes on for all t ( t, t), and so (4.1) R ij + i j f on this interval. From the evolution equation, (4.) dg ij dt 1 (t t) g ij =. = R ij = i j f 1 t t g ij. It follows that the sectional curvatures go lie (t t) 1 (after performing diffeomorphisms), which contradicts the fact that Rm (x, t) = 1. If the injectivity radii of the scaled metrics tend to zero : The flat limit space L can be described as the total space of a flat orthogonal R m -bundle over a flat compact manifold C. The heat ernel u on L will be Gaussian in the fiber directions and will decay exponentially fast to a constant in the base directions, i.e. u(x, τ) (4πτ) m/ e x 4τ 1, where x is vol(c) the fiber norm. With this for u, one finds that v = (m n) ( ln(4πτ)) u. With B the ball around a basepoint of radius τ, the integral of u over B has a positive limit as τ, and so lim τ B v =. When we do the rescaling the allowed range for t goes to. 5. Argument for Theorem I.1.1 Recalling that t [, ɛ ], if φ then if A is sufficiently large, we have 9Aɛ d(y, t) 1Aɛ. We will apply Lemma I.8.3(a) with the parameter r of Lemma I.8.3(a) equal to t. As r ɛ, we have y / B(x, r ). From I.(1.4), on B(x, r ) we have Rm (, t) α t 1 + ɛ. Then from Lemma I.8.3(a), at (y, t) we have ( ) d t d (n 1) 3 (α t 1 + ɛ )t 1/ + t 1/. ( = (n 1) 1 + ) (5.1) 3 α + ɛ t t 1/. It follows that d t d + 1n t. To construct φ, tae the function which is 1 (x 1) on [1, 3/] and (x ) on [3/, ], and smooth it slightly. We have that u is positive for t [, t) and u is constant in t. Then (5.) ( ) hu t = ( h)u 1 (1Aɛ) φ u 1 (1Aɛ) u = 1 (1Aɛ).

19 Similarly, using the result of Corollary I.9.3 that v, (5.3) ( ) 1 hv ( h)v φ v t (1Aɛ) NOTES ON PERELAN S PAPER 19 1 (1Aɛ) φv = In Claim 3 of I.1.1, t [t/, t]. Then on the ball B at time t of radius t t centered at x. Then at time t, (5.4) hv v β. Thus (5.5) hv β e t (Aɛ) β e t t= 1 (1Aɛ) hv. t t 1/ ɛ and so for large A, h will be one B (Aɛ) β ( 1 t ) (Aɛ) β(1 A ). To verify the second equality in the next equation, it is enough to chec that ( (5.6) ) ) f + f he f dv = ( f + h he f dv. This follows from ( ) f + f (5.7) he f dv = = = = = h ( f, (he f ) + f he f) dv ( f, h ) f + f he f dv h f, h h f he f dv f + h h, h h f he f dv ) ( f + h he f dv. To derive the last inequality of the equation (up to unimportant constants), we have and Rh 1. Then h h 1 (1Aɛ) (5.8) Also, (5.9) t uh log h ( h h ) Rh B(x,1Aɛ) ( ) 1 u ɛ (1Aɛ) + 1 u = 1 B(x,1Aɛ) h A + ɛ. u 1 (1 ca ) = ca for an appropriate constant c. The paper now renames ũ u and f f. Then there is a sequence of functions f with support inside an open ball such that u 1 and ( 1 f f + n ) u is bounded

20 BRUCE KLEINER AND JOHN LOTT away from zero by a positive constant. On the other hand, the Euclidean isoperimetric inequality implies the logarithmic Sobolev inequality ( (5.1) 1 ) f f + n u, provided that u = (π) n/ e f satisfies u = 1. A proof, just assuming the Euclidean isoperimetric inequality, is given in [1], where an equivalent form of the inequality is stated [1, I.(8)]. 6. Proof of Corollary I.1. If the corollary is not true, we can center ourselves around the collapsing balls B(x, t) to obtain functions f as before. As in the proof of Theorem I.4.1, the volume condition along with the fact that ( (π) n/ e f 1 means that f, which implies that 1 f f + n ) u. 7. Argument for Corollary I.1.4 Using Theorem I.1.1 and Corollary I.1., we can apply Theorem I.8. starting at time A 1 (ɛr ). 