Volume growth and the topology of manifoldswith nonnegative Ricci curvature

by

Michael Munn

A dissertation submitted to the Graduate Faculty in Mathematics in partial

fulfillment of the requirements for the degree of Doctor of Philosophy, The

City University of New York.

2008

c©2008

Michael Munn

All Rights Reserved

ii

This manuscript has been read and accepted for the Graduate Faculty in

Mathematics in satisfaction of the dissertation requirements for the degree

of Doctor of Philosophy.

Christina Sormani

Date Chair of Examining Committee

Jozef Dodziuk

Date Executive Officer

Isaac Chavel

Jozef Dodziuk

Christina Sormani

Supervisory committee

THE CITY UNIVERSITY OF NEW YORK

iii

Abstract

Volume growth and the topology of manifolds with nonnegative Ricci

curvature

by

Michael Munn

Advisor: Professor Christina Sormani

Let Mn be a complete, open Riemannian manifold with Ric ≥ 0. In 1994,

Grigori Perelman showed that there exists a constant δn > 0, depending only

on the dimension of the manifold, such that if the volume growth satisfies

αM := limr→∞Vol(Bp(r))

ωnrn ≥ 1 − δn, then Mn is contractible. Here we em-

ploy the techniques of Perelman to find specific lower bounds for the volume

growth, α(k, n), depending only on k and n, which guarantee the individual

k-homotopy group of Mn is trivial.

In addition, we extend these results to the setting of metric measure

spaces Y which can be realized as the pointed metric measure limit of a

sequence {(Mni , pi)} of complete, open connected Riemannian manifolds with

RicMi≥ 0, provided the limit space Y satisfies the same lower bounds on

volume growth, i.e. αY > α(k, n).

iv

Acknowledgments

I wish to thank lots of people including but not limited to....

M.M

April of 2008

New York, NY

v

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Almost Equicontinuity and the Construction of Homotopies 10

2.1 Background and Definitions . . . . . . . . . . . . . . . . . . . 10

2.2 Homotopy Construction Theorem . . . . . . . . . . . . . . . . 13

3 Double Induction Argument 16

3.1 Key Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Proof of Main Lemma(k) . . . . . . . . . . . . . . . . . . . . . 19

3.3 Proof of Moving In Lemma(k) . . . . . . . . . . . . . . . . . . 27

3.4 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 33

4 Generalizations to Metric Measure Limits 35

4.1 Background and Definitions . . . . . . . . . . . . . . . . . . . 36

4.2 Generalization of Perelman’s Maximal

Volume Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Generalization of the Excess Estimate . . . . . . . . . . . . . . 43

vi

5 Appendix I 46

5.1 Optimal Constants . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Computing α(k, n) values . . . . . . . . . . . . . . . . . . . . 49

6 Appendix II 58

6.1 Computing α(1, n) . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 Computing α(2, n) . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3 Computing α(3, n) . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vii

Chapter 1

Introduction

Let Mn be an n-dimensional complete Riemannian manifold with nonnega-

tive Ricci curvature. For a base point p ∈ Mn, denote by Bp(r) the open

geodesic ball in Mn centered at p and with radius r. Let Vol(Bp(r)) de-

note the volume of Bp(r) and denote by ωn the volume of the unit ball in

Euclidean space. By the Bishop-Gromov Relative Volume Comparison Theo-

rem, [4, 11], the function r→ Vol(Bp(r))/ωnrn is non-increasing and bounded

above by 1.

Definition 1.0.1. Define αM , the volume growth of Mn, as

αM := limr→∞

Vol(Bp(r))

ωnrn.

The manifold Mn is said to have large (or Euclidean) volume growth if

αM > 0.

The constant αM is a global geometric invariant of Mn, i.e. it is indepen-

dent of base point.

1

Lemma 1.0.2. With Mn be as above, p ∈Mn. The volume growth of Mn,

αM = limr→∞

Vol(Bp(r))

ωnrn,

is independent of the base point p.

Proof. As stated earlier, the Bishop-Gromov Relative Volume Comparison

Theorem, [4, 11], implies that the function r → Vol(Bp(r))/ωnrn is non-

increasing and nonnegative; therefore, the limit exists. Let p, q ∈ Mn be

distinct points in Mn. Note that for any r > 0,

Bq(r) ⊂ Bp(r + d(p, q)); (1.1)

and hence, Vol(Bq(r)) ≤ Vol(Bp(r + d(p, q))). Therefore,

limr→∞

Vol(Bq(r))

ωnrn≤ lim

r→∞

Vol(Bp(r + d(p, q)))

ωnrn(1.2)

= limr→∞

Vol(Bp(r))

ωnrn. (1.3)

Similarly, we can show

limr→∞

Vol(Bp(r))

ωnrn≤ lim

r→∞

Vol(Bq(r))

ωnrn. (1.4)

Thus, the definition of αM is independent of base point.

Note that when αM > 0,

Vol(Bp(r) ≥ αMωnrn, for all p ∈M and for all r > 0.

Also, it should be noted that when αM = 1, Mn is isometric to Rn. This is

a consequence of the Bishop-Gromov Volume Comparison Theorem [4, 11].

In this paper, we study complete manifolds with RicM ≥ 0 and αM > 0.

Anderson [2] and Li [14] have independently shown that the order of π1(Mn)

2

is bounded from above by 1αM

. In particular, if αM > 12, then π1(M

n) = 0.

Furthermore, Zhu [19] has shown that when n = 3, if αM > 0, then M3 is

contractible. It is interesting to note that this is not the case when n = 4

as Menguy [12] has constructed examples of 4-manifolds with large volume

growth and infinite topological type based on an example by Perelman [15].

In 1994, Perelman [16] proved that there exists a small constant δn > 0 which

depends only on the dimension n ≥ 2 of the manifold, such that if αM ≥

1− δn, then Mn is contractible. It was later shown by Cheeger and Colding

[6] that the conditions in Perelman’s theorem are enough to show that Mn

is C1,α diffeomorphic to Rn. In this paper, we follow the method of proof in

Perelman’s theorem. Employing this method, we determine specific bounds

on αM which imply the individual k-th homotopy groups of the manifold are

trivial. We prove

Theorem 1.0.3. Let Mn be a complete Riemannian manifold with Ric ≥ 0.

If

αM > α(k, n),

where α(k, n) are the constants given in Table 5.4, then πk(Mn) = 0.

Remark. Table 5.4 contains values of α(k, n) for 1 ≤ k ≤ 3 and 1 ≤ n ≤

10. In general, the value of α(k, n) is determined by Equation (5.36), where

the function hk,n(x) and the values of δk,n are defined in Definition 3.0.3.

In Section 1.1, we state general results from Riemannian geometry that

will be required for the proof. The key ingredients are the excess estimate of

Abresch-Gromoll, the Bishop-Gromov Volume Comparison Theorem, and a

Maximal Volume Lemma of Perelman [Lemma 1.1.3]. In Chapter 2, we apply

3

the theory of almost equicontinuity from [17] to prove a general Homotopy

Construction Theorem [Theorem 2.2.1] that will be needed when constructing

the homotopies for Theorem 1.0.3.

In Chapter 3, we prove Theorem 1.0.3 using a double induction argu-

ment for the general case. This argument follows Perelman’s except that we

carefully determine the necessary constants to build each step. Perelman’s

double induction argument is built from two lemmas each of which depends

on a parameter k ∈ N. The Main Lemma(k)[Lemma 3.1.1] says that given a

constant c > 1 and an appropriate estimate on volume growth, any given con-

tinuous function f : Sk → Bp(R) can be extended to a continuous function

g : Dk+1 → Bp(cR) . This lemma is proven by defining intermediate functions

gj on finer and finer nets in Dk+1. To define gj on these nets one uses the

Moving In Lemma, described below. To prove the limit g(x) = limj→∞ gj(x)

exists and is continuous, we apply results from Chapter 2.

The Moving In Lemma(k) [Lemma 3.1.2] states that given a constant

d0 > 0 and a map φ : Sk → Bq(ρ) then with an appropriate bound on volume

growth one can move φ inward obtaining a new map φ : Sk → Bq((1− d0)ρ).

The new map φ is uniformly close to the map φ with respect to the radius

ρ. The maps φ and φ are not necessarily homotopic; however, a homotopy is

constructed by controlling precisely the uniform closeness of these maps on

smaller and smaller scales. The Moving In Lemma(k) and Main Lemma(i),

for i = 0, .., k−1, are used to produce finer and finer nets that then converge

on the homotopy required for Main Lemma(k). Moving In Lemma(k) is

proven by constructing the map φ inductively on successive i-skeleta of a

triangulation of Sk. The conclusion of Main Lemma(i), for i = 0, .., k − 1, is

4

needed in the induction step of the proof of Moving In Lemma(k).

The key place in the argument where the volume growth bound is intro-

duced occurs in the proof of the Moving In Lemma; specifically, in producing

a small, thin triangle in an advantageous location. However, due to the

double inductive argument, and the fact that lower dimensional lemmas are

applied on a variety of scales where the choice of c and d0 depend on n and

k, the actual estimate on the volume is produced using inductively defined

functions β(k, c, n) [Definition 3.0.4] and constants Ck,n [Definition 3.0.2].

In Chapter 4, we generalize our results to metric measure spaces satisfying

similar bounds on volume growth and which can be realized as the pointed

Gromov-Hausdorff limits of a sequence {(Mni , pi)} of complete Riemannian

manifolds with nonnegative Ricci curvature. By first generalizing the state-

ments of Lemma 1.1.3 and Theorem 1.1.1 to this setting, it is possible to also

extend the Moving In Lemma of Section 3.1 and ultimately our main result,

Theorem 1.0.3.

In the Appendix I, we complete our analysis of β(k, c, n) to find the

optimal bounds, α(k, n), over all constants c > 1. Through this analysis we

are able to construct a table of values containing the optimal lower bounds

for the volume growth, as stated in Theorem 1.0.3, which guarantee the k-th

homotopy group is trivial. The bounds that we obtain are the best that

can be achieved via Perelman’s method. A portion of this analysis was done

using Mathematica 6. The code for these commands is provided in Appendix

II.

5

1.1 Background

Here we review two facts from the Riemannian geometry of manifolds with

non-negative Ricci curvature. Let Mn be a complete Riemannian manifold

with Ric ≥ 0.

Theorem 1.1.1. [Abresch-Gromoll Excess Theorem]. Let p, q ∈ Mn and

let γ be a minimal geodesic connecting p and q. For any x ∈ Mn, we define

the excess function with respect to p and q as

ep,q(x) = d(p, x) + d(q, x)− d(p, q).

Define h(x) = d(x, γ) and set s(x) = min {d(p, x), d(q, x)}. If h(x) ≤ s(x)/2,

then

ep,q(x) ≤ 8

(h(x)n

s(x)

)1/n−1

= 8

(h(x)

s(x)

)1/n−1

h(x).

This excess estimate is due to Abresch-Gromoll [1] (c.f. [5]).

Definition 1.1.2. For constants c > 1, ε > 0 and n ∈ N, define

γ(c, ε, n) =[1 +

(cε

)n]−1

.

Lemma 1.1.3. [Perelman’s Maximal Volume Lemma]. Let p ∈Mn, R > 0,

for any constants c1 > 1 and ε > 0, if αM > 1 − γ(c1, ε, n), then for every

a ∈ Bp(R), there exist q ∈ Mn \ Bp(c1R) such that d(a, pq) ≤ εR, where pq

denotes a minimal geodesic connecting p and q.

This fact was observed by Perelman in [16]. The proof follows from the

proof of the Bishop-Gromov Volume Comparison Theorem and can also be

found in [20]. Our proof differs in that we utilize a global volume growth

control that Perelman does not need.

6

Proof. Let c2 > c1 > 1 be finite constants. Define Γ ≡ {σ| d(a, σ) ≤ εR} ⊂

Sn−1p (Mn) ⊂ TpM

n. Suppose that for all v ∈ Γ, we have cut(v) < c1R.

In what follows, we determine an upper bound on the volume growth, αM ,

which would allow such a contradiction to occur. In turn, by requesting

the volume growth be bounded below by this upper bound, the lemma will

follow.

By definition, we have

Vol(Bp(c2R)) =

∫Γ

∫ min{cut(v),c2R}

0

AMn(t, v)dtdv

+

∫Sn−1\Γ

∫ min{cut(v),c2R}

0

AMn(t, v)dtdv

≤ Vol(Γ)

∫ c2R

0

A0(t)dt+ Vol(Sn−1 \ Γ)

∫ c2R

0

A0(t)dt

= Vol(Sn−1)

∫ c2R

0

A0(t)dt− Vol(Γ)

((∫ c2R

0

−∫ c1R

0

)A0(t)dt

)= −Vol(Γ)

∫ c2R

c1R

A0(t)dt+ Vol(Sn−1)

∫ c2R

0

A0(t)dt

= −Vol(Γ)

∫ c2R

c1R

A0(t)dt+ Vol(B0(c2R)).

