κ -Noncollapsing Estimates Along The Ricci Flow

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κ -Noncollapsing Estimates Along The Ricci Flow

研 究 生：賴馨華 指導教授：王金龍 教授

中 華 民 國 九十七 年 七 月

97.6.9

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研究生簽名: 賴馨華 學號： 952201011 論文名稱: κ -Noncollapsing estimates along the Ricci Flow 指導教授姓名： 王金龍

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沿著瑞奇流的 κ-noncollapsing _估計

摘要

　在這篇文章裡我們描述了兩種由Perelman_{提出建立沿著瑞奇}

流的κ-noncollapsing定理的方法。第一種方法是使用Perelman entropy_。 第二種方法是利用Perelman’s reduced volume_{的單調性來建立。}

Reduced volume_是對non-collapsing定理更局部的看法，因此我們

學習Perelman的証明中關於龐加萊猜想裡ancient κ-noncollapsing_的 解時(這種解不必是緊緻因此不被總體的量所控制)_{，第二個方法是} 重要的。我們的論述主要是依據Cao-Zhu [6]_，關於Perelman’s W functional_我們參考O. Rothaus [3]_{給予更詳細的說明。}

i

κ-Noncollapsing Estimates Along The Ricci Flow

Abstract

In this paper we report on the two methods pioneered by G. Perel- man [1] to establish his κ-noncollapsing thm of the Ricci flow. The first method uses the Perelman entropy. The second proof uses the monotonicity of the Perelman’s reduced volume. The second proof is important, because the reduced volume is a more localized quan- tity in its definition and so one can in fact establish local versions of the non-collapsing theorem which turn out to be important when we study ancient κ-noncollapsing solutions in Perelman’s proof of the Poincar´e conjecture. Such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy). Our treatment follows closely the cuticle by Cao-Zhu [6], with some more details on Perelman’s W functional by O. Rothaus [3].

ii

中文摘要...i 英文摘要...ii Contents...iii 1. Introduction...p.2 2. Perelman’s Reduce Volume ...p.4 3. No local collapsing theorem...p.11 References...p.20

iii

κ-NONCOLLAPSING ESTIMATES ALONG THE RICCI FLOW

SINHUA LAI

Abstract. In this paper we report on the two methods pioneered by G. Perelman [1] to establish his κ-noncollapsing thm of the Ricci flow. The first method uses the Perelman entropy. The sec- ond proof uses the monotonicity of the Perelman’s reduced volume.

The second proof is important, because the reduced volume is a more localized quantity in its definition and so one can in fact es- tablish local versions of the non-collapsing theorem which turn out to be important when we study ancient κ-noncollapsing solutions in Perelman’s proof of the Poincar´e conjecture. Such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy). Our treatment follows closely the cuticle by Cao-Zhu [6], with some more details on Perelman’s W functional by O. Rothaus [3].

1

1. Introduction

1.1. Contents of this note. Consider a complete Riemannian man- ifold M of dimension n ≥ 3 with the Riemannian metric g_ij. Let g = g(t) be a smooth solution of the Ricci flow

∂g

∂t = −2Ric

on M × [0, T ) for some (finite or infinite) T > 0 with a given initial metric g(0) = g0.

In Section 2.1-2.3, a new monotonic quantity, namely the reduced volume ˜V , is introduced. It is defined in terms of so-called L-geodesics.

Let (p, t₀) be a fixed spacetime point. Define the backward time by τ = t₀ − t. Given a curve γ(τ ) in M defined on 0 ≤ τ ≤ ¯τ (i.e. going backward in real time) with γ(0) = p, its L-length is defined to be

L(γ) = Z ¯τ

0

√τ (| ˙γ(τ )|²_{g(τ )}+ R(γ(τ ), t₀− τ ))dτ.

Let L(q, ¯τ ) be the infimum of L(γ) over curves γ with γ(0) = p and γ(¯τ ) = q. Put

`(q, ¯τ ) = L(q, ¯τ ) 2√

¯ τ . The reduced volume is defined by

V (¯˜ τ ) = Z

M

(4π¯τ )⁻ⁿ²e^{−`(q,¯τ )}dV.