8. Argument for I.1.5 One must assume an upper diameter bound. The idea is to apply the Ricci flow to get noncollapsing manifolds with bounded curvature and diameter, for which the finiteness of diffeomorphism types is nown. One has to now that the Ricci flow exists for a uniform time interval, depending on r and δ. If not, suppose that there is a sequence of solutions {g i } i=1 so that (after a slight rescaling) g i only exists up to time i r, and at time, R r and vol( Ω) n (1 δ i ) c n vol(ω) n 1 for Ω in an r -ball, with δ i. From [9, Theorem 8.1], a sectional curvature of g i must blow up as t i r. Rescaling in space by i/r and time by i /r, we get solutions {ĝ i } i=1 that exist for t [, 1), have R(x) i at time and satisfy vol( Ω) n (1 δ i ) c n vol(ω) n 1 for Ω in an i-ball. Given α >, for i large, we can apply Theorem I.1.1 with ɛ = i 1 and the r of the theorem equal to i, to get Rm(ĝ i ) (x, t) αt whenever t (, 1). This contradicts the assumption that a sectional curvature of ĝ i blows up as t Examples for I.11. There is a κ-noncollapsed ancient solution on the cylinder R S n 1 (r) where the radius satisfies r (t) = r (n )t. On the other hand, the quotient solution on S 1 S n 1 (r) is κ-collapsed for all κ. Bryant s gradient steady soliton is given by g(t) = φ t g, where g = dr + µ(r) dθ is a certain rotationally symmetric metric on R 3. It has sectional curvatures that go lie r 1, and µ(r) r. The gradient function f satisfies R ij + i j f =, with f(r) r. Then for r and r t large, φ t (r, Θ) (r t, Θ). In particular, if R C (R 3 ) is the scalar curvature function of g then R(t, r, Θ) R (r t, Θ).

21 NOTES ON PERELAN S PAPER 1 To chec the conclusion of I.11.8 in this case, given a point (r, Θ) R 3 at time, the scalar curvature goes lie r 1. ultiplying the soliton metric by r 1 and sending t r t gives the asymptotic metric (9.1) d(r/ r ) + r r t r dθ. Putting u = (r r )/ r, the rescaled metric is approximately ( (9.) du u ) t dθ. r Given ɛ >, this will be ɛ-close to the evolving cylinder du + (1 t) dθ provided that u ɛ r, i.e. r r ɛr. To have an ɛ-nec, we want this to hold whenever r r (ɛr 1 ) 1. This will be the case if r ɛ 3. Thus ɛ is approximately (9.3) {(r, Θ) R 3 : r ɛ 3 } and Q = R(x, ) ɛ 3. Then diam( ɛ ) ɛ 3 and at the origin ɛ, R(, ) ɛ. It follows that for the value of κ corresponding to this solution, C(ɛ, κ) must grow at least as fast as ɛ 3 as ɛ. 3. Argument for I.(11.1) Given a Ricci flow solution g(t ) for t [, Ω), put H(X) = R t + R + X, R + t Ric(X, X). The trace Harnac inequality says that if the curvature operator is nonnegative then H(X) for all t >. In particular, we can apply this to the Ricci flow solution corresponding to the piece of the ancient solution starting at an arbitrary time t (, ), with t = t t. Then R t + R t t + X, R + Ric(X, X). Taing t gives R t + X, R + Ric(X, X). In particular, R t. This implies the essential equivalence of the noncollapsing definitions I.4. and I.8.1 for ancient solutions. Namely, if a solution is κ-collapsed in the sense of I.8.1 then it is automatically κ-collapsed in the sense of I.4.. Conversely, if a time-t slice of an ancient solution is collapsed in the sense of I.4. then the fact that R t, together with bounds on distance distortion, implies that it is collapsed in the sense of I.8.1 (possibly for a different value of κ). 31. Argument for I.11. For a fixed τ, I.(7.16) implies that l 1/ C, and so 4τ C (31.1) l 1/ (q) l 1/ (q(τ)) 4τ dist t τ(q, q(τ)). Also from I.(7.16) we have R Cl, and we can plug the previous bound on l into the τ right-hand-side to get a bound on R. Given any τ, we apply these bounds initially at time τ = τ/. From the noncollapsing and the fact that R is nonincreasing in τ, we obtain bounds on the geometry for τ [τ/, τ], at least on a smaller ball. Then as in the proof in I.7.3, we obtain bounds on l for τ [τ/, τ] and on a possibly smaller time-τ ball (but with a uniform bound on how much smaller it is). Then we also get bounds on R in this region.