Here AMn(t, v) denotes the volume element on Mn and A0(t) denotes the

volume element on Rn; that is, A0(t) = tn−1. From the assumption on

the volume growth, we have that Vol(Bp(c2R)) ≥ (1 − γ)Vol(B0(c2R)) and

therefore

(1− γ)Vol(B0(c2R)) ≤ −Vol(Γ)

∫ c2R

c1R

A0(t) + Vol(B0(c2R)) (1.5)

γVol(B0(c2R)) ≥ Vol(Γ)

∫ c2R

c1R

A0(t)dt (1.6)

Vol(Γ) ≤ γVol(B0(c2R))∫ c2Rc1R

A0(t)dt. (1.7)

7

On the other hand, sinceBa(εR) ⊂ AnnΓ(p; 0, c1R), it follows that Vol(Ba(εR)) ≤

Vol(Γ)∫ c1R

0A0(t)dt. Hence

Vol(Ba(εR)) ≤ γVol(B0(c2R))

∫ c1R0

A0(t)dt∫ c2Rc1

A0(t)dt. (1.8)

Furthermore, since Bp(c2R) ⊂ Ba(R + c2R), we know that

Vol(Bp(c2R))

Vol(Ba(εR)≤ Vol(Ba(R + c2R))

Vol(Ba(εR))≤ (R + c2R)n

(εR)n;

and therefore,

Vol(Ba(εR)) ≥ Vol(Ba(R + c2R))(εR)n

(R + c2R)n(1.9)

≥ Vol(Bp(c2R))εn

(1 + c)n(1.10)

≥ (1− γ)Vol(B0(c2R))εn

(1 + c)n. (1.11)

Combining 1.8 and 1.11, we get

(1− γ)Vol(B0(c2R))εn

(1 + c2)n≤ γVol(B0(c2R))

∫ c1R0

A0(t)dt∫ c2Rc1R

A0(t)dt(1.12)(

ε

1 + c2

)n− γ

(ε

1 + c2

)n≤ γ

cn1cn2 − cn1

(1.13)(ε

1 + c2

)n≤ γ

[cn1

cn2 − cn1+

(ε

1 + c2

)n]. (1.14)

By solving 1.14 for γ, we can deduce a lower bound for γ dependent only

on the constants c2, c1, ε and n. That is,

γ ≥(

ε

1 + c2

)n [cn1

cn2 − cn1+

(ε

1 + c2

)n]−1

(1.15)

=

[1 +

cn1cn2 − cn1

(1 + c2ε

)n]−1

. (1.16)

8

Note that, throughout the proof we required a restriction on the volume

growth only within the larger ball Bp(c2R). Since αM is a global restriction

on volume growth, it is possible to take c2 → ∞ and thus refine the lower

bound on γ determined above. Since

limc2→∞

[1 +

cn1cn2 − cn1

(1 + c2ε

)n]−1

=

[1 +

cn1εn

]−1

,

the above lower bound on γ can be expressed more simply as

γ ≥[1 +

cn1εn

]−1

.

Recall that this lower bound on γ provides the upper bound on αM = 1− γ

which leads to the contradiction of the Lemma. Thus, by requiring αM >

1 −[1 +

cn1εn

]−1

, as originally prescribed in the assumption, we have proven

the Lemma.

Remark. Perelman’s Maximal Volume Lemma proves the existence of a

geodesic in Mn of length at least c1R > 1 that are within a fixed distance of

a given point. Consider, for example, the case when Mn = Rn. Given a point

a ∈ Rn, it is possible to find a geodesic of any length (in fact, there exists a

ray) that is arbitrarily close to a. Indeed, letting c1 → ∞ in the expression

for αM , while keeping ε and n fixed, we find that αM → 1. Similary, letting

ε → 0 (with c1, n fixed), forces αM → 1 as well. Recall that by the Bishop-

Gromov Volume Comparison Theorem, αM = 1 implies Mn is isometric to

Rn.

Remark. Allowing the dimension of Mn to increase while keeping ε and

c1 constant also pushes the lower bound on αM closer to 1.

9

Chapter 2

Almost Equicontinuity and the

Construction of Homotopies

In this chapter, we prove a general method of constructing homotopies from

sequences of increasingly refined nets. We begin by reviewing a definition

and theorem from [17].

2.1 Background and Definitions

Definition 2.1.1. [[17], Definition 2.5] A sequence of functions between com-

pact metric spaces fi : Xi → Yi, is said to be almost equicontinuous if there

exists εi decreasing to 0 such that for all ε > 0 there exists δε > 0 such that

dYi(fi(x1), fi(x2)) < ε+ εi, whenever dXi

(x1, x2) < δ. (2.1)

10

Theorem 2.1.2. [[17], Theorem 2.3] If fi : Xi → Yi is almost equicontinuous

between complete length spaces (Xi, xi) → (X, x) and (Yi, yi) → (Y, y) which

converge in the Gromov-Hausdorff sense where X and Y are compact, then

a subsequence of the fi converge to a continuous limit function f : X → Y .

Let X be a complete length space. Let Kj be a sequence of finite cell

decompositions of X, that is X =∐

Xi∈KjXi. Each Kj+1 is a refinement of

Kj.

Definition 2.1.3. For any set A ⊂ X, define

hullj(A) =∐

Xi∈Kj ,Xi∩A6=∅

Xi

Definition 2.1.4. The diameter of the finite cell decomposition Kj, com-

posed of cells Xi, is defined by

diam(Kj) = supXi∈Kj

diam(Xi).

Definition 2.1.5. Let K be a finite cell decomposition of a complete length

space X. A map ψK : X → X which takes all the points in a cell to a point

p in that cell is called a discrete decomposition map of K.

Lemma 2.1.6. Let Kj be a sequence of finite cell decompositions of X and

{ψKj} a sequence of discrete decomposition maps of Kj. This sequence of

maps is almost equicontinuous provided max{diam(σ)|σ ∈ Kj} → 0 as j →

∞.

Proof. For each j, let dj = max{diam(σ)|σ ∈ Kj}. Pick ε > 0 and suppose

x, y ∈ X such that d(x, y) < ε/2. Choose k ∈ N so large that dj < ε/4 for all

j > k. Then, for z ∈ X and d(x, z) = d(y, z),

11

d(ψj(x), ψj(y)) ≤ diam(ψj(Bz(ε/4))

≤ diam(hullj(Bz(ε/4)))

= d(p, q), for some p, q ∈ hullj(Bz(ε/4))

≤ d(p, x) + d(x, y) + d(y, q)

≤ d(p,Bz(ε/4)) + d(x, y) + d(q, Bz(ε/4))

≤ dj + d(x, y) + dj

≤ ε/4 + ε/2 + ε/4, provided j > k

= ε.

Thus, the sequence {ψj} is uniformly almost equicontinuous.

Lemma 2.1.7. The composition of two almost equicontinuous sequences of

maps is again almost equicontinuous; i.e. if {fj} and {gj} are two sequences

of maps which are almost equicontinuous. Then {fj ◦ gj} is also almost

equicontinuous.

Proof. Suppose {fj} and {gj} are two almost equicontinuous sequences of

maps. Since {fj} is almost equicontinous, given ε > 0, there exists δfε >

0 and positive integer Kf such that d(fj(x), fj(y)) ≤ ε for all j > Kf ,

provided d(x, y) < δfε . Choose δf◦gε = δfδgε

and choose a positive integer

K = max{Kf , Kg}, where δgε and Kg are chosen so that when d(a, b) < δgε ,

we have d(gj(a), gj(b)) < δfε , for all j > Kg.

Therefore, if d(a, b) < δfδgε, then d(gj(a), gj(b)) < δfε , for all j > K ≥ Kg

and thus, d(fj(gj(a)), fj(gj(b))) < ε, for all j > K ≥ Kf . Therefore, the

sequence {fj ◦ gj} is almost equicontinuous.

12

Definition 2.1.8. The i-skeleton of a k-dimensional cell decomposition K,

denoted skeli(K) for i = 0, 1, .., k, is defined as the collection of all i-cells

contained in K.

Note that if X = Dk+1 then Sk ⊂ Dk+1 is contained in skelk(K) for any

cell decomposition K of Dk+1.

2.2 Homotopy Construction Theorem

The following theorem is crucial in constructing the homotopies in the man-

ifold setting.

Theorem 2.2.1. (Homotopy Construction Theorem). Let Y be a com-

plete, locally compact metric space, p ∈ Y , R > 0 and f : Sk → Bp(R) ⊂ Y a

continuous map. Given constants c > 1, ω ∈ (0, 1), and a sequence of finite

cell decompositions Kj of Dk+1 with maps fj : skelk(Kj) → Y satisfying the

following three properties

(A) Kj+1 is a subdivision of Kj and fj+1 ≡ fj on Kj and max{diam(σ)|σ ∈

Kj} → 0,

(B) For each (k + 1)-cell, σ ∈ Kj, there exists a point pσ ∈ Bp(cR) ⊂ Y and

a constant Rσ > 0 such that

fj(∂σ) ⊂ Bpσ(Rσ);

and, if σ′ ⊂ σ, where σ′ ∈ Kj+1, σ ∈ Kj, then

Bpσ′(cRσ′) ⊂ Bpσ(cRσ), and Rσ′ ≤ ωRσ, for ω ∈ (0, 1).

13

(C) skelk(K0) = Sk = ∂Dk+1, pσ0 = p, and Rσ0 = R,

then the map f can be continuously extended to a map g : Dk+1 → Bp(cR) ⊂

Y .

Proof. Suppose we have such a sequence of finite cell decompositions Kj of

Dk+1 and continuous maps fj : skelk(Kj) → M satisfying (A), (B), and (C)

above. For any x ∈ Dk+1, choose a sequence of (k + 1)-cells σj ∈ Kj, such

that σj+1 ⊂ σj and x ∈ clos(σj) for all j. Therefore, each point x ∈ Dk+1

determines a sequence of (k + 1)-cells ’converging to’ x. Each of these cells

determines a point, pσj, and a radius, Rσj

> 0, which we assume satisfy the

properties outlined in (A), (B), and (C) above.

As in Perelman’s homotopy construction [16], define g by g(x) = limj→∞ pσj.

If x ∈ skelk(Kj) for some j, set g(x) = fj(x).

For any j and k > 0, note that by (B),

d(pσj, pσj+k

) ≤ cRσj− cRσj+k

≤ cRσj

≤ cωjR.

Since ω ∈ (0, 1), the sequence {pσj} is a Cauchy sequence and thus converges.

Hence, g(x) is well-defined.

Note that ∂Dk+1 = Sk = skelk(K0) and so by the definition of g, for any

x ∈ ∂Dk+1, g(x) = f0(x) = f(x). Thus, g|∂Dk+1 = f .

The continuity of g is not verified in [16]. Here we prove that g is con-

tinuous. Define a sequence of maps gj : Dk+1 → Y by gj(x) = pσjfor each

j.

Claim. The sequence of maps {gj} is uniformly almost equicontinuous.

Proof of Claim. Define a sequence of intermediate maps ψKj: Dk+1 →

14

Dk+1, where ψKjis a discrete decomposition map for Kj. Note that Im(ψKj

)

is a discrete metric space. Define gj : Im(ψKj) → X in such a way that

gj = gj|Im(ψKj).

By (A) we have that max{diam(σ)|σ ∈ Kj} → 0 as j → ∞. Therefore,

the sequence of decomposition maps ψKjis almost equicontinuous by Lemma

2.1.6.

The maps gj are discrete and thus the sequence {gj} is almost equicon-

tinuous.

Since gj = gj ◦ ψKj, by Lemma 2.1.7, the sequence of maps {gj} is also

uniformly almost equicontinuous. This completes the proof of the Claim.

Finally, by Theorem 2.1.2 (see [17] for proof), the limiting map g is con-

tinuous. This completes the proof of Proposition 2.2.1.

15

Chapter 3

Double Induction Argument

In this chapter we use Perelman’s double induction argument outlined in

Chapter 1 to prove Theorem 1.0.3. We introduce a collection of constants

which are defined inductively. We define them here as they are necessary for

the induction statements.

