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Generalizations of the Reduced Distance in the Ricci Flow —
Monotonicity and Applications

presented by

Joerg Enders

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5/08 K.lProj/Acc8Pres/CIRC/DateDue.indd

Generalizations of the Reduced Distance in the Ricci Flow

- Monotonicity and Applications
By

J oerg Enders

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

2008

ABSTRACT

Generalizations of the Reduced Distance in the Ricci Flow

- Monotonicity and Applications
By

J oerg Enders

The evolution of a Riemannian metric by the Ricci flow has been found to be
very powerful in studying and classifying manifolds of certain curvature conditions.
However, the flow typically develops singularities in finite time, which need to be
understood. Quantities monotone in time are a common tool to study geometric
flows near singularities.

We define a reduced distance function based at a point at the singular time
of a Ricci flow on a complete n-dimensional manifold M. Our curvature bound
assumption is weaker than the generic type I condition. We show that the corre-
sponding reduced volume based at singular time is monotone along the flow. Since
the quantity being constant implies that the flow is a gradient shrinking soliton,
type I singularities can be modeled by those special solutions. We also show the
monotonicity of the reduced volume arising from the reduced distance to a compact

submanifold of 1M, and we similarly extend that notion to singular time.

To my parents,

iii

ACKNOWLEDGMENTS

The author would like to thank his advisor, Professor Jon Wolfson, for support,
encouragement and enumerable discussions related to this work. Thanks also go to
the other members of the dissertation committee, Professor Xiaodong Wang for many
conversations regarding this research, as well as Professors Ronald F intushel, Thomas
Parker, and Zhengfang Zhou. The author also extends his gratitude to Professors

Bennett Chow and Lei Ni for comments regarding this project.

iv

TABLE OF CONTENTS

LIST OF FIGURES

1 Introduction

2 Background on Ricci flow
2.1 The equation and its properties .......................

2.2

2.2.1

Special solutions ...............................
Einstein solutions ..............................

2.2.2 Gradient shrinking solitons ........................

2.3
2.4

Perelman’s reduced distance .........................
Reduced volume and monotonicity .....................

3 Reduced distance and monotonicity based at singular time

3.1
3.2
3.3
3.4
3.5

3.5.1

Motivation ...................................

Type A Ricci flow solutions .........................

Reduced distance based at singular time ..................
Reduced volume monotonicity based at singular time ...........
Examples ...................................

Einstein solutions ..............................

3.5.2 Gradient shrinking solitons ........................
3.6 The inverse of Corollary 3.4.3 (iii) ......................

4 Applications of monotonicity based at singular time

4.1
4.2

Rescaling limits of type I Ricci flows are gradient shrinking solitons . . .
Definition of a density ............................

5 Reduced distance and monotonicity based at a submanifold

5.1
5.2
5.3
5.4
5.5

Motivation ...................................
Reduced distance based at a submanifold ..................
Reduced volume monotonicity based at a submanifold ..........
Reduced distance and volume based at a submanifold at singular time . .
Examples ...................................

6 Conclusions

6.1
6.2

Reduced distance based at a point at singular time ............
Reduced distance based at a submanifold ..................

vii

1
1

phi-400N901“

17
17
19
2O
28
33
33
35
40

42
43
48

52
52
54
61
63
65

67
67
69

A Shi’s derivative estimates
B Cheeger-Gromov-Hamilton Compactness Theorem

BIBLIOGRAPHY

vi

71

72

75

2.1
2.2

3.1
3.2
3.3
3.4

4.1
4.2
4.3
4.4

5.1
5.2
5.3

LIST OF FIGURES

£—length of a curve 7 in M" between (qt) and (p,t0) .........

Reduced volume monotonicity based at (p, to) for some Ricci flow g(t)

Convergence of the reduced distances with base times ti approaching T
Construction of 7(t) for a uniform bound on Lp,ti(q, f) ..........
Reduced volume monotonicity based at singular time (p, T) .......

Gradient shrinking soliton with f 74 [ET + c ................

Parabolic rescaling and Cheeger-Gromov-Hamilton convergence .....
Convergence of rescaled reduced volumes based at singular time .....
Convergence of reduced volumes to singular time and density loss . . . .

Relation between density and limit gradient shrinking soliton .......

Formation of a neckpinch ..........................
A symmetric neckpinch ............................