The remarkable fact is that if g is a Ricci flow solution then ˜V is non- increasing in ¯τ , i.e. nondecreasing in real time t. The proof of mono- tonicity uses a subtle cancelation between the ¯τ -derivative of `(γ(¯τ ), ¯τ ) along an L-geodesic and the Jacobian of the so-called L-exponential map.

In Section 2.3, a modified ”entropy” functional W(g, f, τ ) is intro- duced. It is nondecreasing in t provided that g is a Ricci flow solution, τ = t0− t and (4πτ )ⁿ²e^−f satisfies the conjugate heat equation.

In Section 3.1, the entropy functional W is used to prove a no local collapsing theorem. The statement is that if g is a given Ricci flow on a finite time interval [0, T ) then for any (scale) ρ, there is a number κ > 0 so that if B_t(x, r) is a time-t ball with radius r less than ρ, then

|Rm| ≤ 1

r² ⇒ V ol(B_t(x, r)) ≥ κrⁿ on B_t(x, r).

κ-NONCOLLAPSING ESTIMATES 3

The method of proof is to show that if r⁻ⁿV ol(Bt(x, r)) is very small than the evaluation of W at time t is very negative, which contradicts the monotonicity of W.

In Section 3.2 we will use a cut-off argument to extend the no local collapsing theorem to any complete solution with bounded curvature.

In some sense, the second no local collapsing theorem gives a good relative estimate of the volume element for the Ricci flow.

1.2. Historical remarks. Historically, in the 1980’s, it was Richard Hamilton who initiated the program of using the Ricci flow to solve the Poincar´e conjecture as well as the hyperbolicity conjecture of three dimensional manifolds. His idea is to do surgeries on the manifold when the curvature tends to blow-up in some region during the Ricci flow. In order to perform surgeries Hamilton needs to classify the neighborhood of the blow-up region. He used a standard parabolic scaling of the Ricci flow to perform the blow-up analysis and he also proved a convergence theorem of the rescalled region when a “Little Loop Lemma” holds.

Unfortunately his proof of the Little Loop Lemma turns out to be incorrect and it becomes the first main obstacle to carry out Hamilton’s program.

In [1], G. Perelman made a breakthrough in Hamilton’s program.

Among many other things, Perelman formulated and proved the No Local Collapsing Theorem which in particular implies the Little Loop Lemma as a corollary. The L-geodesic, reduced length ` as well as the reduced volume ˜V are all due to Perelman. Since Perelman’s paper was written in a rather dense manner, it is highly desirable to have more transparent proofs, with more details filled in, of the results proved or announced in [1]. Since then several nice articles had appeared aiming at understanding Perelman’s argument.

Our purpose here is simply to understand Perelman’s No Local Col- lapsing Theorems, both the compact and non-compact cases. Our treatment follows closely the cuticle by Cao-Zhu [6], with some more details on Perelman’s W functional by using results in O. Rothaus [3].

It is the author’s hope that this note will be helpful as a supplementary reading for people who wants to read Perelman’s work.

2. Perelman’s Reduce Volume

2.1. The L-geodesics. We write the Ricci flow in the backward ver- sion

∂g_ij

∂τ = 2R_ij

on a manifold M with τ = τ (t) satisfying ^dτ_dt = −1. We always as- sume that either M is compact or g_ij(τ ) are complete and have uni- formly bounded curvature. The L-length of a (smooth) space curve γ : [τ1, τ2] → M is defined by

L(γ) = Z τ2

τ1

√τ (R(γ(τ )) + | ˙γ(τ )|²)dτ,

where the scalar curvature R(γ(τ )) and the norm | ˙γ(τ )| are evaluated using the metric at time t = t0− τ. Here τ1 > 0.