Definition 3.0.2. For k, n ∈ N and i = 0, 1, .., k, define constants Ck,n(i)

iteratively as follows:

Ck,n(i) = (16k)n−1 (1 + 10Ck,n(i− 1)n + 3 + 10Ck,n(i− 1)) , i ≥ 1 (3.1)

and Ck,n(0) = 1. We denote Ck,n = Ck,n(k).

Definition 3.0.3. Define a function

hk,n(x) =

[1− 10k+2Ck,nx

(1 +

x

2k

)k]−1

. (3.2)

This function hk,n has a vertical asymptote at x = δk,n for some small

value δk,n > 0, where 10k+2Ck,nδk,n

(1 +

δk,n

2k

)k= 1. Note that hk,n : (0, δk,n) →

16

(1,∞) is a smooth, one-to-one, onto, increasing function. Thus h−1k,n : (1,∞) →

(0, δk,n) is well-defined.

Toward proving Theorem 1.0.3, we need to build the homotopy as de-

scribed earlier. This requires control on the volume growth of Mn. We now

define the expression β(k, c, n) which we will use to control the volume growth

of Mn.

Definition 3.0.4. For constants, c > 1 and k, n ∈ N, the value of β(k, c, n)

represents a minimum volume growth necessary to guarantee that any con-

tinuous map f : Sk → Bp(R) has a continuous extension g : Dk+1 → Bp(cR).

Define

β(k, c, n) = max{ 1− γ(c, h−1k,n(c), n); (3.3)

β(j, 1 +h−1k,n(c)

2k, n), j = 1, .., k − 1}, (3.4)

where β(0, c, n) = 0 for any c and β(1, c, n) = 1 − γ(c, h−11,n(c), n). Recall

that γ(c, d, n) = [1 + cn

dn ]−1 [Definition 1.1.2] was used in proving Perelman’s

Maximal Volume Lemma [Lemma 1.1.3].

3.1 Key Lemmas

In this section we state the Main Lemma and the Moving In Lemma. These

are similar to the lemmas used in Perelman’s paper [16] except that we are

controlling the constants carefully so as to determine their best values.

Lemma 3.1.1. (Main Lemma(k)). Let Mn be a complete Riemannian

manifold with Ric ≥ 0 and let p ∈ Mn and R > 0. For any constant c > 1

17

and k, n ∈ N, if

αM ≥ β(k, c, n), (3.5)

then any continuous map f : Sk → Bp(R) can be continuously extended to a

map g : Dk+1 → Bp(cR).

Lemma 3.1.2. [Moving In Lemma(k)]. Let Mn be a Riemannian mani-

fold with Ric ≥ 0. For any constant d0 ∈ (0, δk,n) and k, n ∈ N if

αM ≥ β(k, hk,n(d0), n), (3.6)

then given q ∈ Mn, ρ > 0, a continuous map φ : Sk → Bq(ρ) and a triangu-

lation T k of Sk such that diam(φ(∆k)) ≤ d0ρ for all ∆k ∈ T k, there exists a

continuous map φ : Sk → Bq((1− d0)ρ) such that

diam(φ(∆k) ∪ φ(∆k)) ≤ 10−k−1

(1 +

d0

2k

)−k(1− hk,n(d0)

−1)ρ. (3.7)

In the next two sections we prove these lemmas. In Section 3.2 we prove

Main Lemma(k) assuming Moving In Lemma(k) and Main Lemma(j) for

j = 1, .., k− 1. In Section 3.3 we prove Moving In Lemma(k) assuming Main

Lemma(i), i = 0, .., k − 1. In Section 3.4 we apply these lemmas to prove

Theorem 1.0.3. Before proceeding we prove Main Lemma (0).

Lemma 3.1.3. [Main Lemma(0)]. Let X be a complete length space and

let p ∈ X, R > 0. For any constant c > 1, any continuous map f : S0 →

Bp(R) ⊂ X can be continuously extended to a map g : D1 → Bp(cR) ⊂ X.

Proof. The image f(S0) consists of two points, p1, p2 ∈ X. Since X is a

complete length space, it is possible to find length minimizing geodesics σi

connecting pi to p, for i = 1, 2. Define g so that Im(g) = σ1 ∪ σ2 and

18

g(−1) = p1 and g(1) = p2. Thus, g is a continuous extension of the map f

and by construction Im(g) ⊂ Bp(cR) ⊂ X.

3.2 Proof of Main Lemma(k)

Proof. The proof is by induction on k. When k = 0, the result follows from

Lemma 3.1.3. No assumption on volume growth is necessary. Assume now

that Main Lemma(i) holds for i = 1, .., k − 1: Given any constants ci > 1,

a continuous map f : Si → Bp(R) has a continuous extension to a map

g : Di+1 → Bp(ciR) provided αM ≥ β(i, ci, n). We will now show that the

result is true for dimension k.

Let f : Sk → Bp(R) ⊂ Mn be a continuous map. Choose c > 1 and

suppose αM ≥ β(k, c, n). Our goal now is to show that the map f : Sk →

Bp(R) has a continuous extension. To do this we will show that there exits a

sequence of finite cell decompositions, Kj, of Dk+1 and maps fj that satisfy

the hypothesis of the Homotopy Construction Theorem [Theorem 2.2.1] and

thus create the homotopy g : Dk+1 → Bp(cR).

For j = 0, define K0 to be the cell decomposition consisting of a single

cell (i.e. K0∼= Dk+1) so skelk(K0) = Sk. Recall that we use the notation

skelk(Kj) to denote the union of the boundaries of the cell decomposition of

Kj [Definition 2.1.8].

As in [16], inductively define Kj+1 given Kj in the following way. For

a (k + 1)-cell, σ ∈ Kj, note that σ is homeomorphic to a disk so it can be

viewed in polar coordinates as(Sk × (0, 1]

)∪ {0}. Let T kσ be a triangulation

of Sk, where Sk ∼= ∂σ and diamσ(∆k) < 1/k for all ∆k ∈ T kσ . Define Kj+1 so

19

that

σ∩ skelk(Kj+1) = (Sk×{1})∪ (Sk×{1/2})∪(skelk−1(T

kσ )× [1/2, 1]

). (3.8)

This inductive construction of theKj provides us with a sequence of finite cell

decompositions of Dk+1. Note that with an appropriate selection of Sk×{1/2}

this sequence of decompositions satisfies Condition A on cell decompositions

as required by the Homotopy Construction Theorem [Theorem 2.2.1] because

max{diam(σ)|σ ∈ Kj} → 0.

Next, we define the continuous maps fj : skelk(Kj) → Mn. Begin by

setting f0 ≡ f . In this way, f0 : skelk(K0) → Bp(R) ⊂ Mn and the initial-

izing hypothesis (C) of Theorem 2.2.1 is satisfied. We verify the rest of the

hypothesis inductively.

Suppose fj satisfies hypotheses (A) and (B) of Theorem 2.2.1. It remains

to define fj+1 and check that hypotheses (A) and (B) hold for this fj+1. We

describe the process to define fj+1 on the refinement of a single (k + 1)-cell

σ ∈ Kj. To define fj+1 on all of skelk(Kj+1), repeat this process on each

(k + 1)-cell of Kj.

Given a (k + 1)-cell σ ∈ Kj, by hypothesis (B), there exists a point pσ ∈

Bp(cR) ⊂Mn and a constant Rσ > 0 such that fj(∂σ) ⊂ Bpσ(Rσ). As before,

view σ as(Sk × (0, 1]

)∪ {0}, and think of fj as a map fj : Sk → Bpσ(Rσ).

Define fj+1 : skelk(Kj+1) →Mn in three stages.

First we set

fj+1 ≡ fj on Sk × {1}, (3.9)

which is all that is required to satisfy hypothesis (A).

We claim that we can apply the Moving In Lemma(k) to the map fj. Set

d0 = h−1k,n(c) and keep k, n as before. The volume growth assumption (3.6) is

20

satisfied since αM ≥ β(k, c, n) = β(k, hk,n(d0), n). Take q = pσ, ρ = Rσ, and

φ = fj and take a sufficiently fine triangulation, T kσ , of Sk ∼= ∂σ such that

diam(fj(∆k)) ≤ d0Rσ for all ∆k ∈ T kσ . Applying the Moving In Lemma(k)

[Lemma 3.1.2], we obtain a map fj : Sk → Bpσ((1− d0)Rσ). We set

fj+1 ≡ fj on Sk × {1/2}. (3.10)

This completes the second stage of our construction of fj+1. Furthermore,

by (3.7),

diam(fj(∆k) ∪ fj(∆k)) ≤ 10−k−1

(1 +

d0

2k

)−k(1− (hk,n(d0))

−1)Rσ (3.11)

= 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ, (3.12)

for all ∆k ∈ T kσ .

For the third stage and to complete the definition of fj+1 on σ∩skelk(Kj+1),

it remains to define fj+1 on skeli(Tkσ ) × [1/2, 1] for i = 0, 1, .., k − 1. Below

we describe this procedure (inductively) for a single k-simplex ∆k of the tri-

angulation T kσ . Here we use the induction hypothesis and assume the Main

Lemma(j) is true for j = 1, .., k − 1. First, we apply Lemma 3.1.3 to the 0-

skeleton [note that Lemma 3.1.3 is an analog of Main Lemma(0)]. Then, we

apply Main Lemma [Lemma 3.1.1] repeatedly starting with the 1-dimension

skeleton and continuing to the (k − 1)-dimension skeleton.

Let ∆0 ∈ T kσ be a 0-simplex. Consider the map fj+1,0 on S0 defined by

fj+1,0(−1) = fj+1(∆0 × {1}) and fj+1,0(1) = fj+1(∆

0 × {1/2}). On these

components, the map fj+1,0 is obtained from (3.9) and (3.10). We want to

21

define fj+1 on ∆0 × [1/2, 1]. Note that,

diam(Im(fj+1,0)) = d(fj+1,0(−1), fj+1,0(1)) (3.13)

= diam(fj+1(∆0 × {1}) ∪ fj+1(∆

0 × {1/2})) (3.14)

= diam(fj(∆0) ∪ fj(∆0)) (3.15)

≤ diam(fj(∆k) ∪ fj(∆k)) (3.16)

≤ 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ. (3.17)

In this last line we have applied (3.12).

If we set

Rj+1,0 = 1/2 · 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ, (3.18)

then, by our estimate on the diameter of its image, we have

fj+1,0 : S0 → Bpj+1,0(Rj+1,0), (3.19)

for some point pj+1,0 ∈ Mn. We now apply Main Lemma(0) [Lemma 3.1.3]

taking c = 1 + d0/2k, p = pj+1,0, R = Rj+1,0 and f = fj+1,0. Clearly,

the hypotheses of Main Lemma(0) are satisfied since β(0, c, n) = 0 and Mn

is a complete Riemannian manifold. Therefore, there exists a continuous

extension

gj+1,1 : D1 → Bpj+1,0

((1 +

d0

2k

)Rj+1,0

)(3.20)

22

and we use it to define fj+1 on skel0(Tkσ )× [1/2, 1]. Furthermore,

diam(fj+1(∆0 × [1/2, 1])) = diam(Im(gj+1,1)) (3.21)

≤ 2 ·(

1 +d0

2k

)Rj+1,0 (3.22)

≤ 2 ·(

1 +d0

2k

)· 1/2 · (3.23)(

10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ

)(3.24)

≤ 10−k−1

(1 +

d0

2k

)−k+1

(1− c−1)Rσ. (3.25)

We will use induction on i to define fj+1 on ∆i × [1/2, 1], for 0 ≤ i < k.