The .C—distance from ((1.0 to a submanifold (S,t0) ............

vii

12
15

18
21
29
39

46
49
50

53
55

CHAPTER 1

Introduction

Given an n—dimensional Riemannian manifold M", Hamilton [16] introduced the
evolution of an initial metric 9(0) on M n through a family of metrics g(t) according
to

M = —2Ricg(t).

(it
Such a family (M",g(t)), t E [0,T) is called a Ricci flow. The analysis of this

quasilinear second order (weakly) parabolic partial differential equation requires un-
derstanding the competition between the heat equation type (smoothing) behavior
of the evolution and the development of singularities. In general, Ricci flows in di-
mensions n 2 3 develop singularities and do not limit to a constant curvature metric
(after renormalization). To classify manifolds in a particular case in dimension 4,
Hamilton [19] introduced the concept of “Ricci flow with surgery” to continue the
flow beyond singularities. Pursuing this idea in arbitrary dimensions without strong
curvature assumptions requires a careful analysis of the singularities developing in
Ricci flow. This is where one of Perelman’s main contributions comes in. For any
p E M n and O < to less than the singular time T, the “reduced distance” lpato arises
from a space-time version of Riemannian geometry adapted to the Ricci flow in all

dimensions [26]. It is a locally Lipschitz function on M n x [0, t0] and satisfies impor-

tant partial differential inequalities. Those imply that the “reduced volume based at

(17,130)”

Vp,t0({) 1= A171 (47r(t0 — 5))_ge—lp’t0(q’{) dtrolg(t-)(q)
is monotone nondecreasing in I along the Ricci flow on [0, t0]. Moreover, Vp¢0® is
constant in f if and only if (M ’7’, g(t)) is isometric to Euclidean space with the flat
(non—evolving) metric.

The reduced volume is the key analytic tool to prove the “No Local Collapsing
Theorem”, which is essential to extract Cheeger-Gromov type limit flows from a
sequence of rescaled Ricci flows around a singularity. For limit flows with nonnegative
curvature operator arising from finite time singularities, Perelman uses the reduced
distance again to obtain blow-down limits that are gradient shrinking solitons, i.e.
only evolve by shrinking in scale and changing by diffeomorphisms generated by a
gradient vector field. By classifying gradient shrinking solitons in dimension 3, this
fully explains the structure of singularities and allows one to perform surgery to
continue the flow [27].

Perelman’s argument is 3-dimensional in several key steps. On the other hand,
Hamilton conjectured that gradient shrinking solitons arise as blow-up solutions for
type I singularities, i.e. as rescaling limits around singularities of Ricci flows satisfying

the (generic) curvature bound

_9__

SUP lng(t)]g(t) S T _ t-

M"

If such a rescaling limit is compact, the conjecture follows from [29]. The proof
uses the monotonicity of Perelman’s entropy functional: the scaling properties of the
entropy imply that the functional is constant on the limit flow. which is precisely the
case on gradient shrinking solitons.

Unlike the entropy functional, the reduced volume is defined for any complete

Ricci flow (of any dimension). After presenting the necessary background in Chapter

2

2, we extend in Chapter 3 the reduced distance and volume to the singular time T
to allow for gradient shrinking solitons other than flat Euclidean space to appear in
the equality case of the monotonicity formula. To do this, we introduce a new (mild)

curvature bound, which is more general than the type I assumption:

Definition 1.0.1 Let T < 00. A complete n— dimensional Ricciflow (Mn,g(t)) on
[0,T) is said to be of type A if there exist C > 0 and r E [1, %) such that for all
t e [0,T)

C
R . < —.
31111.3] mg(t)lg(t) - (T— tjr

To our knowledge. it is not known whether there are closed maximal Ricci flows
on [0, T) that are not of type A. In [11], and in this dissertation in Section 3.3, we

prove the following main

Theorem 1.0.2 Let (M n, g(t)) be a complete n— dimensional Ricci flow on [0,T)
of type A. Also let p E M n and t,- /' T. Then there exists a locally Lipschitz reduced
distance based at singular time lpflw : M n x (0, T) -—+ R, which is a sabsequential
limit

, CP...<A/I"><(0.T>>x 1

Ni ’ PT

 

and for all (q, B E M n x (0,T) satisfies the partial differential inequality

8 t 2 n
_0—ilT”T(q’ 0 _ A9(0119»T(‘ba + lVgOlpma, algm — Rg(t‘)(Q) + m 2 0
in the sense of distributions.