22 BRUCE KLEINER AND JOHN LOTT As the rescaled solutions are uniformly noncollapsing and have uniform curvature bounds on balls, Appendix D implies that we can tae a subsequence that converges to a solution g ij (τ), 1 τ <. We may assume that we have locally Lipschitz convergence of l. We define the reduced volume V (τ) for the limit solution using the limit function l. We claim that for any τ, the number V (τ) for the limit solution is the limit of numbers Ṽ (τ i) for the original solution, with τ i. To see the convergence, note that the monotonicity of I.7.1 implies that in general, the integrand for Ṽ satisfies (31.) τ n/ e l(x,τ) J(X, τ) const. e X /4. Thus we can apply dominated convergence, along with the pointwise convergence of the integrands of Ṽ (τ i) to the integrand of V (τ). As there is a uniform upper bound on l on an appropriate ball around q(τ i ), and a lower volume bound on the ball, it follows that as i, Ṽ (τ i ) is uniformly bounded away from zero. From this argument and the monotonicity of Ṽ, V (τ) is a positive constant as a function of τ. As the monotonicity of V (τ) follows from I.(7.13), we must have equality in I.(7.13) (as a distributional equation). This implies equality in I.(7.1), which implies equality in I.(7.14). Writing I.(7.14) as (31.3) (4 R) e l = l n e l, τ elliptic theory (presumably) gives smoothness of l. One then wishes to say that equality in I.(7.14) implies equality in I.(7.1), which implies that one has a gradient shrining soliton. There is an apparent problem with this argument, as the use of I.(7.1) implicitly assumes that the solution is defined for all τ, which we do not now. However, one can instead use Proposition I.(9.1), with f = l. Equality in (7.14) implies that v =, so I.(9.1) directly gives the gradient shrining soliton equation. (The problem with the argument using I.(7.1), and its resolution using I.(9.1), were pointed out by the UCSB group.) If the gradient shrining solition g ij (τ) is flat then i j l = g ij τ into the equality I.(7.14) gives l = l τ and l = n. Putting this τ. It follows that the level sets of l are distance spheres of radius 4τ. From the smoothness of l, must be R n and I.(7.13) and I.(7.14) imply that with an appropriate choice of origin, l = x. This gives the contradiction. 4τ 3. Argument for I.11.4 We refer to Appendix A, and also recall that Hamilton [7] has proven a Harnac inequality for Ricci flows with nonnegative curvature operator. For ancient solutions it says that for t 1 < t, x 1, x, we have ( ) (3.1) R(x, t ) exp d t 1 (x 1, x ) R(x 1, t 1 ). (t t 1 ) For later use, we note that if R(x, t) = for some (x, t) then the solution is flat. Now suppose (, g(t)) is a κ-solution on an n-manifold, where n, pic p, and consider the time t slice (, g(t )). Let R be the asymptotic scalar curvature ratio. We will be considering a sequence of pointed rescaled Ricci flows (, x, g (t)), where x is a sequence with d t (x, p), g (t) := R(x, t )g( t+t R(x,t ), and t (, ]. )