Assume we have defined fj+1 = fj on all simplices ∆i ∈ T kσ and we have

defined fj+1 on all possible ∆i−1 × [1/2, 1] so that

diam(fj+1(∆i−1 × [1/2, 1])) ≤ 10i−1−k

(1 +

d0

2k

)i−k(1− c−1)Rσ. (3.26)

Note that this holds for i = 1 by (3.25). Also, note that (3.12) implies

diam(fj+1(∆i × {1}) ∪ fj+1(∆

i × {1/2})) (3.27)

= diam(fj(∆i) ∪ fj(∆i)) (3.28)

≤ diam(fj(∆k) ∪ fj(∆k)) (3.29)

≤ 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ.(3.30)

We now build a new map fj+1,i+1 on ∆i × [1/2, 1]. View

(∆i × {1}) ∪ (∆i × {1/2}) ∪ (∂∆i × [1/2, 1]) as Si. Since ∂∆i × [1/2, 1] is a

collection of ∆i−1 × [1/2, 1], we have a map

fj+1,i : Si → Bpj+1,i(Rj+1,i), (3.31)

23

for some point pj+1,i ∈Mn and where by (3.26) and (3.30) we have

2Rj+1,i = diam(fj+1|∆i×{1} ∪ fj+1|∆i×{1/2}) + (3.32)

diam(fj+1(∂∆i × [1/2, 1])) (3.33)

≤ diam(fj(∆i) ∪ fj(∆i)) + (3.34)

diam(fj+1(∆i−1 × [1/2, 1])) (3.35)

≤ 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ + (3.36)

10i−1−k(

1 +d0

2k

)i−k(1− c−1)Rσ (3.37)

≤ 10i−k(

1 +d0

2k

)−k+i(1− c−1)Rσ. (3.38)

Therefore,

diam(Im(fj+1,i)) ≤ 10i−k(

1 +d0

2k

)−k+i(1− c−1)Rσ. (3.39)

Apply Main Lemma(i) taking c = 1+d0/2k and k, n as before. This is allowed

because the volume growth requirement for Main Lemma(i) is satisfied by

(3.4) and because the volume growth satifies

αM ≥ β(k, c, n) (3.40)

≥ β(i, 1 +h−1k,n(c)

2k, n) (3.41)

= β(i, 1 +d0

2k, n). (3.42)

Therefore, there exists a continuous extension

gj+1,i+1 : Di+1 → Bpj+1,i((1 + d0/2k)Rj+1,i) (3.43)

24

of the continuous map fj+1,i. This extension defines fj+1 on skeli(Tkσ )×[1/2, 1]

and we have the bound

diam(fj+1(∆i × [1/2, 1])) = diam(Im(gj+1,i+1)) (3.44)

≤ 2 ·(

1 +d0

2k

)·Rj+1,i (3.45)

= 2 ·(

1 +d0

2k

)· 1/2 · (3.46)

10i−k(

1 +d0

2k

)−k+i(1− c−1)Rσ(3.47)

= 10i−k(

1 +d0

2k

)−k+i+1

(1− c−1)Rσ. (3.48)

Furthermore, we have the bound

diam(fj+1(∆i × [1/2, 1])) ≤ 10i−k

(1 +

d0

2k

)i+1−k

(1− c−1)Rσ, (3.49)

for all ∆i ⊂ ∆k, i = 0, 1, .., k − 1, which implies our induction hypothesis on

i. Thus, we have defined fj+1 on skeli(Tkσ )× [1/2, 1] for each i = 0, 1, ..k− 1.

We now complete the proof by showing that the hypotheses (A) and (B)

of the Homotopy Construction Theorem [Theorem 2.2.1] hold for the function

fj+1.

Hypothesis (A) follows immediately from this construction since each

Kj+1 is a subdivision of the previous Kj and by definition fj+1 ≡ fj on Kj.

To check (B) holds, let σ′ ∈ Kj+1 and suppose σ′ ∼= ∆k× [1/2, 1] for some

25

∆k ∈ Sk. Notice that

diam(fj+1(∂σ′)) ≤ diam(fj+1|∆k×{1} ∪ fj+1|∆k×{1/2}) + (3.50)

diam(fj+1(∂∆k × [1/2, 1])) (3.51)

≤ diam(fj(∆k) ∪ fj(∆k)) + (3.52)

diam(fj+1(∆k−1 × [1/2, 1])) (3.53)

≤ 10−k−1

(1 +

d0

2k

)−k(1− c−1)Rσ + (3.54)

10−1(1− c−1)Rσ, (3.55)

where the last line follows from (3.12) and (3.49) with i = k − 1.

Set

Rσ′ = 1/2 · [10−k−1

(1 +

d0

2k

)−k(1− c−1) + 10−1(1− c−1)]Rσ. (3.56)

Then, by (3.55), there exists a point pσ′ ∈ Mn such that fj+1(∂σ′) ⊂

Bpσ′(Rσ′).

To verify Bpσ′(cRσ′) ⊂ Bpσ(cRσ), let x ∈ Bpσ′

(cRσ′) and notice that for

q ∈ f(∆k × {1/2}) ⊂ Bpσ′(Rσ′),

d(x, pσ) ≤ d(x, q) + d(q, pσ) (3.57)

≤ 2 · 1/2(1− c−1)cRσ + (1− d0)Rσ (3.58)

≤ (c− 1)Rσ + (1− d0)Rσ (3.59)

< cRσ. (3.60)

Therefore, Bpσ′(cRσ′) ⊂ Bpσ(cRσ).

Furthermore, sinceBpσ′(cRσ′) ⊂ Bpσ(cRσ) for all nested sequences σ′ ⊂ σ,

26

it follows that

d(pσ′ , p) ≤ d(pσ′ , pσ) + ...+ d(pσ− , p) (3.61)

≤ cRσ − cRσ′ + ...+ cR− cRσ− (3.62)

= cR− cRσ′ (3.63)

< cR. (3.64)

Thus, pσ′ ∈ Bp(cR) as required.

Lastly, we have Rσ′ ≤ ωRσ for

ω = 1/2 ·

[10−k−1

(1 +

d0

2k

)−k(1− c−1) + 10−1(1− c−1)

]. (3.65)

Note that ω ∈ (0, 1) because k ≥ 1 and d0 < 1.

Thus, we have constructed a sequence of maps fj : skelk(Kj) → Mn

satisfying the hypotheses of the Homotopy Construction Theorem [Theorem

2.2.1]. Therefore, the map f can be continuously extended to a map

g : Dk+1 → Bp(cR) ⊂Mn. This completes the proof of Main Lemma(k).

3.3 Proof of Moving In Lemma(k)

We now prove Moving In Lemma(k) assuming that Main Lemma(j) is true

for j = 0, .., k − 1.

Proof. Recall that αM ≥ β(k, hk,n(d0), n) and we are given q ∈ Mn, ρ > 0,

a continuous map φ : Sk → Bq(ρ) and a triangulation T k of Sk such that

diam(φ(∆k)) ≤ d0ρ for all ∆k ∈ T k. We must show that there exists a

continuous map φ : Sk → Bq((1− d0)ρ) such that

diam(φ(∆k) ∪ φ(∆k)) ≤ 10−k−1

(1 +

d0

2k

)−k(1− hk,n(d0)

−1)ρ. (3.66)

27

We will construct φ inductively on skeli(Tk) for i = 0, .., k in such a way

that φ(∆i)) ≡ φ(∆i) if φ(∆i) ⊂ Bq((1 − 2d0)ρ); and, if φ * Bq((1 − 2d0)ρ),

then

φ(∆i) ⊂ Bq((1− d0(2− i/k))ρ), (3.67)

diam(φ(∆i) ∪ φ(∆i)) ≤ 10diρ, (3.68)

for all ∆i ⊂ T k, i = 0, .., k. The constants di > 0 satisfy

d0 + 10di ≤ bi(di+1 − 3d0 − 10di) (3.69)

d0 + 10di ≤ bi(c− 1 + d0(2− i/k)) (3.70)

8b1

n−1

i (d0 + 10di) ≤ d0

2k(3.71)

10dk ≤ 10−k−1(1 + d0/2k)−k(1− hk,n(d0)

−1), (3.72)

for some constants bi ∈ (0, 1/2]. The existence of such constants di and bi

is proven in Lemma 5.1.1. Note that (3.68) and (3.72) together immediately

imply (3.66). Thus, we need only define φ so that the above conditions are

obeyed. To do so, we construct φ successively on the i-skeleta of Tk.

Begin with the case i = 0. Let ∆0 ∈ skel0(Tk) and assume φ(∆0) /∈

Bq((1−2d0)ρ), else we are done. Let σ∆0 denote a length minimizing geodesic

from φ(∆0) to q and define φ(∆0) = σ∆0((1 − 2d0)ρ). In this way, φ(∆0) ∈

Bq((1− 2d0)ρ) and (3.67) is satisfied for i = 0. Furthermore,

diam(φ(∆0) ∪ φ(∆0)) = d(φ(∆0), φ(∆0)) (3.73)

= d(q, φ(∆0))− d(q, φ(∆0)) (3.74)

≤ ρ− (1− 2d0)ρ = 2d0ρ ≤ 10d0ρ. (3.75)

Thus, (3.68) is also satisfied when i = 0.

28

Now assume that φ is defined on skeli(Tk) and that (3.67) and (3.68) for

0 ≤ i ≤ k − 1. We now construct φ on skeli+1(Tk). Let ∆i+1 ⊂ skeli+1(T

k).

As before, suppose φ(∆i+1) * Bq((1 − 2d0)ρ), else we are done by simply

setting φ(∆i+1) ≡ φ(∆i+1).

Next apply Perelman’s Maximal Volume Lemma [Lemma 1.1.3], taking

c1 = hk,n(d0), ε = d0, and p = q, R = ρ. Since, by our hypothesis,

αM ≥ β(k, hk,n(d0), n) (3.76)

= max

{1− γ(hk,n(d0), d0, n); β

(j, 1 +

d0

2k, n

), j = 1, .., k − 1

}(3.77)

≥ 1− γ(hk,n(d0), d0, n), (3.78)

there exists a point r∆ ∈Mn\Bq(hk,n(d0)ρ) such that d(φ(∆i+1), qr∆) ≤ d0ρ.

Recall, qr∆ denotes a minimal geodesic connecting q and r∆. Let σ∆ be a

length minimizing geodesic from q to r∆ and define a point q∆ = σ∆((1 −

di+1)ρ). For any x ∈ ∂∆i+1, the triangle with vertices φ(x), q∆, and r∆ is

small and thin. To verify this, we use the the induction hypothesis that φ

has already been defined on skeli(Tk), 0 ≤ i ≤ k− 1, and that the properties

(3.67),(3.68) are satisfied in dimension i.

Note that,

d(φ(x), q∆r∆) ≤ d(φ(x), q∆r∆) + d(φ(x), φ(x)) (3.79)

≤ d0ρ+ diam(φ(∆i) ∪ φ(∆i)) (3.80)

≤ d0ρ+ 10diρ. (3.81)

29

And

d(φ(x), q∆) ≥ d(q∆, φ(x))− d(φ(x), φ(x)) (3.82)

≥ d(q, φ(x))− d(q∆, q)− diam(φ(∆i) ∪ φ(∆i)) (3.83)

≥ (1− 2d0)ρ− diam(φ(∆i+1)− (1− di+1)ρ− 10diρ (3.84)

≥ (1− 2d0)ρ− d0ρ− (1− di+1)ρ− 10diρ (3.85)

≥ (di+1 − 3d0 − 10di)ρ. (3.86)

And finally,

d(φ(x), r∆) ≥ d(r∆, q)− d(q, φ(x)) (3.87)

≥ Cρ− d(q, φ(∆i)) (3.88)

≥ cρ− (1− d0(2− i/k))ρ (3.89)

= (c− 1 + d0(2− i/k))ρ. (3.90)

The inequalties (3.69) and (3.70) guarantee that the triangle with vertices

φ(x), q∆, and r∆ is small and thin for some constants 0 < bi ≤ 1/2.

According to the excess estimate of Abresch-Gromoll [Theorem 1.1.1], we

have that for any x ∈ ∂∆i+1, with i = 0, 1, .., k − 1,

eq∆,r∆(φ(x)) = d(φ(x), q∆) + d(φ(x), r∆)− d(q∆, r∆) (3.91)

≤ 8

(d(φ(x), q∆r∆)

min{d(φ(x), q∆), d(φ(x), r∆)}

) 1n−1

d(φ(x), q∆r∆)(3.92)

≤ 8b1

n−1

i (d0 + 10di)ρ. (3.93)

Also, by the triangle inequality,

d(q, q∆) + d(q∆, r∆) = d(q, r∆) ≤ d(q, φ(x)) + d(φ(x), r∆). (3.94)

30

Adding (3.93) and (3.94), we get

d(φ(x), q∆) ≤ 8b1

n−1

i (d0 + 10di)ρ+ d(q∆, r∆)− d(φ(x), r∆) (3.95)

≤ 8b1

n−1

i (d0 + 10di)ρ+ d(q, φ(x)) + d(φ(x), r∆) (3.96)

−d(q, q∆)− d(φ(x), r∆) (3.97)

≤ 8b1

n−1

i (d0 + 10di)ρ+ (1− d0(2− i/k))ρ− (1− di+1ρ) (3.98)

=

(8b

1n−1

i (d0 + 10di) + di+1 − d0(2− i/k))

)ρ (3.99)

It then follows from (3.71) that, for all x ∈ ∂∆i+1,

d(φ(x), q∆) ≤ (d0

2k+di+1d0(2− i/k))ρ =

(di+1 − d0(2−

2i+ 1

2k)

)ρ. (3.100)

Now apply the Main Lemma [Lemma 3.1.1] in dimension i taking

p = q∆, (3.101)

R =

(di+1 − d0

(2− 2i+ 1

2k

))ρ, (3.102)

c = 1 + d0/2k; (3.103)

and letting f = φ. Since

αM ≥ β(k, hk,n(d0), n) (3.104)

= max

{1− γ(hk,n(d0), d0, n); β

(j, 1 +

d0

2k, n

), j = 1, .., k − 1

}(3.105)

≥ β(i, 1 +d0

2k, n) (3.106)

by our hypothesis, there exists a continuous extension of φ from ∂∆i+1 to

∆i+1. Furthermore,

d(φ(∆i+1), q∆) ≤ (1 + d0/2k)

(di+1 − d0(2−

2i+ 1

2k)

)(3.107)

≤(di+1 − d0(2−

i+ 1

k)

)ρ, (3.108)

31

provided di < 1, which is guaranteed by the fact that the di’s are increasing

in i and, by (3.72), dk < 1. Therefore, by the triangle inequality,

d(φ(∆i+1), q) ≤ d(φ(∆i+1), q∆) + d(q∆, q) (3.109)

≤(di+1 − d0(2−

i+ 1

k)

)ρ+ (1− di+1) ρ (3.110)

=

(1− d0(2−

i+ 1

k)

)ρ. (3.111)

Thus, (3.67) is satisfied for i + 1 for any choice of di, bi satisfying the

inequalities (3.69), (3.70) and (3.71).