To derive the L-geodesic equation, as in the standard Riemannian geometry we consider an 1-parameter family of curves γ_s : [τ₁, τ₂] → M , parametrized by s ∈ (−, ). Equivalently, we have a map γ(s, τ )e with s ∈ (−, ) and τ ∈ [τ₁, τ₂]. Putting X = ^∂_∂τ^eγ and Y = ^∂_∂s^eγ, we have [X, Y ] = 0. This implies that ∇_XY = ∇_YX. Writing δ_Y as shorthand for _ds^d|_s=0, and restricting to the curve γ(τ ) = eγ(0, τ ). We have (δ_YY )(τ ) = Y (τ ) and (δ_YX)(τ ) = (∇_XY )(τ ). Then

δ_Y(L)

= Z τ2

τ1

√τ (h∇R, Y i + 2hX, ∇YXi)dτ

= Z τ2

τ1

√τ (h∇R, Y i + 2hX, ∇_XY i)dτ

= Z τ2

τ1

√τ (h∇R, Y i + 2 d

dτhX, Y i − 2h∇_XX, Y i − 4Ric(X, Y ))dτ

= 2√

τ hX, Y i   

τ2

τ1

+ Z τ2

τ1

√τ hY, ∇R − 2∇_XX − 4Ric(·, X) − 1 τXidτ.

Hence the L-geodesic equation is

∇_XX −1

2∇R + 1

2τX + 2Ric(X, ·) = 0

where the 1-form Ric(X, ·) has been identified with the corresponding dual vector field.

Give any p, q ∈ M and τ2 > τ1 > 0 there exists a L-shortest geodesic γ : [τ₁, τ₂] → M such that γ(τ₁) = p, γ(τ₂) = q and satisfies the L- geodesic equation. Multiplying √

τ to the- L-geodesic equation, we

κ-NONCOLLAPSING ESTIMATES 5

get

∇_X(√

τ X) =

√τ

2 ∇R − 2√

τ Ric(X, ·) on [τ₁, τ₂].

That is,

(1) d

dτ(√

τ X) =

√τ

2 ∇R − 2Ric(√

τ X, ·) on [τ₁, τ₂].

Thus, if a continuous curve defined on [τ₁, τ₂] satisfying the L-geodesic equation (1) for any subinterval 0 < τ₁ < τ < τ₂, then v = lim

τ →0⁺

√τ X(τ ) exists. This allows us to extend the definition of the L-length to in- clude the case τ₁ = 0 for all those (continuous) curves γ : [0, τ₂] → M which are smooth on (0, τ₂] and have limits lim

τ →0+

√τ ˙γ(τ ).

This means that for a fixed p ∈ M, by taking τ1 = 0 and γ(0) = p, the vector v = lim

τ →0

√τ X(τ ) is well-defined in TPM. The L-exponential map L exp_τ : T_PM → M sends v to γ(τ ).

2.2. Perelman’s reduced volume. The function L(q, ¯τ ) is the infi- mum of the L-length among curves γ with γ(0) = p and γ(¯τ ) = q.

When we perform standard variational calculations of the function L, we can get the following results (see [2], 18-22):

(2) dL(γ(¯τ ), ¯τ ) d¯τ =√

¯

τ (R(γ(τ ))) + |X(¯τ )|²) and

(3) ¯τ³²(R(γ(¯τ ) + |X(¯τ )|²) = −K + 1

2L(q, ¯τ ), where

K = Z τ¯

0

τ³²H(X(τ ))dτ and

H(X) = −R_τ − 1

τ − 2h∇R, Xi + 2Ric(X, X).

Also

(4) L_τ¯(q, ¯τ ) = 2√

¯

τ R(q) − 1

2¯τL(q, ¯τ ) + 1

¯ τK, (5) ∆L ≤ n

√τ¯ − 2√

¯ τ R − 1

¯ τ

Z ¯τ 0

τ³²H(X)dτ = n

√τ¯− 2√

¯ τ R − 1

¯ τK and

(6) L¯_τ¯+ ∆ ¯L ≤ 2n where L(q, τ ) = 2¯ √

τ L(q, τ ).

Moreover,

(7) d|Y |²

dτ   τ =¯τ

≤ 1

¯ τ − 1

√τ¯ Z τ¯

0

√τ H(X, ˜Y )dτ.