We then define for t E (0, T) a reduced volume based at singular time (p, T)
by
~ —" —1 ,t
Vp,T(t') := an ((47T(T -— n) 7e 2),qu lavozgwq),
and Show any reduced volume based at singular time is monotone nondecreasing in

t. This uses the differential inequality in the theorem. Moreover, VET“) S 1 for all

3

t E (0, T), and if Vp,T(t_) is constant in t then (111", g(t)) is a gradient shrinking
soliton (see the precise statement in Section 3.4).

In Chapter 4 we illustrate how the usefulness of the reduced volume for analyzing
the structure of singularities is increased by generalizing the reduced volume defined at
smooth points in space-time to points at the singular time T. It also is a more direct
analogue to Huisken’s well-known monotonicity formula for the mean curvature flow
[20]. We show how one can use the reduced volume monotonicity based at singular
time to prove Hamilton’s conjecture without the restriction of compact blow-ups.
This has recently been done in [23], where similar estimates to define lp,T have been
obtained under the assumptions that the Ricci flow is of type I (and It— noncollapsed).

As a further generalization, in Chapter 5, we define and prove the monotonicity of
the reduced volume arising from a reduced distance to a lower-dimensional submani-
fold S of M n. It turns out that this quantity can also be extended to singular base
time. Our discussion is motivated by examples of singular sets, which are of lower
dimension than M n.

The dissertation finishes with conclusions in Chapter 6. We mention an alternative
definition of a reduced distance based at singular time, as well as future directions
of further understanding and employing the generalizations of Perelman’s reduced

distance introduced in Chapters 3 and 5.

CHAPTER 2

Background on Ricci flow

In this chapter we will present the background material on Ricci flow needed in the
subsequent chapters. After a discussion of important solutions to the Ricci flow, we
will mainly focus on the reduced distance and its properties. The natural way the
reduced distance arises is backward in time. While the forward time formulation
chosen here is different from Perelman’s original paper [26], it prepares the discussion
in the later chapters. We finish the chapter with a discussion of the reduced volume

and its monotonicity.

2.1 The equation and its properties

Let (M", g(t)), t E [0, T), 0 < T < 00 be a 1-parameter family of complete oriented
connected n-dimensional Riemannian manifolds with bounded (sectional) curvature

solving the equation

(99 .
where Ricgu) denotes the Ricci curvature tensor of the metric g(t). We call a family
with the assumptions as above a Ricci flow on [0,T) throughout this thesis. In

particular, we will always assume that the curvature for each g(t) is bounded.

It follows from work by Hamilton [16], DeTurck [9], Shi [31][32] as well as Chen
and Zhu [5] that for a given complete Riemannian manifold with bounded curva-
ture (M n, 90), there exists a unique solution to this quasilinear second order weakly
parabolic equation (2.1) with g(0) = 90 on a time interval [0, T) and the solution is
maximal if and only if

,1)”; 331,3 [ng(t)'g(t) = 00’

where nglt) denotes the full curvature tensor of g(t).
If we let p E M", and {2:2} normal coordinates near p E M", then we can
compute at p that

(99]..
-8—;] = _2Rij = 3A92‘j» (2.2)

where Ri j are the components of the Ricci tensor, and A is the coordinate Laplacian.
While equation (2.2) only holds at p, and hence doesn’t prove short—time existence, it
indicates that the Ricci flow is the “heat equation for the Riemannian metric”. This
explains the many results of special curvature cone conditions that are preserved un-
der the flow, as well as convergence to asymptotically round metrics under positivity
assumptions, e.g. (2-)positive curvature operator [2]. In general, however, the non-
linearity of the equation causes the formation of singularities in finite time, before
the manifold tends to constant curvature. This happens even when the topology is

simple.

2.2 Special solutions

In this section we will discuss special solutions to the Ricci flow (2.1). In particular,
gradient shrinking solitons will play a key role in the further discussions, namely in
the equality case of the monotone quantities.

The proper fixed points of the equation are Ricci flat solutions: Any 90 with

Ricg0 E 0 gives rise to a solution g(t) 2 go. The metric does not evolve with time.
A trivial example is the flat metric an on Euclidean space R".
Because of the scaling and diffeomorphism invariance of the Ricci tensor, we next

consider generalized fixed points of the equation.