Furthermore,

diam(φ(∆i+1 ∪ φ(∆i+1)) ≤ diam(φ(∂∆i+1) ∪ φ(∂∆i+1)) + (3.112)

diam(φ(∆i+1)) + diam(φ(∆i+1))(3.113)

≤ 10diρ+ d0ρ+ 2

(di+1 − d0(2−

i+ 1

k

)ρ(3.114)

=

(2di+1 + d0

(−3 +

2(i+ 1)

k

)+ 10di

)ρ(3.115)

≤ (2di+1 + 10di − d0)ρ. (3.116)

The inequality (3.69) and the fact that 0 < bi ≤ 1/2 imply that

diam(φ(∆i+1) ∪ φ(∆i+1)) ≤ 10di+1ρ. (3.117)

So, (3.68) are satisfied for dimension i+1. Therefore, φ has been defined

so that (3.67) and (3.68) are satisfied for i = 0, .., k. When i = k, (3.67)

implies

φ(∆k) ⊂ Bq((1− d0)ρ), ∀∆k ∈ T k. (3.118)

32

Thus, we have constructed the map φ : Sk → Bq((1−d0)ρ); and furthermore,

diam(φ(∆k) ∪ φ(∆k)) ≤ 10dkρ (3.119)

≤ 10−k−1

(1 +

d0

2k

)−k (1− hk,n(d0)

−1), (3.120)

where the last inequality follows from (3.68).

3.4 Proof of the Main Theorem

In this section we prove Theorem 1.0.3 using Main Lemma(k).

As a direct consequence of Main Lemma(k) [Lemma 3.1.1], we have

Proposition 3.4.1. Let Mn be a complete Riemannian manifold with Ric ≥

0. For k ∈ N, there exists a constant δk(n) > 0 such that if αM ≥ 1− δk(n),

then πk(Mn) = 0.

Proof. Choose some c > 1 and set δk(n) = 1 − β(k, c, n). The conclusion

then follows from Lemma 3.1.1.

Thus, we recover Perelman’s result [16]:

Lemma 3.4.2. [[16], Theorem 2]. Let Mn be a complete Riemannian man-

ifold with Ric ≥ 0. There exists a constant δn > 0 such that if αM ≥ 1− δn,

then Mn is contractible.

Proof. Choose some c > 1 and set δn = 1 − maxk=1,..,n β(k, c, n). Then

Lemma 3.1.1 implies πk(Mn) = 0 for all positive values k. Hence, Mn is

contractible by the Whitehead Theorem [18].

33

Remark. In the Appendix we use the expression for β(k, c, n) from

Definition 3.0.4 to find the ‘best’ value (depending only on k and n) of αM

which guarantees that πk(Mn) = 0. This is the lower bound for αM as stated

in Theorem 1.0.3.

We now prove Theorem 1.0.3.

Proof. Let

α(k, n) = infc∈(1,∞)

β(k, c, n).

By assumption, αM > α(k, n) and thus there exists c0 > 1 such that αM ≥

β(k, c0, n). The result follows by applying Main Lemma(k) with c = c0. In

the appendix we compute values of α(k, n).

34

Chapter 4

Generalizations to Metric

Measure Limits

In this chapter, we generalize the results of the previous chapters to metric

measure spaces, Y , which can be realized as the pointed Gromov-Hausdorff

limit of sequences, {(Mni , pi)}, of complete, connected Riemannian manifolds

all of whose Ricci curvatures are nonnegative, RicMi≥ 0. It is possible to

define a limiting measure on such limit spaces (as we will see) and by requiring

the volume growth of this measure to be maximal, one should expect to

retrieve the results of the previous sections. In other words, suppose we have

a sequence of complete, open pointed Riemannian manifolds {(Mni , pi)} each

with nonnegative Ricci curvature. By Gromov’s Precompactness theorem

[11], this collection of manifolds is pre-compact in the Gromov-Hausdorff

topology. Thus, it is possible to extract a subsequence of theMni ’s converging

in the pointed Gromov-Hausdorff sense to a complete length space (Y m, p∞).

Cheeger-Colding have shown [6] that in the so-called ‘non-collapsing’ case,

35

the additional lower bounds on volume allow one to define a (not necessarily

unique) measure on the limit space (Y, p∞) such that a further subsequence

on the Mni ’s converge in the metric measure sense to Y [Theorem 4.1.3]. By

requiring the volume growth of this limiting measure to be bounded below

by the constants obtained in Table 5.4, what can be said of the topology

of the limit space Y m? We find that versions of our two crucial lemmas

[Lemma 1.1.3 and Theorem 1.1.1] hold in this more general setting and thus

the results of Theorem [1.0.3] can be extended to such metric measure limits.

4.1 Background and Definitions

We begin by briefly discussing some notions of Gromov-Hausdorff distance

and convergence. Roughly speaking, the Gromov-Hausdorff distance defines

a metric on the class of isometry classes of compact metric spaces, where the

distance between isometric spaces is zero. More formally,

Definition 4.1.1. [[11], [3] Definition 7.3.10] Let X and Y be compact metric

spaces. The Gromov-Hausdorff distance between them is denoted dGH(X, Y )

and, for an r > 0, we say dGH(X, Y ) < r if and only if there exists a

metric space Z and subspaces X ′ and Y ′ of Z which are isometric to X and

Y respectively and such that dH(X ′, Y ′) < r. That is to say, the Gromov-

Hausdorff distance is the infimum of all r > 0 for which the above Z,X ′, and

Y ′ exist. Here dH denotes the Hausdorff distance between subsets of Z.

One says that a sequence of compact metric spaces {Xn}∞n=1 converges in

the Gromov-Hausdorff sense [11, 3] to a compact metric space X, denoted

XnGH−−→ X, if dGH(Xn, X) → 0. As previously mentioned, Gromov’s Pre-

36

compactness [11] theorem implies that every such sequence has a converging

subsequence whose limit is a locally compact length space. Thus, points in

the limit space can be connected by a path minimizing geodesic. However,

not every path minimizing geodesic in the limit space is realized as the limit

of geodesics along the sequence.

Definition 4.1.2. Let Y be a metric measure limit of a sequence of Rie-

mannian manifolds Mni . A geodesic path, σ, in Y is called a limit geodesic

if it can be realized as the limit of geodesics σi ∈ Mni contained within the

sequence of manifolds.

Note that every pair of points in Y has a limit geodesic of minimizing

length connecting them. For the purposes of the proof of the Moving In

Lemma, it is enough to prove our versions of Perelman’s Maximal Volume

Lemma and the Abresch-Gromoll excess estimate on small thin triangles

formed in the limit spaces from limit geodesics.

For noncompact, metric spaces, we use a slightly more general definition

of convergence. Namely, it is necessary to keep track of a sequence of points

pi ∈ Xi through the convergence. A pointed metric space is a pair (X, p)

consisting of a metric space X and a point p ∈ X. Further, a sequence of

noncompact pointed metric spaces (Xi, pi) converge in the pointed Gromov-

Hausdorff sense to (X, p) if for any r > 0, the compact metric spaces Bxi(r)

converge in the Gromov-Hausdorff sense to Bp(r).

This is equivalent to the following: for every r > 0 and η > 0, there exists

an N > 0 such that for all i > N , there exists a (not necessarily continuous)

map fi : Bpi(r) → X satisfying

1) fi(pi) = p∞;

37

2) supx1,x2∈Bpi (r)|d(fi(x1), fi(x2))− d(x1, x2)| < η;

3) Tε(fi(Bpi(r))) ⊃ Bp∞(r − η).

Often this type of pointed convergence is denoted (Xn, pn) −−→GH

(X, p).

Gromov-Hausdorff convergence defines a very weak topology. In fact, in

general, one only has that the limit of a sequence of length spaces is again a

length space.

In [6], the authors examine the structure of spaces Y , which can be

realized as the pointed Gromov-Hausdorff limits of sequences of complete,

connected Riemannian manifolds, {(Mni , pi)}, whose Ricci curvatures have

a definite lower bound. Among other things, they construct renormalized

limit measures, ν, on the limit space Y and show that such a measure satis-

fies an analog of the Bishop-Gromov Volume Comparison Theorem, namely

for y ∈ Y ,ν(By(r))

ωnrn↓;

(see also [11]). Note that it is possible for such limit spaces to be col-

lapsed. The sequence is said to be non-collapsed if there exists a lower

bound Vol(Bpi(1)) ≥ v > 0. Otherwise, the sequence is said to collapse; i.e.

limi→∞ Vol(Bpi(1)) = 0. For any sequence, collapsed or not, it is possible to

find a subsequence for which the renormalized limit measure exists. These

renormalized limit measures were also constructed in [10]. This limit mea-

sure is obtained by taking the limits of normalized Riemannian measures on

a suitable subsequence Mnj . In [6], Cheeger-Colding show that

Theorem 4.1.3. Given any sequence of pointed manifolds {(Mni , pi)}, for

which RicMi≥ 0 holds, there is a subsequence, {(Mn

j , pj)}, convergent to

some (Y m, y) in the pointed Gromov-Hausdorff sense, and a continuous func-

38

tion ν : Y m → R+ → R+, such that if qj ∈ Mnj , z ∈ Y m, and qj → z, then

for all R > 0,VolMj

(Bqj(R))

VolMj(Bqj(1))

→ ν(Bz(R)) (4.1)

Colding proved that when the sequence is noncollapsing, there is no need

to renormalize the volume or take a subsequence. In this case, the limit

measure is unique.

This function ν is precisely the renormalized limit measure of the space

Y m. In fact, this limit measure can be realized as a unique Radon measure

on Y m. Furthermore, for all z ∈ Y m and 0 < r1 ≤ r2, the renormalied limit

measure ν satisfies the following Bishop-Gromov type volume comparison

ν(Bz(r1))

ν(Bz(r2))≥ Vn,0(r1)

Vn,0(r2)=

(r1r2

)n(4.2)

With this notion of measure for the limiting space Y , we can generalize

the notion of volume growth to this class of metric measure limit spaces.

Definition 4.1.4. Let (Y, p) be the pointed Gromov-Hausdorff limit of a

sequence, {(Mni , pi)}, of complete, connected Riemannian manifolds all of

whose Ricci curvatures are nonnegative, RicMi≥ 0. Let ν denote the renor-

malized limit measure of (Y, p) as defined above. Set

αY := limr→∞

ν(Bp(r))

ωnrn.

When dealing with Riemannain manifolds, the Main Theorem [Theorem

1.0.3] is ultimately proven using both the Main Lemma [Lemma 3.1.2] and

the Moving In Lemma [Lemma 3.1.1]. However, taking a closer look at their

respective proofs, only the Moving-In Lemma requires the additional struc-

ture of a smooth manifold. That is to say, the consequences of the Main

39

Lemma would still hold true in an arbitrary complete length space equipped

with, for example, a ‘Moving In Property’ providing the content of the Mov-

ing In Lemma. Thus, in generalizing our results to metric measure limits, we

need only focus on extending the statement of the Moving In Lemma. Recall

from Section 3.3, the only ingredients of Riemannian geometry that were

used in proving the Moving In Lemma were Perelman’s Maximal Volume

Lemma [Lemma 1.1.3] and the Abresch-Gromoll excess estimate [Theorem

1.1.1]. In what follows, we prove analogs of these results for metric measure

limits.