Defining the reduced length by

`(q, τ ) = L(q, τ ) 2√

τ and the reduced volume by

V (τ ) =˜ Z

M

(4πτ )⁻ⁿ²e^{−`(q,τ )}dq.

The goal is to show that ˜V (τ ) is nonincreasing in τ , i.e. nondecreasing in t. To do this one uses the L-exponential map to write ˜V (τ ) as an integral over T_pM :

V (τ ) =˜ Z

TpM

(4πτ )⁻ⁿ²e^{−`(L expτ(v),τ )}J (v, τ )χ_τ(v)dv,

where J (v, τ ) = det d(L_expτ)_v is the Jacobian factor in the change of variables and χ_τ is a cutoff function related to the L-cut locus of p.

To show that ˜V (τ ) is nonincreasing in τ , it suffices to show that τ⁻ⁿ²e^{`(L expτ(v),τ )}J (v, τ )

is nonincreasing in τ , or equivalently that

−n

2log(τ ) − `(L exp_τ(v), τ ) + log J (v, τ ) is nonincreasing in τ. Hence it is necessary to compute

d`(L exp_τ(v), τ )

dτ and dJ (v, τ ) dτ .

The fact that ˜V (τ ) is nonincreasing in τ is then used to show that the Ricci flow solution cannot collapse near p.

Theorem 2.1 (Monotonicity of Perelman’s reduced volume). Let g_ij be a family of complete metrics evolved by the Ricci flow _∂τ^∂ g_ij = 2R_ij on a manifold M with bounded curvature. Fix a point p in M and let `(q, τ ) be the reduced distance from (p, 0). Then Perelman’s reduced volume

V (τ ) =˜ Z

M

(4πτ )⁻ⁿ²e^{−`(q,τ )}dV_τ(q) is nonincreasing in τ .

κ-NONCOLLAPSING ESTIMATES 7

Proof. Fix p ∈ M. From the discussion above, we can write V (τ ) =˜

Z

TpM

(4πτ )⁻ⁿ²e^{−`(L expτ(v),τ )}J (v, τ )χ_τ(v)dv,

where J (v, τ ) = det d(L_expτ)_v is the Jacobian factor in the change of variable and χ_τ is a cutoff function related to the L-cut locus of p.

We first show that for each v, the expression

−n

2log(τ ) − `(L exp_τ(v), τ ) + log J (v, τ )

is nonincreasing in τ. Let γ be the L-geodesic with initial vector v ∈ T_pM. From (2) and (3),

(8) d`(γ(τ ), τ ) dτ

τ =¯τ = − 1

2¯τ`(γ(¯τ ) +1

2(R(γ(¯τ ) + |X(¯τ )|²) = −1

2¯τ⁻³²K.

Next, let {Y_i}ⁿ_i=1 be a basis for the Jacobi fields along γ that vanish at τ = 0. We can write

log J (v, τ )² = log det((d(L exp_τ)v)^∗d(L exp_τ)v = log det(S(τ ))+const., where S is the matrix

S_ij(τ ) = hY_i(τ ), Y_j(τ )i.

Then

d log J (v, τ )

dτ = 1

2Tr S⁻¹dS

dτ

 .

To compute the derivative at τ = ¯τ , we can choose a basis so that S(¯τ ) = I_n, i.e. hY_i(¯τ ), Y_j(¯τ )i = δ_ij then using (7) and the same method as in ([2], 21),

(9) dlnJ (v, τ ) dτ

τ =¯τ = 1 2

n

X

i=1

d|Y_i|² dτ

τ =¯τ ≤ n 2¯τ − 1

2τ¯⁻³²K.

From (8) and (9), we deduce that τ −n

2e^{−`(L expτ(v),τ )}J (v, τ )

is nonincreasing in τ. Finally, if τ ≤ τ⁰ then Ω_τ⁰ ⊂ Ω_τ, so χ_τ(v) is nonincreasing in τ. Hence ˜V (τ ) is nonincreasing in τ. 