2.2.1 Einstein solutions

Let (Mn,go), n > 1, be an Einstein manifold, i.e. 90 satisfies

. Rgo
RZCQO = 790.

where R90 denotes the constant scalar curvature of go. Then

g(t) = (1 - 3R... t) go (2.3)

is a solution to the Ricci flow, where g(t) is Einstein for each t. It only changes by

scaling. Einstein manifolds are fixed points of the volume normalized Ricci flow

(99 .

_2_ W Randwlgit)
n Vol(g(t))

 

g(t). (24)

HR >0,tl.‘lt' 2.3 .'t tht' 't 1—," .W'thT:=
90 1e souion( )exisson e 1me1n erva ( 00 m6) 1

m—n ,wethen et
90 g

n 1

Rg(t) = — and Ricgu) = m

2(T _ t) 9(0- (2-5)

Example 2.2.1 The shrinking round n-sphere (Sn,g(t)), n > 1, on the maxi-

mal time interval (——00, T) is given by

g(t) = 2m —1><T — ogsn.

where gSn denotes the standard round metric on S" of sectional curvature I.

2.2.2 Gradient shrinking solitons

Motivated by Perelman’s perspective in [26] and [27], we make the following

Definition 2.2.2 A Riemannian manifold (M n, g) is called a gradient shrinking

soliton if there exists a potential function f : M —a R such that
. 1
Ricg + Vngf — 59 = O, (2.6)
where nggf denotes the covariant Hessian of f. We will typically write (Mn, 9, f).

Remark 2.2.3 By making the definition as above, we assume that gradient shrinking
solitons come at a certain scale: A more general definition requires (Mn,g,f) to
satisfy

A
R’ng + Vngf — 59 = 0,

where /\ > 0. We can always scale such an f and g to obtain (2.6).

If V9 f determines a complete vector field, then we get a corresponding Ricci
flow g(t) on (—oo,T) with g(T - 1) = 9, called gradient shrinking soliton in
canonical form, in the following way [7]: The l-parameter family of diffeomorphisms
(fit of M n generated by integrating the vector field Til—3V9 f (with ¢T—1 = id) gives
us

g(t) = (T - t)¢2‘9- (2-7)

With the potential function f (t) = at); f, the solution g(t) satisfies the analogue of

(2.6) on (——oo,T), i.e.

may“) + vgltlvgltlm) — 2—Eri_—t)g(t) = 0. (2.8)
Also, f(t) satisfies
3f _ t 2
a _ [V9( )f(t)]g(t)' (2.9)

As we will always assume that (M n, g) has bounded curvature, it follows from (2.6)
that V9 f is at most linear, and in particular complete. Given a gradient shrinking
soliton, we can therefore always obtain the corresponding gradient shrinking soliton

in canonical form.

Remark 2.2.4 We would like to make the following clarification, which we have not
seen carefully discussed in the literature: If we have a Ricci flow g(t) on [0, T) which
together with a I-parameter family of functions f (t) satisfies ( 2.8), this clearly defines
a gradient shrinking soliton according to Definition 2.2.2 by considering the equation
at t = T — 1. If V9(T—1) f (T — 1) is complete, we can conclude by uniqueness of
the Ricci flow [5/ that the corresponding Ricci flow in canonical form equals g(t) on
[T — 1,T). Moreover, if (2.9) holds for the original family of functions f (t), then
the given triple (Mn,g(t),f(t)) is the gradient shrinking soliton in canonical form

constructed from g(T — 1) as explained above.

Solitons are “self-similar” solutions, since they evolve only by scaling and diffeo-
morphism. Their existence is to be expected due to the diffeomorphism invariance
of the Ricci flow equation (2.1). We can regard them as generalized fixed points, i.e.
fixed points of the volume normalized Ricci (2.4) flow on the space of metrics modulo

the diffeomorphism group.

Without going into a more detailed discussion, we mention that by changing the
minus sign in (2.6) to “+” (or dropping the metric term), one can analogously define
gradient expanding (or steady) solitons. Also, if the vector field generating the
diffeomorphisms ¢t is not a gradient vector field, one gets a more general notion of
solitons. It follows from [26] (in the compact case) and [23] (in the complete case)
that any soliton is a gradient shrinking soliton for some (possibly different) vector

field.