4.2 Generalization of Perelman’s Maximal

Volume Lemma

Proposition 4.2.1. Let (Y, p∞) be the pointed metric measure limit of a

sequence of Riemannian manifolds {(Mni , pi)} with RicMn

i≥ 0 and assume

that αY > 1− γ(c1, ε, n). Then for any a∞ ∈ Bp∞(R), R > 0, there exists a

point q∞ ∈ M∞ \ Bp∞(c1R) such that dM∞(a∞, p∞q∞) ≤ εR, where p∞q∞ is

a minimizing limit geodesic in Y connecting p∞ and q∞.

Recall that the expression γ(c1, ε, n) is defined by Definition 1.1.2. Fur-

thermore, requiring that αY ≥ v > 0 guarantees that the sequence in question

is non-collapsing. Recall, this implies the measures do not need to be renor-

malized and the volumes of balls in Mni converge to balls of the same radius

in Y .

Proof. Let a∞ ∈ Bp∞ and choose δ > 0 such that Ba∞(δ) ⊂ Bp∞(R). By

40

property (3) above, choosing η < δ/2, we have, for i > Nη,

Tη(fi(Bpi(R))) ⊃ Bp∞(R− η). (4.3)

Since clearly, a∞ ∈ Bp∞(R−δ/2) ⊂ Bp∞(R−η), we have a∞ ∈ Tδ/2(fi(Bpi(R)))

for i sufficiently large. Letting η ↓ 0, we can construct a sequence of points

ai ∈ Bpi(R) and maps fi : Bpi

(r) → Y such that fi(ai) → a∞ ∈ Y . There-

fore, a∞, and in fact any point in Y , can be realized as the limit of a sequence

of points in Mni .

Ultimately, we would like to use Perelman’s Maximal Volume Lemma

on elements of the limiting sequence to show that the same result holds on

the limit space. However, it is possible that the manifolds in the sequence

{(Mni , pi)} are compact and converge in the metric measure sense to a non-

compact Y . With this in mind, it is necessary to appeal to a more general

form of Perelman’s Maximal Volume Lemma as proved in his original pa-

per [16]. With everything else remaining the same, the original statement

assumes only Vol(Bp(c2R)) ≥ (1 − γ)ωnrn, for some c2 > c1 > 1, rather

than a universal bound on the volume growth. The same proof holds with

neglecting the final step of allowing c2 to tend to infinity.

By Theorem 4.1.3, for i sufficiently large the volume of balls Bpi(r) ⊂

(Mni , pi) can be approximated by the volume of balls of the same radius in

the limit space (Y, p). That is to say, for any ε > 0, there exists anN > 0 such

that |ν(Bp(r))−VolMi(Bpi

(r))| < ε for all i > N . Since, αY > 1−γ(c1, ε, n))

and ν(Bp(r))

ωnrn is nonincreasing as a function of r, it is possible to approximate

the volume of balls in the manifolds Mni which are sufficiently close to Y .

41

Namely, for constants c2 > c1 > 1 and i sufficiently large,

VolMi(Bpi

(c2R)) > ν(Bpi(c2R))− ε (4.4)

> (1− γ(c1, ε, n))ωn(c2R)n − ε. (4.5)

Therefore, VolMi(Bpi

(c2R)) ≥ (1−γ(c1, ε, n))ωn(c2R)n and by Perelman’s

Maximal Volume Lemma, as originally stated in [16] and described above,

for each point ai ∈ Bpi(R) there exists a point qi ∈ Mi \ Bpi

(c1R) such

that dMi(ai, piqi) < εR. Here dMi

denotes the distance function on Mni and

recall ab denotes a minimal geodesic connecting a to b. In fact, since the

points qi lie on geodesics emanating from pi, it is possible to find points

qi ∈ Bpi(2R) \ Bpi

(R) satisfying di(ai, piqi) < εR. Again, by the properties

pointed convergence, for all η > 0 and i sufficiently large, there exists a map

fi : Bpi(R) → Y such that

dGH(Bfi(ai)(εR), Ba∞(εR)) < η.

By controlling the location of the balls Bfi(ai)(εR) in relation to the points

p∞ and a∞, it is possible to also control the location of the points fj(qj).

That is to say, for all j > i, the points {fj(qj)} lie a compact sector of

Bp∞(2R) \ Bp∞(R) and it is possible to extract a convergent subsequence

{qjk} such that fjk(qjk) → q∞ ∈ Bp∞(2R) \Bp∞(R) ⊂ Y \Bp∞(R).

The limit space Y is a complete length space; and thus, there exists a

minimum length geodesic connecting the points p∞ and q∞, denoted p∞q∞.

It remains only to show that this minimal geodesic path lies within εR of the

point a∞. In fact, it is possible to realize this geodesic path in Y as the limit

of geodesics piqi in Mni . Furthermore, since each of these paths lies within

42

εR of the respective points ai, and the points ai are ‘converging’ to the point

a∞, the limiting geodesic path (after passing to an appropriate subsequence)

must also lie with εR of a∞; that is, dY (a∞, p∞q∞) ≤ εR as required. This

completes the proof.

4.3 Generalization of the Excess Estimate

Next, we generalize the Abresch-Gromoll excess estimate of Section 1.1 to

metric measure limits of Riemannian manifolds with nonnegative Ricci cur-

vature. Note that in Section 4.2 we produced limit geodesic when prov-

ing Proposition 4.2.1. Furthermore, throughout the proof if the Moving In

Lemma [Lemma 3.1.2] and Main Lemma [Lemma 3.1.1] it is possible to use

limit geodesics between pairs of points. With this in mind, it is only neces-

sary to prove the excess theorem for small, thin triangles which are formed

from limit geodesics.

We can now prove the following generalization

Proposition 4.3.1. Let (Y, y) be the pointed metric measure limit of a se-

quence of Riemannian manifolds {(Mni ,mi)} with RicMn

i≥ 0; p∞, q∞ ∈ Y .

Define, for any x∞ ∈ Y ,

ep∞,q∞(x∞) = dY (p∞, x∞) + dY (q∞, x∞)− dY (p∞, q∞).

Set s(x∞) = min{dY (p∞, x∞), dY (q∞, x∞)} and h(x∞) = dY (x∞, p∞q∞),

where p∞q∞ denotes a limit geodesic in Y . If h(x∞) ≤ s(x∞)/2, then

ep∞,q∞(x∞) ≤ 8

(h(x∞)n

s(x∞)

)1/n−1

.

43

Proof. Let ε > 0 and choose 0 < η < ε/3. By property (2) of the definition

of pointed Gromov-Hausdorff convergence, there exists a constant Nη > 0

such that for all r > 0 and i > Nη, there is a map fi : Bmi(r) → Y such that

supx1,x2∈Bpi (r)

|d(fi(x1), fi(x2))− d(x1, x2)| < η.

This implies that, for any ε > 0,

|ep∞,q∞(x∞)− epi,qi(xi)| < 3η < ε, (4.6)

for i sufficiently large, i > Nη. Furthermore, each element of the sequence

(Mni , pi) has RicMi

≥ 0 and so by applying the Abresch-Gromoll excess esti-

mate, we find that ep∞,q∞(x∞) < epi,qi(xi) + ε ≤ 8(hn(xi)s(xi)

)1/n−1

+ ε.

Note that s(x∞) → s(xi) and, since we required the geodesic p∞q∞ be a

limit geodesic of Y , we also have (after passing to a subsequence if necessary)

h(xi) → h(x∞). Thus, for any ε′ > 0,

ep∞,q∞(x∞) < 8

(h(x∞)n

s(x∞)

)1/n−1

+ ε′. (4.7)

Thus, since ε′ > 0 was arbitrary, we have proven the theorem, namely,

ep∞,q∞(x∞) ≤ 8(h(x∞)n

s(x∞)

)1/n−1

.

With these two propositions, it is possible to prove the results of the

Moving In Lemma (Lemma 3.1.2) as stated for metric measure limits of

complete, open Riemannian manifolds with nonnegative Ricci curvature so

long as the volume growth of the limit is bounded below by β(c, k−1k,n(c), n),

where the expression β(., ., .) and the function hk,n(x) are as defined in Chap-

ter 3. In fact, when applying this generalized excess estimate [Proposition

44

4.3.1] to the limit spaces, the geodesics that are produced from the general-

ized version of Perlman’s Maximal Volume Lemma [Proposition 4.2.1] when

creating the small, thin triangles, are precisely those realized as the limit of

geodesics in the manifolds of the limiting sequence. Lastly, recall that Homo-

topy Construction Theorem holds for locally compact metric spaces. Thus,

the homotopies can be constructed in this more general setting and so the

proof can be reproduced verbatim substituting Proposition 4.2.1 for Lemma

1.1.3 and Proposition 4.3.1 for Theorem 1.1.1 as necessary.

Theorem 4.3.2. Let (Y, p) be the pointed metric measure limit of a sequence

{(Mni , pi)} of complete, open Riemannian manifolds satisfying RicMn

i≥ 0

and assume αY > α(k, n). Then πk(Y, p) = 0.

45

Chapter 5

Appendix I

The constants Ck,n explicitly determine the function hk,n(x) defined in Chap-

ter 3. In this appendix, we show that the constants Ck,n as defined are op-

timal and use the definition of hk,n to compute explicit values for α(k, n) as

stated in Theorem 1.0.3.

5.1 Optimal Constants

Recall Definition 3.0.2 of Ck,n(i):

Ck,n(i) = (16k)n−1(1 + 10Ck,n(i− 1))n + 3 + 10Ck,n(i− 1), i ≥ 1 (5.1)

and Ck,n(0) = 1. Denote Ck,n = Ck,n(k)

The constants Ck,n grow large very quickly. Preliminary values for Ck,n

where 1 ≤ k ≤ 3 and 1 ≤ n ≤ 8 are listed in Table 5.1.

Lemma 5.1.1. If di = Ck,n(i)d0 and bi = [16k(1 + 10Ck,n(i))]−(n−1), then

d0 + 10di = bi(di+1 − 3d0 − 10di), (5.2)

46

Table 5.1: Table of Ck,n values for 1 ≤ k ≤ 3, 1 ≤ n ≤ 10

k = 1 k = 2 k = 3

n = 1 24 - -

n = 2 384 1.89× 108 -

n = 3 6144 1.52× 1017 1.36× 1060

n = 4 98304 1.25× 1029 1.00× 10133

n = 5 1.57× 106 1.06× 1044 9.53× 10248

n = 6 2.51× 107 9.15× 1061 1.43× 10418

n = 7 4.03× 108 8.10× 1082 4.12× 10650

n = 8 6.44× 109 7.35× 10106 2.80× 10956

n = 9 1.03× 1011 6.82× 10133 5.50× 101345

n = 10 1.65× 1012 6.49× 10163 3.81× 101828

and

8b1

n−1

i (d0 + 10di) =d0

2k, (5.3)

for i = 0, 1, .., k. Furthermore, (3.70) and (3.72) hold as well.

Proof. The proofs of (5.2) and (5.3) are by induction in i. When i = 0 the

conclusion holds. Assume the conclusion holds for i < k. It remains to verify

the conclusion for i+ 1. Note that

bi+1(di+2 − 3d0 − 10di+1)

= [16k(1 + 10Ck,n(i+ 1))]n−1(Ck,n(i+ 2)d0 − 3d0 − 10Ck,n(i+ 1)d0)

= (1 + 10Ck,n(i+ 1))d0 = d0 + 10di+1.

47

Similarly, for the second equation we get

8b1

n−1

i+1 (d0 + 10di+1) = 8[16k(1 + 10Ck,n(i+ 1))]−1(1 + 10Ck,n(i+ 1))d0 =d0

2k.

To verify that (3.72) holds, note that dk = Ck,n(k)d0 = Ck,nd0 and, by

the definition of hk,n(d0) [Definition 3.0.3], we have exactly (3.72).

Lastly, both (3.72) and (5.2) imply (3.70). Note that, setting hk,n(d0) = c,

(3.72) implies

dk ≤ 10−k−2(1 + d0/2k)−k(1− c−1) (5.4)

= 10−k−2(1 + d0/2k)−k1/c(c− 1) (5.5)

≤ c− 1, (5.6)

where the last inequality follows because 10−k−2 < 1, (1+ d0/2k)−k < 1, and

1/c < 1.