2.3. Perelman’s W functional.

Definition 2.2. Perelman’s W functional is defined by W(g_ij, f, τ ) =

Z

M

[τ (R + |∇f |²) + f − n](4πτ )⁻ⁿ²e^−fdv

where g_ij is a Riemannian metric, f is a smooth function on M, and τ is a positive scalar parameter. The functional W is invariant under simultaneous scaling of τ and g_ij (or equivalently the parabolic scaling), and invariant under diffeomorphism. Namely, for any positive number a and any diffeomorphism ϕ

W(aϕ^∗g_ij, ϕ^∗f, aτ ) = W(g_ij, f, τ ).

Now we set

µ(g_ij, τ ) = inf{W(g_ij, f, τ )|f ∈ C^∞(M ), 1 (4πτ )^n/2

Z

e^−fdV = 1}.

Note that if we let u = e^{−f /2}, then the functional W can be expressed as

W(g_ij, f, τ ) = Z

M

[τ (Ru²+ 4|∇u|²) − u²log u²− nu²](4πτ )⁻ⁿ²dV and the constraintR

M(4πτ )⁻ⁿ²e^−fdV = 1 becomes R

Mu²(4πτ )⁻ⁿ²dV = 1.

Lemma 2.3.

µ(g_ij, τ ) = inf{W(g, f, τ )|f ∈ C^∞(M ), 1 (4πτ )^n/2

Z

e^−fdV = 1}

is finite and nondecreasing, where g_ij is a Riemannian metric, f is a smooth function on M and τ is a positive scalar parameter.

In order to prove this ,we will use the following Lemmas. Let Ω be a domain (open connected) in M and C₀^∞(Ω) be the space of real valued infinitely differentiable functions, compactly supported in Ω.

The Sovolev space H₀¹(Ω) is the closure of C₀^∞(Ω) in the norm k f k²=

Z

Ω

f²+ Z

Ω

|∇f |²,

where the integrations use the volume element arising from the Rie- mannian structure and ∇f, |∇f |², are also determined by the Riemann- ian structure. Let 4 be the Laplace-Beltrami operator. For any real valued measurable function f on Ω, we say that f ∈ L^p+(Ω) if |f |^q is integrable on Ω for some q > p. We use k f k_q to denote the L^q(Ω) norm of f . Let H be a non-negative measurable function on Ω,

κ-NONCOLLAPSING ESTIMATES 9

for which log H ∈ Lⁿ²⁺. Let ρ be a positive real number and define a_ρ(H) as the infimum of

Z

(ρ|∇|²− f²log f²+ f²log H) for f ∈ H¹₀,

subject to the proviso R f² = 1. (The integral is well defined, since f ∈ H₀¹ ⇒ f ∈ L^2n/(n−2).)

Lemma 2.4. R (ρ|∇f |² + f²log H) is bounded below if f ∈ H₀¹ and R f² = 1.

Proof. We may write log H = U + V, where U has its Lⁿ² norm as small as we want, say

k U k n 2 ≤ 

and V is bounded, say |V | ≤ D. By the Sobolev theory, there exists a constant C independent of f such that

k f k ²ⁿ

(n−2)≤ C k f k . But now

Z

f²U ≤k f k²2n (n−2)

k V kⁿ

2≤ C² ∈k f k² . So

Z

(ρ|∇f |²+ f²log H)

= ρ k f k² −ρ + Z

f²V ≥ ρ k f k² −ρ − D − C² k f k², which is bounded below as long as C² < ρ.

 Lemma 2.5. For f ∈ H₀¹,R f² = 1, the functionalR (ρ|∇f |²−f²log f²) is bounded below.

Proof. Pick  satisfying 0 <  < _(n−2)² . Then by Jensen’s inequality for the logarithm,

Z

f²log f² = 1

 Z

f²log |f |^2 ≤ (2 + 2) log k f k_2+2, and by the Sobolev theory,

k f k_2+2≤ C k f k .

Hence Z

(ρ|∇f |²− f²log f²) = ρ k f k² −ρ − Z

f²log f²

≥ ρ k f k² −ρ − (2 + 2)

 log C k f k,

which is bounded below since k f k≥ 1. 

Lemma 2.6. a_ρ(H) is finite.