Example 2.2.5 By {2.5) and Remark 2.2.4 we see that any Einstein solution with
positive scalar curvature can be regarded as a gradient shrinking soliton in canonical

form with f E 0 (or f such that Vf is a Killing vector field) and scaled so that

R _1
710-2-

Example 2.2.6 Consider the non-evolving Ricci flow (R",g(t) = an). With

$2
f(x,t) = #7437, we have

Wm = ”72—59mm

which makes flat Euclidean space into a gradient shrinking soliton in canonical form,

called Gaussian soliton. It will arise in an important context in Section 2.4.
Example 2.2.7 Consider the round shrinking cylinder
(Sn—k X Rk,g(t) = 2(7). — k —1)(T _ tlgsn—k + ng),

where n — k _>_ 2. It follows easily from examples 22.1,225 and 2.2.6 that this is
a solution to Ricci flow which is a gradient shrinking soliton in canonical form with

potential
2
[3” [9R]:

f(6,$,t) = m,

where 6 E Sn_k and 1: E Rk.

Example 2.2.8 Example 2.2.7 is a special case of the following: We can construct
gradient shrinking solitons as products of Einstein solutions Nn—k of positive scalar
curvature with flat Euclidean space Rk. It is an interesting question to ask when

gradient shrinking solitons are of that form [28].

We will now discuss an important equation satisfied by gradient shrinking solitons:
Let (M n, g(t), f (t)) be a gradient shrinking soliton in canonical form on (—oo,T).

Taking the trace of equation (2.8) gives

72.

Rg(t) + Ag(t)f(t) - m = 0. (2.10)

10

We conclude from equations (2.9) and (2.10) that.

 

(9
— af — Ag(t)f(t) + [vg(t)f(t)l3(t) — Rg(t) + 2(Tn_ t) = 0 (2'11)
In Perelman’s point of view [26], if we let
u(:r, t) := (4m — aria—fl“), (2.12)

where :1: E M n and t E (—oo,T), then a straight forward computation shows that
(2.11) is equivalent to
* —
Dg(t)u(:r,, t) — 0. (2.13)
flere

,. a
D90) = 7n " Ag(t) + Rye)

‘ . . ‘ __ a . .
denotes the formal adjomt of the heat operator Ely“) -— 32 — Ag“) under the Ricc1
flow. This observation will play a key role in the equality case of the reduced volume
monotonicity.

We will also need the following fact: If (M n, g(t), f (t)) is a gradient shrinking

soliton in canonical form, then it follows from the contracted second Bianchi identity

and (2.8) that there exists a constants C(t), such that

R9“) + |v9(t)f(t)|§(t) — Lt) = C(t). (2.14)

2.3 Perelman’s reduced distance

In this section we will briefly discuss Perelman’s reduced distance for the Ricci flow.
Contrary to [26], we will use forward time notation rather than backward time, since
it will come more natural when we consider a sequence of different base-times in
Chapter 3. The monotonicity of Perelman’s reduced volume will then be described in
Section 2.4. While the results are due to Perelman, there are by now several references

detailing his work, e.g. [30], [21], [36], [6], [22].

11

Definition 2.3.1 Let (ll/I",g(t)) be a Ricci flow on [0,T). For any curve

7 : [t-,t0] —+ 1W, where 0 < t< to < T, we define the .C-length of 7 by

t
= f; O ,Ft0'—"'i(|qv(t)|§(t) +Rg(t)(7(t)))dt.

Figure 2.1 illustrates this definition.

 

\
/

r

O r
eel->-
0..

0

Figure 2.1: £—length of a curve 7 in M" between (qf) and (p, to).

At a curve 7, the first variation of .C with fixed endpoints is given by
(SI/5(7) = <2 to — tr > I 0 (2.15)

t0 1 1
—/t_0 2 tO— Mvgigflt)—2—(t0—_57(t)—2Ricg(t)(7(t),-)#—§V9(t)Rg(t),Y(t))dt.