Therefore, since 1 ≤ i < k,

bi(c− 1 + d0(2− i/k)) ≥ bi(dk + d0(2− i/k)) (5.7)

≥ bi(dk + d0) (5.8)

≥ bi · dk (5.9)

≥ bi · di+1 (5.10)

≥ bi(di+1 − 3d0 − 10di) (5.11)

= d0 + 10di, (5.12)

where the last equality follows from (5.2). Thus, (3.70) holds and this com-

pletes the proof.

Remark. So we see that the constants Ck,n(i) suffice for the proof of The-

orem 1.0.3. Next we show that these constants provide the optimal choice.

48

Lemma 5.1.2. If (3.69) and (3.71) hold for all i ≥ 0, then

di ≥ Ck,n(i)

and

bi ≤1

[16k(1 + 10Ck,n(i))]n−1.

Proof. The proof is by induction on i. When i = 0 the conclusion holds.

From (3.69) and assuming the conclusion holds for i, we have

di+1 ≥ 1

bi(d0 + 10di) + 3d0 + 10di (5.13)

≥ [(16k)n−1(1 + 10Ck,n(i))n + 3 + 10Ck,n(i)]d0 (5.14)

= Ck,n(i+ 1)d0. (5.15)

Using this lower bound for di+1 and (3.71, we get

bi+1 ≤(d0

2k

1

d0 + 10di+1

)n−1

(5.16)

=

(1

16k(1 + 10Ck,n(i+ 1))

)n−1

. (5.17)

This completes the proof.

5.2 Computing α(k, n) values

The term β(k, c, n) denotes the minimal volume growth necessary to guar-

antee that any continuous map f : Sk → Bp(R) has a continuous extension

g : Dk+1 → Bp(cR) (see Definition 3.0.4). Recall that the expression for

β(k, c, n) is iteratively defined.

49

By definition,

β(k, c, n) = max

{1− γ

(c, h−1

k,n (c) , n); (5.18)

β

(j, 1 +

h−1k,n (c)

2k, n

), j = 1, .., k − 1

}. (5.19)

Ultimately we are not concerned with the location of the homotopy map.

Thus we have a certain amount of freedom when choosing which c value

to take. To determine the optimal bound on volume growth guaranteeing

πk(Mn) = 0, it is necessary to choose the ‘best’ value of c for β(k, c, n); that

is, the c which makes β(k, c, n) the smallest. Set α(k, n) = infc>1 β(k, c, n).

In order to compute explicit values for α(k, n), we must successively sim-

plify the components of β(k, c, n). Ultimately, because of its iterative defini-

tion, it is possible to express β(k, c, n) as the maximum of a collection of γ

terms. Using the definition of γ(c, ε, n), we can then compute specific values

for α(k, n). Here we describe in detail the method to compute α(k, n) and

compile a table of these values for k = 1, 2, 3 and n = 1, ..., 10.

To begin, we have

β(1, c, n) = 1− γ(c, h−1

1,n (c) , n). (5.20)

By definition, when k = 2

β(2, c, n) = max

{1− γ

(c, h−1

2,n (c) , n), (5.21)

β

(1, 1 +

h−12,n (c)

4, n

)}. (5.22)

To evaluate this expression for β(2, c, n), simplify the β(1, 1 +

h−12,n(c)

4, n)

term by setting c = 1 +h−12,n(c)

4and applying (5.20). Therefore,

50

β(2, c, n) =max

{1− γ

(c, h−1

2,n (c) , n), (5.23)

1− γ

(1 +

h−12,n (c)

4, h−1

1,n

(1 +

h−12,n (c)

4

), n

)}. (5.24)

Similarly, to evaluate β(3, c, n) we have, by definition,

β(3, c, n) = max

{1− γ

(c, h−1

3,n (c) , n); (5.25)

β

(j, 1 +

h−13,n (c)

6, n

), j = 1, 2

}(5.26)

= max

{1− γ

(c, h−1

3,n (c) , n), (5.27)

β

(1, 1 +

h−13,n (c)

6, n

), (5.28)

β

(2, 1 +

h−13,n (c)

6, n

)}. (5.29)

Substituting β(1, 1 +

h−13,n(c)

6, n)

with the expression obtained by setting

c = 1 +h−13,n(c)

6and evaluating (5.20) yields

β(3, c, n) = max

{1− γ

(c, h−1

3,n (c) , n), (5.30)

1− γ

(1 +

h−13,n (c)

6, h−1

1,n

(1 +

h−13,n (c)

6

), n

), (5.31)

β

(2, 1 +

h−13,n (c)

6, n

)}. (5.32)

Finally, apply (5.23) with c = 1 +h−13,n(c)

6to simplify the remaining

β(2, 1 +

h−13,n(c)

6, n)

term. We get

51

β(3, c, n)

= max

{1− γ

(c, h−1

3,n (c) , n),

1− γ

(1 +

h−13,n (c)

6, h−1

1,n

(1 +

h−13,n (c)

6

), n

),

1− γ

(1 +

h−13,n (c)

6, h−1

2,n

(1 +

h−13,n (c)

6

), n

),

1− γ

1 +h−1

2,n

(1 +

h−13,n(c)

6

)4

, h−11,n

1 +h−1

2,n

(1 +

h−13,n(c)

6

)4

, n

}.Because of the successive nesting, when completely expanded, the ex-

pression β(k, c, n) can be written as the maximum of 2k−1 terms of the form

1−γ(., ., n). However, given the nature of the functions hk,n(x) and the behav-

ior of γ(c, h−1k,n(c), n) when 1 < c < 2, the maximum of this collection of 1−γ

terms is determined by the the maximum of the leading 1 − γ(c, h−1k,n(c), n)

term and the final 1− γ term containing the most iterations. That is to say,

for all k and n,

β(k, c, n)

= max

{1− γ

(c, h−1

k,n (c) , n); β

(j, 1 +

h−1k,n (c)

2k, n

), j = 1, .., k − 1

}

52

= max

{1− γ

(c, h−1

k,n (c) , n),

1− γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

}

;

which in turn can be written as

β(k, c, n) = max

{1− γ

(c, h−1

k,n (c) , n),

1− γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

}

= 1−min

{γ(c, h−1

k,n (c) , n),

γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

}.

Recall that the constants δk,n [Definition 3.0.3] represent the location

of the vertical asymptote x = δk,n of the function hk,n(x). Therefore, the

function h−1k,n is bounded above by the constant δk,n; that is, h−1

k,n(c) < δk,n for

all c > 1. In Table 5.2, we list values of δk,n for 1 ≤ k ≤ 3 and 1 ≤ n ≤ 10.

For fixed k, n, the function γ(c, h−1k,n(c), n) is increasing as a function of c

when 1 < c < 2. Further, we have that for all c > 1 and k, n.

53

Table 5.2: Table of δk,n values for 1 ≤ k ≤ 3, 1 ≤ n ≤ 10

k = 1 k = 2 k = 3

n = 1 4.17× 10−5 - -

n = 2 2.60× 10−6 5.29× 10−13 -

n = 3 1.63× 10−7 6.58× 10−22 7.34× 10−66

n = 4 1.02× 10−8 7.98× 10−34 9.96× 10−139

n = 5 6.36× 10−10 9.45× 10−49 1.05× 10−254

n = 6 3.97× 10−11 1.09× 10−66 7.01× 10−424

n = 7 2.48× 10−12 1.23× 10−87 2.43× 10−656

n = 8 1.55× 10−13 1.36× 10−111 3.57× 10−962

n = 9 9.70× 10−15 1.47× 10−138 1.81× 10−1351

n = 10 6.06× 10−16 1.54× 10−168 2.62× 10−1834

h−1k,n(c) < δk,n (5.33)

h−1k,n(c)/4 < δk,n/4 (5.34)

1 + h−1k,n(c)/4 < 1 + δk,n/4 << 2. (5.35)

Define εk,n as

εk,n = limc→∞

γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

54

= γ

1 + . . .h−1k−1,n

(1 +

δk,n

2k

)2(k − 1)

, h−11,n

1 + . . .h−1k−1,n

(1 +

δk,n

2k

)2(k − 1)

, n

=

1 +

1 + . . .h−1

k−1,n

(1+

δk,n2k

)2(k−1)

h−11,n

(1 + . . .

h−1k−1,n

(1+

δk,n2k

)2(k−1)

)n

−1

.

With this simplification, it is possible to explicitly compute the values of

εk,n. Table 5.3 below lists values of εk,n for k = 1, 2, 3 and n = 1, ..., 10. These

values were computing using Mathematica 6.0 and the source code for these

computations as well as additional exposition can be found in Appendix II.

Table 5.3: Table of εk,n values for 1 ≤ k ≤ 3, 1 ≤ n ≤ 10

k = 1 k = 2 k = 3

n = 1 1.04× 10−5 - -

n = 2 4.24× 10−13 1.89× 10−37 -

n = 3 6.74× 10−23 1.92× 10−86 3.52× 10−284

n = 4 4.18× 10−35 1.70× 10−167 1.29× 10−722

n = 5 1.01× 10−49 7.64× 10−290 1.25× 10−1563

n = 6 9.61× 10−67 1.64× 10−462 4.16× 10−3006

n = 7 3.56× 10−86 1.55× 10−694 2.75× 10−5289

n = 8 5.14× 10−108 6.06× 10−995 9.42× 10−8693

n = 9 2.90× 10−132 9.08× 10−1373 1.94× 10−13536

n = 10 6.41× 10−159 4.87× 10−1837 1.24× 10−20180

55

The value α(k, n), as described in Theorem 1.0.3, represents the optimal

lower bound for the volume growth guaranteeing πk(Mn) = 0. We can then

set α(k, n) = 1− εk,n. Table 5.4 contains the values of α(k, n) for k = 1, 2, 3

and n = 1, ..., 10.

Table 5.4: Table of α(k, n) values for 1 ≤ k ≤ 3, 1 ≤ n ≤ 10

k = 1 k = 2 k = 3

n = 1 1− 1.04× 10−5 - -

n = 2 1− 4.24× 10−13 1− 1.89× 10−37 -

n = 3 1− 6.74× 10−23 1− 1.92× 10−86 1− 3.52× 10−284

n = 4 1− 4.18× 10−35 1− 1.70× 10−167 1− 1.29× 10−722

n = 5 1− 1.01× 10−49 1− 7.64× 10−290 1− 1.25× 10−1563

n = 6 1− 9.61× 10−67 1− 1.64× 10−462 1− 4.16× 10−3006

n = 7 1− 3.56× 10−86 1− 1.55× 10−694 1− 2.75× 10−5289

n = 8 1− 5.14× 10−108 1− 6.06× 10−995 1− 9.42× 10−8693

n = 9 1− 2.90× 10−132 1− 9.08× 10−1373 1− 1.94× 10−13536

n = 10 1− 6.41× 10−159 1− 4.87× 10−1837 1− 1.24× 10−20180

In general, for n ≥ 2, α(1, n) = 1−[1 + 2

h−11,n(2)

]−1

; and for k > 1, we have

α(k, n) = 1− εk,n (5.36)

= 1−

1 +

1 + . . .h−1

k−1,n

(1+

δk,n2k

)2(k−1)

h−11,n

(1 + . . .

h−1k−1,n

(1+

δk,n2k

)2(k−1)

)n

−1

. (5.37)

These are the bounds are the best that can be achieved via Perelman’s

method.

56

Combining this information with previous results of Anderson [2], Li [14],

Cohn-Vossen [9] and Zhu [19] we can refine the table above.

Table 5.5: Table of revised α(k, n) values for 1 ≤ k ≤ 3, 1 ≤ n ≤ 10

k = 1 k = 2 k = 3

n = 1 − - -

n = 2 0 0 -

n = 3 0 0 0

n = 4 1/2 1− 1.70× 10−167 1− 1.29× 10−722

n = 5 1/2 1− 7.64× 10−290 1− 1.25× 10−1563

n = 6 1/2 1− 1.64× 10−462 1− 4.16× 10−3006

n = 7 1/2 1− 1.55× 10−694 1− 2.75× 10−5289

n = 8 1/2 1− 6.06× 10−995 1− 9.42× 10−8693

n = 9 1/2 1− 9.08× 10−1373 1− 1.94× 10−13536

n = 10 1/2 1− 4.87× 10−1837 1− 1.24× 10−20180

57

Chapter 6

Appendix II

The evaluation of the constants given in the previous chapter were computed

using Mathematica 6.0 software. In this section we include the Mathematica

source code and give some additional explanation.

Recall, Perelman proved that the existence of a small constant εn > 0

such that knowing the volume growth of the manifold is bounded below by

1 − εn implies the manifold is contractible. Here we will apply the analysis

of Perelman’s method in steps to find small constants εk,n with the property

that if the volume growth of the manifold is greater than 1 − εk,n then the

k-th homotopy group is trivial. That is we take α(k, n) = 1− εk,n.