Proof. It follows from Z

(ρ|∇f |²− f²log f² + f²log H)

= Z

(ρ

2|∇f |²+ f²log H) + Z

(ρ

2|∇f |²− f²log f²).

 The following two results are proved in [3]:

Theorem 2.7. a_ρ(H) is an attained minimum.

Theorem 2.8. A minimizer f for a_ρ(H) is continuous on ¯Ω.

Any f ∈ H₀¹ withR f² = 1 which attains the minimum of a_ρ(H) will be called a minimizer for a_ρ(H).

Remark 1. We can show that µ(g_ij, τ ) is achieved by a smooth mini- mizer f from Theorem 2.8.

Proof of Lemma 2.3. We use Lemma 2.6 to show that µ(g_ij, τ − t) is finite. We can get that µ(g_ij(t), τ − t) is nondecreasing along the Ricci

flow follows from [6, Corollary 1.5.9.] 

κ-NONCOLLAPSING ESTIMATES 11

3. No local collapsing theorems

Definition 3.1. Let κ, γ be two positive constants and let g_ij(t), 0 ≤ t < T , be a solution to the Ricci flow on an n-dimensional manifold M . We call the solution g_ij(t) is κ-noncollapsed at (x₀, t₀) ∈ M × [0, T ) on the scale γ if it satisfies the following property:

Whenever

|Rm|(x, t) ≤ r⁻² for all x ∈ B_t0(x_o, r) and t ∈ [t₀− r², t₀], we have

V ol(B_t0(x₀, r))) ≥ κrⁿ.

Here Bt0(x0, r) is the geodesic ball centered at x0 ∈ M and of radius r with respect to the metric gij(t0). We now use the W − f unctional to prove the no local collapsing theorem.

3.1. No local collapsing theorem I.

Theorem 3.2 (No local collapsing theorem I). Suppose that M is a compact Riemannian manif old and g_ij(t), 0 ≤ t < T < ∞, is a solution to the Ricci f low. Then the solution gij(t) is κ-noncollapsed at (x₀, t₀) ∈ M × [0, T ) on the scale γ ∈ (0,√

T ].

Proof. We want to prove

(10) V ol_t0(B_t0(x₀, a)) ≥ κaⁿ for all 0 < a ≤ r. Recall that

(11) µ(g_ij, τ ) = inf{W (g_ij, f, τ )|

Z

M

(4πτ )⁻ⁿ²e^−fdv = 1}.

Note that: Since µ(gij(t), τ − t) is nondecreasing in t by Lemma 2.3, if we assume that g_ij(t) = g_ij(0) for all t ∈ R then µ(g(0), 2T ) ≤ µ(g(0), τ ) for all 0 ≤ τ ≤ 2T .

Let f be the minimizer of µ(g(0), 2T ). By Theorem 2.8, we know that f is smooth. Since M is compact,we get |µ(g(0), 2T )| ≤ a for some a ∈ R. Let

µ₀ = inf

0≤τ ≤2Tµ(g_ij(0), τ ) ≥ µ(g_ij(0), 2T ) > −∞.

By Lemma 2.3, we have

(12) µ(g_ij(t₀), b) ≥ µ(g_ij(0), t₀+ b) ≥ µ₀

for 0 < b ≤ r². Let 0 < ζ ≤ 1 be a positive smooth function on R where ζ(s) = 1 for |s| ≤ ¹₂, |ζ⁰|²/ζ ≤ 20 and ζ(s) is very close to zero for |s| ≥ 1. Define a function f on M by

(4πr²)⁻ⁿ²e^{−f (x)} = e^−c(4πr²)⁻ⁿ²ζ(d_t0(x, x₀) r ),

where the constant c is chosen so that R

M(4πr²)⁻ⁿ²e^−fdv_t0 = 1. Then we use (12) to get

(13)