Y(t) is the variational vector field along 7(t) with Y(t) = Y(t0) = 0 and # de-
notes the metric dual, which we will omit. The Euler-Lagrange equation is called
.C-geodesic equation and given by

—— 7t'( )— 2Ricg(t7)(7(t),-)—-1V9(th (t) = 0. (2.16)
2(t0 — t) 2 9
Smooth solutions to (2.16) are called .C-geodesics. Multiplying by to — t, we can

rewrite (2.16) to get rid of the unbounded coefficient of the second term:

1
Vf/(f77177u )( tO—t 7(t )—2,/t0—t tRicg( tO— t7(t), )—§(t0—t)Vg(t)Rg(t) = .

12

Using the direct method in the calculus of variations after a change of variables in
.C, as well as the ODE (2.17) together with the assumed curvature bound and Shi’s

derivative estimates [32] (see also Appendix A), one obtains the following

Proposition 2.3.2 For any (q,l) and (p, to) with p,q E M" and 0 < i< to < T
there exists an L— minimizing £- geodesic 7 : [t,t0] ——> M with 7(0 = q and

7(t0) = p. Moreover, the initial vector limt/‘to ,/t0 — t7(t) e TpM exists.
Definition 2.3.3 For (q, t) and (p, to) as in the Proposition 2.3.2, we define

(i) the C— distance from (at) to (p, to)
Lptolq (10:: inf{£(r) l7 lt tol 441.7(5) =q 7 (=t0) 19}

(ii) the reduced distance based at (p, to)

_;_;___1;,t0((I(QD

t _
[NOW ‘1‘)“ 2/—_"—_t__0 —t
(iii) and

n
Up.t0(q (LIT—D (47T( (t0 - 5))78—1'qpt0m’0

We will fix (p, to) E M" x (0,T) and regard LPatO’ [Pato and ”19,150 as functions on
space-time M" x (0,t0).
By studying the second variation of L and obtaining a Laplacian comparison

theorem for the reduced distance, Perelman derives the following three differential

(in)equalities:

Theorem 2.3.4 Fix (p, to) E A!" x (0,T). Then for all (qf) E A!" X (0,t0) the
following hold:

8 t 2 _ n
_B—flpvtOval — Ag(alp,t0(qy i) + [V9(-)lp,t0(q,i)]g(a - Rg(t)(Q)+ m _ >0

(2.18)

13

or equivalently , ngvp to (q, (t) S 0. (2.19)

 

. ,t (00
_lvgdthOwaaa + Rg(t)(‘1)+ p 0:? _t_ "+ 2A gmipyomfi g o. (2.20)
431 ( i) +1v9®1 ( m2 — R (()+ ”(q W” =0 (2 21)
0t— p,t0 q. p,t0 (1. g(t) g(t7 1 to _f - -

Remark 2.3.5 It can be shown (see e.g. [36]) that Lp,t0(q, D is locally Lipschitz on
M" x (0, t0). Hence, the same is true for lp,t0(q, D and up,t0(q, D, and the functions
are diflerentiable a.e. in q and f. The (in)equalities in Theorem 2.3.4 therefore hold
in the barrier sense, or in the weak sense. In particular, they hold in the sense of
distributions. For {2.18) this means that for any nonnegative (,b E C00 t(MnX X(0, t0))

Tl

T 8
f0 Li1((‘ailp.to + lvg(t)’p.to|§(t) _Rg(t) + m) 05 (222)
+v9ltlzp¢0 - ngcp) dvolgmdt 2 0.

In fact, [p,t0(q,t—) is smooth away from a closed subset C C 1W” x (0,t0), and C n

M" X {t} has zero Lebesgue measure in IV!" x {t} for every t E (0, to).

2.4 Reduced volume and monotonicity

We can now define Perelman’s reduced volume.

Definition 2.4.1 Let (Mn,g(t)) be a Ricci flow on [0,T), and (p, to) E Alnx (0,T).

Then for each i E (0, to) we define the reduced volume based at (p, to) by

Vp,t0(£) 5: [M 'L’p,t0(qa 1?) delg({)(Q) = ./M (47r(t0 " al—ge—lp’tom,” dvolg(t7(q)-

Example 2.4.2 For the Gaussian soliton (Example 2.2.6) one computes

lq— pl2
1]), t0(q (qt-)— _

4(m-0’

and hence

Vp,t0(0= fn ((WO-0) _‘4—‘0‘dq=1

14

The monotonicity of the reduced volume along the Ricci flow (see also Figure 2.2)

is now essentially a consequence of inequality (2.19) in Theorem 2.3.4:

Corollary 2.4.3 (Monotonicity of the reduced volume)

Under the some assumptions as in Definition 2.4.1, we have
argnmwza
(a) 11m,— /,0 mem = 1,

(iii) Vpdoall = Vp,t0(t—2) for 0 < t1 < £2 < to if and only if (A1",g(t)) is isometric

to the Gaussian soliton (R'n,an) and Vpato (t) E 1.