First we define the iterative constants Ck,n(i), for i ≥ 1 and create a table

containing Ck,n for 1 ≤ n ≤ 10 and 1 ≤ k ≤ 6.

58

In this and each table that follows, each cell represents the (n, k)-th entry.

Each row represents fixed n and each column represents fixed n.

Next we define the functions hk,n(x) and find the vertical asymptote δk,n

of this function. These asymptotes are displayed in the table below. These

values are relevant because the ‘small constant’ d0 as stated in the Moving

In Lemma [Lemma 3.1.2] is bounded above by δk,n; i.e. d0 ∈ (0, δk,n). The

table below gives values of δk,n for 1 ≤ n ≤ 10 and 1 ≤ k ≤ 6.

59

Next we define a constant d[k, n, c], for c > 1. This constant denotes

h−1k,n(c) on the domain (0, δk,n). Note that on this domain the function hk,n(x)

is a smooth, one-to-one, onto increasing function. Thus, the inverse is well-

defined. Note that the value of c must be greater than 1 or the output will

return “The value of c must be greater than 1”.

Next we define the terms γ(c, d, n), the expression given from Perelman’s

Maximal Volume Lemma [Lemma 1.1.3].

With these definitions in place, we can now begin to examine the behavior

of γ(c, h−1k,n(c), n) as the value of c varies.

60

6.1 Computing α(1, n)

To simplify the situation above, set k = 1, n = 2. We graph below γ(c, d[1, 2, c], 2)

as a function of c; recall d[1, 2, c] = h−11,2(c).

Since β(1, c, 2) = 1 − γ(c, h−11,2(c), 2), the ‘best’ value for c will be when

γ(c, h−11,2(c), 2) is largest. Note that, in general, the expression γ(c, h−1

k,n(c), n)

attains its maximum when ch−1

k,n(c)attains its minimum, approximately at

c = 2. To analyze this behavior more closely, set Hk,n(c) := ch−1

k,n(c), and

compute

H ′k,n(c) =

d

dcHk,n(c) =

h−1k,n(c)− c

h′k,n(h−1k,n(c))[

h−1k,n(c)

]2 .

For c > 1, set s := h−1k,n(c) ∈ (0, δk,n) and note that

61

h′k,n(s) =d

ds

[1− 10k+2Ck,ns

(1 +

s

2k

)k]−1

= −[hk,n(s)]2 ·[−10k+2Ck,n

(1 +

s

2k

)k− 10k+2Ck,nsk

(1 +

s

2k

)k−1 1

2k

]= [hk,n(s)]

2 ·[10k+2Ck,n

(1 +

s

2k

)k+ 10k+2Ck,n

s

2

(1 +

s

2k

)k−1]

To find the critical values of Hk,n we find the values of c where h−1k,n(c) =

ch′k,n(h−1

k,n(c)). To simplify, re-write this equation in terms of s rather than c.

Note this is allowed since hk,n is a 1-1, onto function from (0, δk,n) to (1,∞)

[Definition 3.0.3]. Define Hk,n(s) := s− hk,n(s)

h′k,n(s). We define the above functions

in Mathematica as follows.

We can now use Mathematica to solve Hk,n(s) = 0 for s. In fact, we use

a command which will find the roots of Hk,n(s) near a prescribed value of s.

From the graph of γ above, we expect this root to be approximately at c = 2

and thus have Mathematica search for a root near s = h−1k,n(2). Recall, from

our Mathematica definitions above, that h−1k,n(2) is denoted by d[k, n, 2].

Choosing values for k and n, keeping k ≤ n, we can then find the value s

which solves Hk,n(s) = 0 and thus is a critical value for Hk,n. For example,

with k = 1 and n = 2 we use the following command.

62

The root found here is precisely h−11,2(2) which we can verify by applying

h1,2 to the above value. We get

In fact, keeping k = 1 and letting the dimension n vary, we find that root

of Hk,n(s) consistently occurs at s = h−11,n(2). We can verify this for values of

n ≤ 10 by creating a grid where each entry is h1,n evaluated at the root of

H1,n(s).

In fact, we can find the same result for much larger values of n. How-

ever, the information provided here from Perelman’s method is already much

weaker than what is previously known from Anderson [2] and Li [14]. It is

also interesting to note that it seems the expression γ(c, h−1k,n(c), n) does in-

deed attain its maximum at c = 2 for all values of k ≥ 1 and n ≥ 2. However,

as we will see later, for larger values of k, more γ terms are involved in the

expression of β(k, c, n) and it happens that we need only devote this much

63

attention to finding the maximum of γ(c, h−1k,n(c), n) when k = 1.

Lastly, note that the function H ′′k,n(c) is concave up for c > 1. A direct

computation yields

H ′′k,n(c) =

1(h−1k,n(c)

)3[ch′′(h−1k,n(c)

)(h′(h−1k,n(c)

))3 +2

h′(h−1k,n(c)

) ( c

h−1k,n(c)

− 1

)]

Note that h′k,n(h−1k,n(c)

)> 0, h′′k,n

(h−1k,n(c)

)> 0, and h−1

k,n(c) > 0 for c > 1.

Furthermore, for c > 1, we have 0 < h−1k,n(c) < δk,n << 1 and thus it follows

that c/h−1k,n(c) > 1. Therefore, H ′′

k,n(c) > 0, for all c > 1.

Thus, Hk,n(c) has a minimum at c = 2 and therefore, γ(c, h−1k,n(c), n) is

maximized precisely when c = 2. Finally, this yields, for n ≥ 2,

α(1, n) = 1− γ(2, h−1k,n(2), n) (6.1)

= 1−

[1 +

2

h−11,n(2)

]−1

. (6.2)

For k = 1, n = 2, we compute,

Thus, we can take ε1,2 = 4.24× 10−13 and α(1, 2) = 1− 4.24× 10−13. In

a similar way, we can compute the values of ε1,n and α(1, n) for all n. Below

we use Mathematica to define ε1,n = γ(2, h−11,n(2), n).

The grid below provides the values of ε1,n for 1 ≤ n ≤ 10.

64

As we stated before, Perelman’s method provides a lower bound on vol-

ume growth that is much stronger than that of Anderson and Li. In order

to find some new information of how volume growth affects the topology of

manifolds with Ric ≥ 0, we now focus on π2(Mn); i.e. setting k = 2.

6.2 Computing α(2, n)

We begin by finding α2,2; that is, we want to compute infc>1 β(2, c, 2). By

definition, β(2, c, 2) is the maximum of the two expressions

1− γ(c, h−1

2,2(c), 2), and

1− γ

(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2

).

The maximum of these expressions is equivalent 1 minus the minimum of

γ(c, h−1

2,2(c), 2), and

γ

(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2

).

65

As a function of c, the minimum of these two expressions is a piecewise

function. To find the ‘best’ value of c we must find the c where this minimum

is the largest. That is to say, in order to minimize

1−min{γ(c, h−1

2,2(c), 2), γ

(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2

)},

we must maximize min{γ(c, h−1

2,2(c), 2), γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2)}.

So, we want to find c where the minimum of these two expressions is the

largest.

To better understand the situation we graph these expressions as a func-

tion of c. First we graph γ(c, h−1

2,2(c), 2)

on the domain 1.01 ≤ c ≤ 10.

Next we graph γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2)

on the same domain.

66

It seems as if the second graph has a horizontal asymptote which we can

find by taking a limit or evaluating the function at extremely large values of

c.

So,

limc→∞

γ

(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2

)= lim

c→∞

1 +

1 +h−12,2(c)

4

h−11,2

(1 +

h−12,2(c)

4

)

2−1

≈ 1.18771× 10−37

The existence of such an asymptote ultimately relies on the behavior of

the function h2,2(x); and in general, hk,n(x). Below we graph h2,2(x) on the

domain 0 ≤ x ≤ δ2,2.

67

Recall, the function h2,2 has a vertical asymptote at δ2,2 ≈ 5.29× 10−13;

i.e., and in general, limx→δk,nhk,n(x) = ∞. Thus, the expression limx→∞ 1 +

h−1k,n(x) = 1 + δk,n. In this case, with k, n = 2, we have

limx→∞

1 + h−12,2(x)

4= 1 +

δ2,24≈ 1 + 1.32× 10−13.

When evaluated throughout the expression γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2),

the horizontal asymptote arises naturally.

Below we graph the function γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2)

on the

same axes as its horizontal asymptote y = 1.18771× 10−37.

68

The expression γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2)

has a horizontal asymp-

tote at approximately y = 1.18771 × 10−37 which is significantly less than

the maximum of γ(c, h−1

2,2(c), 2)

which is achieved when c = 2. In addition,

as c goes to infinity, the graph of γ(c, h−1

2,2(c), 2)

goes to zero. This is evident

from the graph above. More rigourously, note that since, for all k, n, we have

limc→∞ h−1k,n(c) = δk,n << 1 it follows that limc→∞ c/h−1

k,n(c) = ∞. Thus,

since the expression γ is continuous in c,

limc→∞

γ(c, h−1k,n(c), n) = lim

c→∞

[1 +

(c

h−1k,n(c)

)n]−1

= 0

The maximum of these two functions must be bounded above by the hor-

izontal asymptote of γ(1 +

h−12,2(c)

4, h−1

1,2

(1 +

h−12,2(c)

4

), 2). Although we don’t

know exactly what value of c achieves this maximum, this, in fact, is not

69

necessary. The important point is that we do know the maximum is arbi-

trarily close to (and bounded above by) the horizontal asymptote, which for

k, n = 2 is roughly 1.18771 × 10−37. Therefore, ε2,2 = 1.18771 × 10−37 and

thus α(2, 2) = 1− 1.18771× 10−37.

In fact, these horizontal asymptotes are precisely why we don’t need to

worry about whether or not γ(c, h−1k,n(c), n) achieves a maximum at c = 2 for

k > 1. Once k > 1, the horizontal asymptote of the second (or last) γ term

will dominate the minimum of the two (and any other) expressions. In a

sense, when k > 1 we don’t have the chance to reach the maximum of the

first γ expression of β because the horizontal asymptote of the next gamma

expressions. In general, the more iterations of the hk,n’s there are the closer

the horizontal asymptote is pushed to zero. This justifies the simplification

of β(k, c, n) found in Section 5.2. Namely,

70

β(k, c, n) = max

{1− γ

(c, h−1

k,n (c) , n),

1− γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

}

= 1−min

{γ(c, h−1

k,n (c) , n),

γ

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, h−11,n

1 + . . .

h−1k−1,n

(1 +

h−1k,n(c)

2k

)2(k − 1)

, n

}.

Rather than focusing on the entire collection of intermediate β(j, c, n)

terms, we need only focus on the first γ expression (involving only the h−1k,n(c)

term) and the last γ expression from the expansion (involving the largest

number of iterations of the h−1k,n terms).

With this in mind, we can now compute values of ε2,n and thus α(2, n)

for n ≥ 3 for n = 2, 3, ..., 10. First we create a table of values of δ2,n/4

for n = 2, 3, ..., 10, since these are the values we are concerned with when

evaluating the expressions of γ.

71

Using these values we can now evaluate the expression γ(1 + δ2,n

4, h−1

1,n

(1 + δ2,n

4

), n)

for n = 3, ..., 10.

Computing ε2,3:

Computing ε2,4:

Computing ε2,5:

Computing ε2,6:

72

Computing ε2,7:

Computing ε2,8:

Computing ε2,9:

Computing ε2,10:

Finally, we compile all the information we have collected so far concerning

εk,n for k = 1 and n = 1, 2, ..., 10 into a single Table.

73

6.3 Computing α(3, n)

As before, when computing values of ε3,n for n ≥ 3 we only focus on the final

γ containing the most iterations of the h−1k,n’s. As before, we start by creating

a list of values representing δ3,n/6.

74

Just as before, we use these values to evaluate the expression

γ

1 +h−1

2,n

(1 +

h−13,n(δ3,n)

6

)4

, h−11,n

1 +h−1

2,n

(1 +

h−13,n(δ3,n)

6

)4

, n

for n = 3, ..., 10.

Computing ε3,3:

Computing ε3,4:

Computing ε3,5:

Computing ε3,6:

75

Computing ε3,7:

Computing ε3,8:

Computing ε3,9:

Computing ε3,10:

76

We compile all the information we have collected so far concerning εk,n

for k = 1 and n = 1, 2, ..., 10 into a single Table.

The rest of the terms εk,n can be computed using the exact same method

as above. Note that higher values of k require more iterations of the h−1k,n

functions when evaluating the final γ expression.

77

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