W(g_ij(t₀), f, r²) = Z

M

[r²(|∇f |²+ R) + f − n](4πr²)⁻ⁿ²e^−fdv_t0 ≥ µ₀. By (13), we get

(14) (c − n) + Z

M

[r²(|∇f0|²+ R) − log ζ](4πr²)⁻ⁿ²e^−fdvt0 ≥ µ0. Since

Z

M

(r²R)(4πr²)⁻ⁿ²e^−fdv_t0

= Z

B_t0(x0,r)

(r²R)(4πr²)⁻ⁿ²e^−fdv_t0 + Z

M \B_t0(x0,r)

r²R(4πr²)⁻ⁿ²e^−fdv_t0

= Z

B_t0(x0,r)

(4πr²)⁻ⁿ²e^−fdv_t0 + Z

M \B_t0(x0,r)

r²R(4πr²)⁻ⁿ²e^−cζdv_t0 ≤ 2,

|∇f₀|² = |∇(− log ζ)|² = (ζ⁰)² ζ² · 1

r² and

Z

M

((ζ⁰)²

ζ² − log ζ)(4πr²)⁻ⁿ²e^−fdv_t0

= Z

B_t0(x0,r)

((ζ⁰)²

ζ − ζ log ζ)(4πr²)⁻ⁿ²e^−cdv_t0

≤ 2(20 + e)(4πr²)⁻ⁿ²e^−cV ol(B_t0(x₀, r), thus (13) is reduced to

c ≥ −2(20 + e)V ol(B_t0(x₀, r))

V ol(B_t0(x₀,^r₂)) + (n − 2) + µ₀. Note that

1 = Z

M

(4πr²)⁻ⁿ²e^−cζ(d_t0(x, x₀) r )dv_t0

≥ Z

B_t0(x0,^r₂)

(4πr²)⁻ⁿ²e^−cdv_t0

= (4πr²)⁻ⁿ²e^−cV ol(B_t0(x₀,r 2)).

κ-NONCOLLAPSING ESTIMATES 13

Note also that 1 =

Z

M

(4πr²)⁻ⁿ²e^−fdv_t0

= Z

M

(4πr²)⁻ⁿ²e^−cζ(d_t0(x, x₀) r )dv_t0

≤ 2 Z

B_t0(x0,r)

e^−c(4πr²)⁻ⁿ²dvt0.

Thus we get

V ol(B_t0(x₀, r)) ≥ 1

2e^c(4πr²)ⁿ². Let us set

K = min{1

2exp(−2(20 + e)3⁻ⁿ+ (n − 2) + µ₀),1 2α_n} where α_n is the volume of the unit ball in Rⁿ. Then we obtain

V ol(B_t0(x₀, r))) ≥ 1

2e^c(4πr²)ⁿ²

≥ 1

2(4π)ⁿ²exp(−2(20 + e)3⁻ⁿ+ (n − 2) + µ0)rⁿ

≥ Krⁿ

provided that V ol(B_t0(x₀,^r₂)) ≥ 3⁻ⁿV ol(B_t0(x₀, r)).

The above argument also works for any smaller radius a ≤ r. Thus we have proved

(15) V ol(B_t0(x₀, a)) ≥ Kaⁿ

for a ∈ (0, r] and V ol(Bt0(x0,^a₂)) ≥ 3⁻ⁿV ol(Bt0(x0, a)). Now we argue by contradiction to prove the assertion (9) for any a ∈ (0, r].

Suppose (10)a fails for some a ∈ (0, r]. Then by (15) we have

V ol(B_t0(x₀,a

2)) < 3⁻ⁿV ol(B_t0(x₀, a))

< 3⁻ⁿKaⁿ

< K(a 2)ⁿ. Thus (10)^a

2 also fails. By induction we get V ol(B_t0(x₀, a

2^k)) < K( a 2^k)ⁿ

for all k ≥ 1. But this contradicts to the limit

κ→∞lim V ol(B_t0(x₀, a 2^k))/( a

2^k)ⁿ= α_n.

 3.2. No Local Collapsing Theorem II.. In this section we will use a cut-off argument to extend the no local collapsing theorem to any complete solution with bounded curvature. In some sense, the second no local collapsing theorem gives a good relative estimate of the volume element for the Ricci flow.