23 NOTES ON PERELAN S PAPER 3 Case 1: R =. In this case we may assume there are sequences x, r > such r that d(x, p),, R(x d(x,p) )r, and R(x) R(x ) for all x B(x, r ), where distances and scalar curvatures are taen at time t [9, Lemma.]. Consider the sequence of pointed, rescaled Ricci flows (, x, g (t)) as above. By (3.1) we have R (x, t) for all t, and all x with d (x, x ) r, where d is the distance function for g () and R (, ) is the scalar curvature of g ( ). The κ-noncollapsed assumption now implies that there are an ɛ > and a sequence ρ such that the injectivity radius of g (t) is at least ɛ at all points (x, t) with d (x, x ) < ρ, ρ t, and so we may extract a pointed limit solution (, x, g (t)), t (, ], of the sequence of pointed Ricci flows. By relative volume comparison, (, x, g ()) has positive asymptotic volume growth. By Appendix A, the Riemannian manifold (, x, g ()) is isometric to an Alexandrov space which splits off a line, which means that it is a Riemannian product R N. This implies a product structure for earlier times [5, Section 8]. Now when n =, we have a contradiction, since R(x, ) = 1 but (, g ()) is a product surface, and must therefore be flat. When n > we obtain a κ-solution on an (n 1)-manifold with positive asymptotic volume growth at time zero, and by induction this is impossible. Case : < R <. Here we choose a sequence x such that d(x, p) and R(x, t )d (x, p) R. Consider the rescaled pointed solution (, x, g (t)) for t (, ] as above. We have R (x, ) = 1, and for all b >, for sufficiently large, we have R (x, t) R (x, ) R for all x such that d d (x,p) (x, p) > b. Fixing b < R < B, and applying the derivative estimates and κ-noncollapsing assumptions as in case 1, we may extract a point limit flow (, x, g (t)) from the sequence (, x, g (t)) where := {x b < d (x, p) < B}. Note that (, g ()) is isometric to an annular portion of a nonflat metric cone, since (, x, g ()) Gromov-Hausdorff converges to the Tits cone C T (, g(t )). When n = this contradicts the fact that R (x, ) = 1. When n 3, we will derive a contradiction from Hamilton s curvature evolution equation (3.) Rm t = Rm +Q(Rm). Choose an orthonormal frame e 1,..., e n in the tangent space of (, g ()) at x such that e 1 points radially outward (w.r.t. to the cone structure), and e, e 3 span a -plane with strictly positive curvature. Let P denote the -plane spanned by e 1 and e. In terms of the curvature operator, the fact that Rm (e 1, e, e, e 1 ) = is equivalent to P, Rm P =. As the curvature operator is nonnegative, it follows that Rm P =. (In fact, this is true for any metric cone.) Then (3.3) P, ( ei Rm )P = e i P, Rm P ei P, Rm P P, Rm ei P =. Similarly, (3.4) P, ( Rm )P = i P, ( ei ei ei e i ) Rm P = i ei P, Rm ei P. However, e3 P has a nonradial component 1 e r e 3. Thus ( Rm )(e 1, e, e, e 1 ) >. The zeroth order quadratic term Q(Rm) appearing in (3.) is nonnegative when Rm is nonnegative, so we conclude that t Rm (e 1, e, e, e 1 ) > at t =. This means that Rm ( ɛ)(e 1, e, e, e 1 ) < for ɛ > sufficiently small, which is impossible.