Lemma 3.3 (Perelman). Suppose we have a solution to the Ricci flow (gij)t= −2Rij.

(a) Suppose that Ric(x, t0) ≤ (n − 1)K for distt0(x, x0) < r0. Then the distance function d(x, t) = dist_t(x, x₀) satisfies at t = t₀ outside B(x₀, r₀) the differential inequality

dt− ∆d ≥ −(n − 1)(2

3Kr0+ r⁻¹₀ ).

(The inequality is understood in the barrier sense when necessary.) (b) Suppose Ric(x, t₀) ≤ (n − 1)K when

distt0(x, x0) < r0 or distt0(x, x1) < r0. Then at t = t₀,

d

dtdist_t(x₀, x₁) ≥ −2(n − 1)(2

3Kr₀+ r₀⁻¹).

Proof of Lemma (a). Let r : [0, d(x, t₀)] → M be a shortest nor- mal geodesic from x₀ to x with respect to the metric g_ij(t₀). Let {X, e₁, . . . , e_n−1} be an orthonormal basis of T_x0M . Extend this basis parallelly along γ to form a parallel orthonormal basis { X, e₁, · ·

·, e_n−1} along γ. We consider that x and x₀ are not conjugate to each other in the metric g_ij(t₀).

Let X_i(s), i = 1, . . . , n − 1, be the Jacobi fields along γ such that Xi(0) = 0, Xi(d(x, t0)) = ei(d(x, t0)) and [Xi, X] = 0 for i = 1, . . . , n − 1. Then we have (see for example [4])

∆_t0(x, x₀) =

n−1

X

i=1

Z d(x,t0) 0

(|∇_XX_i|²− R(X, X_i, X, X_i))ds Define vector fields Y_i, i = 1, . . . , n − 1, along γ as follows:

Y_i(s) = f (s)e_i(s),

κ-NONCOLLAPSING ESTIMATES 15

where f (s) = _r^s

0 if s ∈ [0, r0] and f (s) = 1 if s ∈ [r0, d(x, t0)] then we can see that

Y_i(0) = 0 = X_i(0),

Y_i(d(x, t₀)) = e_i(d(x, t₀)) = X_i(d(x, t₀)).

Thus by using the standard index comparison theorem (see for example [5]) we have

∇d_t0(x, x₀)

=

n−1

X

i=1

Z d(x,t0) 0

(|∇X∇Xi|²− R(X, Xi, X, Xi))ds

≤

n−1

X

i1

Z d(x,t0) 0

(|∇_XY |² − R(X, Y_i, X, Y_i))ds

= Z r0

0

1

r₀²(n − 1 − s²Ric(X, X))ds +

Z d(X,t0) r0

(−Ric(X, X))ds

= − Z

r

Ric(X, X) + Z r0

0

((n − 1)

r²₀ + (1 − s²

r₀²Ric(X, X))ds

≤ − Z

r

Ric(X, X) + (n − 1)(2

3Kr₀+ r⁻¹₀ ).

On the other hand,

∂

∂td_t(x, x₀) = ∂

∂t

Z d(x,t0) 0

pg_ijXⁱX^jds = − Z

r

Ric(X, X)ds.

Thus we get the desired result. 

Proof of Lemma (b). The proof is divided into three cases.

Case 1: d_t0(x₀, x₁) ≥ 2r₀.

Let γ be a normalized minimal geodesic from x₀to x₁ and X(s) = ^dr_ds. If any piecewise-smooth vector field V along γ that vanishes at the endpoints, the second variation formula gives

Z d(x0,x1) 0

(|∇_XV |²+ hR(V, X)V, Xi)ds ≥ 0.

Let e_i(s)ⁿ⁻¹_i=1 be a parallel orthonormal frame along γ that is perpen- dicular to X. Put V_i(s) = f (s)e_i(s), where

f (s) =







s

r0 if 0 ≤ s ≤ r₀,

1 if r₀ ≤ s ≤ d(x₀, x₁) − r₀,

d(x0,x1)−s

r0 if d(x₀, x₁) − r₀ ≤ s ≤ d(x₀, x₁).