24 4 BRUCE KLEINER AND JOHN LOTT Case 3: R =. From [13], the universal cover of any noncompact nonnegatively curved complete manifold with R = splits isometrically as a Euclidean space times a surface. With the additional assumption that V >, if n 3 then it follows that is flat. To give a direct argument in our case, let us tae any sequence x with d(x, p). Set r := d t (x, p), let g (t) := r g(r (t + t)) for t (, ], and let d be the distance function associated with g (). For any < b < B, let (b, B) := {x < b < d (x, p) < B}. Since R =, we get that sup x (b,b) Rm (x, ) as. Invoing the κ- noncollapsed assumption and the Harnac inequality as in the previous cases, we conclude that (, d, p) Gromov-Hausdorff converges to a metric cone (, d, p ) (the Tits cone C T (, g(t ))) which is flat and smooth away from the vertex p, and the convergence is smooth away from p ; furthermore, we may extract a limiting smooth, ancient, incomplete, time independent, Ricci flow ( \{p }, g ( )). The unit sphere in C T (, g(t )) defines a compact smooth hypersurface S in ( \ {p }, g ()) with principal curvatures 1. Therefore we have a sequence S of smooth hypersurfaces with principal curvatures 1 w.r.t. g () as. For sufficiently large, for all t ( 1, ], the inward principal curvatures of S with respect to g (t) are close to 1 for all t [ 1, ]; this implies that S bounds a domain B diffeomorphic to the closed n-ball (see Appendix A), and the diameter of B w.r.t. g (t) is < 3 (by Riccati comparison). Applying the Harnac inequality to y S and points x B, we see that sup x B Rm (x, 1) as. Thus (B, g ( 1), p) Gromov-Hausdorff converges to a flat manifold (B, g ( 1), p ) with convex boundary, which must be isometric to a Euclidean unit ball since all principal curvature of its boundary are = 1. This implies that S is isometric to the standard (n 1)-sphere, and C T (, g(t )) is isometric to n-dimensional Euclidean space. Thus (, g(t )) is isometric to R n, which contradicts the definition of a κ-solution. 33. Corollaries of I.11.4 Corollary If B(x, r ) is a ball in a time slice of a κ-solution, then the normalized volume r n vol(b(x, r )) is controlled (i.e. bounded away from zero) the normalized scalar curvature rr(x ) is controlled (i.e. bounded above).. If B(x, r ) is a ball in a time slice of a κ-solution, then the normalized volume r n vol(b(x, r )) is almost maximal the normalized scalar curvature rr(x ) is almost zero. 3. (Precompactness) If (, g ( ), (x, t )) is a sequence of pointed κ-solutions (without the assumption that R(x, t ) = 1) and for some r >, the r-balls B(x, r) (, g (t )) have controlled normalized volume, then a subsequence converges to an ancient solution (, g ( ), (x, )) which has nonnegative curvature operator, and is κ-noncollapsed (though a priori the curvature may be unbounded on a given time slice). 4. There is a constant η = η(κ) such that for every κ-solution (, g( )), and all x, we have R (x, t) ηr 3, R t ηr. ore generally, there are scale invariant bounds on all derivatives on curvature tensor which depend only on κ. 5. There is a function α : [, ) [, ) depending only on κ such that lim s α(s) =, and for every κ-solution (, g( )) and x, y, we have R(y)d (x, y) α(r(x)d (x, y)).

25 NOTES ON PERELAN S PAPER 5 Proof. Assertion 1, =. Suppose we have a sequence of κ-solutions (, g ( )), and sequences t (, ], x, r >, such that at time t, the normalized volume of B(x, r ) is c >, and R(x, t )r. By Appendix B, for each, we can find y B(x, 5r ), r r, such that R(y, t ) r R(x, t )r, and R(z, t ) R(y, t ) for all z B(y, r ). Note that by relative volume comparison, (33.) vol(b(y, r )) r n vol(b(y, 1r )) (1r ) n vol(b(x, r )) (1r ) n c 1 n. Rescaling the sequence of pointed solutions (, g ( ), (y, t )) by R(y, t ), we get a sequence satisfying the hypotheses of Appendix D (we use here the fact that R t for an ancient solution), so it accumulates on a limit flow (, g ( ), (y, )) which is a κ-solution. By (33.), the asymptotic volume ratio of (, g ()) is c >. This contradicts I n Assertion 3. By relative volume comparison, it follows that every r-ball in (, g (t )) has normalized volume bounded below by a (-independent) function of its distance to x. By 1, this implies that the curvature of (, g (t )) is bounded by a -independent function of the distance to x, and hence we can apply Appendix D to extract a smoothly converging subsequence. Assertion 1, =. Suppose we have a sequence (, g ( )) of κ-solutions, and sequences x, r >, such that R(x, t )r < c for all, but r n vol(b(x, r )). For large, we can choose r (, r ) such that r n vol(b(x, r )) = 1c n where c n is the volume of the unit Euclidean n-ball. By relative volume comparison, r r. Applying 3, we see that the pointed sequence (, g ( ), (x, t )), rescaled by the factor r, accumulates on a pointed ancient solution (, g ( ), (x, )), such that the ball B(x, 1) (, g ) has normalized volume 1c n at t =. Suppose the ball B(x, 1) (, g ()) were flat. Then by Hamilton s Harnac inequality (applied to the approximators) we would have R (x, t) = for all x, t, i.e. (, g (t)) would be a time independent flat manifold. But flat manifolds other than Euclidean space have zero asymptotic volume, which contradicts the assumption that the sequence (, g ( )) is κ-noncollapsed. Thus B(x, 1) (, g ()) is not flat, which means, by the Harnac inequality the scalar curvature of g () is strictly positive everywhere. Therefore, with respect to g we have ( ) ( ) lim inf R(x, t )r = lim inf (R(x, t )) r ) r r const. lim inf =, r r which is a contradiction. Assertion, =. Apply 1, the precompactness criterion, and the fact that a nonnegativelycurved manifold whose balls have normalized volume c n must be flat. Assertion, =. Apply 1, the precompactness criterion, and the Hamilton s Harnac inequality (to the approximators). Assertion 4. This follows by rescaling g so that R(x, t) = 1, applying 1 and 3.

26 6 BRUCE KLEINER AND JOHN LOTT Assertion 5. The quantity R(z)d (u, v) is scale invariant. If the assertion failed, we would have sequences (, g ( )), x, y, such that R(y ) = 1, d(x, y ) remains bounded, but the curvature at x blows up. This contradicts An alternate proof of I.11.3 using I.11.4 and Corollary 33.1 Lemma There is a constant v = v(κ) > such that if (, g( )) is a -dimensional κ-solution, x, y and r = d(x, y) then (34.) vol(b t (x, r)) vr. Proof. If the lemma were not true then there would be a sequence (, g ( )) of -dimensional κ-solutions, and sequences x, y, t R such that r vol(b t (x, r )), where r = d(x, y ). Let z be the midpoint of a shortest segment from x to y in the t -time slice (, g (t )). For large, choose r (, r /) such that (34.3) r vol(b t (z, r )) = π, i.e. half the area of the unit dis in R. As π (34.4) = r vol(b t (z, r )) r vol(b t (x, r )) = ( r /r ) r vol(b t (x, r )), =. Then by part 3 of Corollary 33.1, the sequence of pointed Ricci, accumulates on a Ricci flow (, g ( ), (z, )). The segments from z to x and y accumulate on a line in (, g ()), and hence (, g ()) splits off a line. By (34.3), (, g ()) cannot be isometric to R, and hence r it follows that lim r flows (, g ( ), (z, t )), when rescaled by r must be a cylinder. Considering the approximating Ricci flows, we get a contradiction to the κ-noncollapsed assumption. Lemma 34.1 implies that the asymptotic volume of any noncompact -dimensional κ- solution is at least v >. By I.11.4 (including the -dimensional case in section 3) we therefore conclude that every -dimensional κ-solution is compact. Consider the family F of -dimensional κ-solutions (, g( ), (x, )) with diam(, g()) = 1. By Lemma 34.1, there is uniform lower bound on the volume of the t = time slices of κ-solutions in F. Thus F is compact in the smooth topology by part 3 of Corollary 33.1 (the precompactness leads to compactness in view of the diameter bound). This implies (recall that R > ) that there is a constant K 1 such that every time slice of every -dimensional κ-solution has K-pinched curvature. Hamilton has shown that volume-normalized Ricci flow on compact surfaces with positively pinched initial data converges exponentially fast to a constant curvature metric [6]. His argument shows that there is a small ɛ >, depending continuously on the initial data, so that when the volume of the (unnormalized) solution has been reduced by a factor of at least ɛ, the pinching is at most the square root of the initial pinching. By the compactness of the family F, this ɛ can be chosen uniformly when we tae the initial data to be the t = time slice of a κ-solution in F. Now let K be the worst pinching of a -dimensional κ-solution, and let (, g( )) be a κ-solution where the curvature pinching of (, g()) is K. Choosing t < such that ɛ vol(, g(t)) = vol(, g()), the previous paragraph implies the curvature pinching of

27 NOTES ON PERELAN S PAPER 7 (, g(t)) is at least K. This would contradict the fact that K is the upper bound on the pinching for all κ-solutions, unless K = Argument for I.11.5 Compactly contained means that B(x, r ) has compact closure. If the corollary is not true, there are an ɛ > and sequences of solutions with vol(b(x, A / Q ) > ɛ(a / Q ) n for large, where A. The blowup limit has V() >. Suppose that for each κ >, there are a point (x κ, t κ ) and a radius r κ so that Rm(x κ, t κ ) rκ on the time-t κ ball B(x κ, r κ ), and vol(b(x κ, r κ )) < κ rκ. n From the Bishop-Gromov inequality, V(t κ ) < κ. As is said in I, letting R be the supremum of the scalar curvature and taing κ gives V() =, which is a contradiction. 36. Argument for I.11.6 To argue as in the proof of Claim 1 of I.1.1, put r = 1 and suppose that g ij (t) satisfies the Ricci flow equation for t [t, ], with R(x, t) > C(1 + (t t ) 1 ) for some (x, t) satisfying d t (x, x ) < 1. Following the notation of the proof of I.1.1, put A = 4 λc1/ and α = λ C 1/, where we will tae λ to be a sufficiently small number that only depends on n. Put (36.1) α = {(x, t) : R(x, t) α(t t ) 1 }. As in Claim 1 of I.1.1, with λ small enough, one can find (x, t) α with d t (x, x ) < 1 3 such that (36.) R(x, t) R(x, t) whenever (36.3) (x, t) α, t [t, t], d t (x, x ) d t (x, x ) + AR 1/ (x, t). We want to show that (36.) holds whenever (36.4) t 1 αq 1 t t, dist t (x, x) 1 1 AQ 1/, where Q = R(x, t). As in the proof of Claim of I.1.1, we have a uniform bound R(x, t) Q on the time-t distance ball B(x, dist t (x, x ) + AQ 1/ ), provided that t 1 αq 1 t t. We want to show that if (x, t) satisfies (36.4) then it satisfies (36.) or (36.3). If (x, t) satisfies (36.4) and does not satisfy (36.) then it belongs to α and x lies in the time-t ball B(x, dist t (x, x ) AQ 1/ ). Applying Lemma I.8.3(b) with r = 1 Q 1/ gives (36.5) dist t (x, x) dist t (x, x) const.(n) αq 1/ λ const.(n)aq 1/. Taing λ small enough, we have (36.6) dist t (x, x ) dist t (x, x ) + 1 AQ 1/, and so it is consistent to use the above curvature bound in the application of Lemma I.8.3(b). Hence (36.) holds whenever t 1 αq 1 t t and dist t (x, x) 1 1 AQ 1/. Renaming A gives that (36.) holds whenever t AQ 1 t t and dist t (x, x) AQ 1/, where A tends to infinity with C.