Vp,t0 (t)

- fit/(.0) for (fl/.90).) :3- (3."..vi5tn)

  

 

 

fl-—---------.-—-....-----—

0 tL o T

Figure 2.2: Reduced volume monotonicity based at (p, to) for some Ricci flow
(Miamonnn.

The reduced distance and reduced volume monotonicity are one of the most impor-
tant analytic ingredients for Perelman‘s study of singularities in the Ricci flow [26].
First, the reduced volume is used to prove the “No Local Collapsing Theorem”, which

is necessary to guarantee Cheeger-Gromov-Hamilton convergence of rescaled flows to

15

a singularity model. In a next step, the reduced distance is again the essential tool to
construct asymptotic solitons for ancient solutions. Those help in the classification
of singularities. In the next chapter, we will extend the reduced distance and volume

to base points at singular time.

16

CHAPTER 3

Reduced distance and
monotonicity based at singular

time

3. 1 Motivation

In the previous chapter, we defined the reduced volume Vplto for smooth points in
space-time (p, to). We now discuss our main motivation to allow the base time to in
the reduced volume to be the singular time T of a Ricci flow.

Results of the type of Corollary 2.4.3 are known for other monotone quantities
in geometric evolution equations, e.g. for Perelman’s /\— and u— functionals for
the Ricci flow [26], or Huisken’s monotonicity formula for the mean curvature flow
[20]. In all of those, solitons arise in the equality case. Note also that the equality
of Harnack expressions in the Ricci flow similarly identifies gradient expanding and
steady solitons, see e.g. [17], [4], [24].

In the proof of Corollary 2.4.3 (iii), it is the fact that at time to the curvature is

bounded, that limits the equality case of the monotonicity to the Gaussian soliton.

17

If we are able to base the reduced distance and volume at singular time (p, T) for a
maximal Ricci flow on [0, T), depending on the base point p E M n, we expect to get
gradient shrinking solitons other than the Gaussian soliton, whenever this generalized
reduced volume is constant. We will prove this in Corollary 3.4.3. Before that, in
Section 3.3, we will define a reduced distance based at singular time and study its
properties. Our approach is as follows:

Let (Mn,g(t)) be a maximal Ricci flow on [0,T). Let t,- / T and p E M". Then
for all (q, t) E M" x (0. T) the reduced distance [p,tif‘h D is defined for large enough i

and the differential inequality (2.18) holds for each such i. This raises two questions:
(i) Does there exist a good limit lpflw := limt,/T lpgi?
(ii) Does a differential inequality analogous to (2.18) hold for lp,T?

In Figure 3.1, we sketch the idea to define a reduced distance based at a point p

at singular time T.

 

0 5 ti ti+1 T

Figure 3.1: Convergence of the reduced distances lp,ti(q,t7 from q and p based at
times t,- approaching the singular time T.

18

While the discussion is motivated by a base point p in the singular set (see Defi-
nition 4.2.2) in order to apply the quantity lpgw to the analysis of singularities as we
discuss in Chapter 4, this will not be a requirement in our main result in this section.
We will address the issue of the choice of base points in Sections 3.5.2 and 4.2. In
general, to answer the two questions above positively, we need to mildly strengthen

the bounded curvature assumption to a new curvature condition.

3.2 Type A Ricci flow solutions

Definition 3.2.1 A Ricci flow (Mn,g(t)) on [0,T) is said to be of type A if there

exist C > 0 and r E [1, 3) such that for all t E [0,T)

C
SUP 'ng(t)'g(t) 5 (gm—,7,"
Note that for r = 1 this includes the well-known type I condition. From the

maximum principle for lng(t)l (t) it follows that for a closed maximal Ricci flow

9
on the interval [0, T)

1
yrs lng(t)|g(t) Z _8(T _ t)’

which implies that curvature blow-up with r < 1 is impossible. On the other hand,
the type I condition is assumed to be generic. To our knowledge it is not known
whether there are closed maximal Ricci flows which are not of type A. The only
known example of a type II (i.e. not of type I) closed Ricci flow is the degenerate
neckpinch [15], but its curvature blow-up rate is not known. In [8], an example of a
1
—t

type II Ricci flow on R2 is shown to blow up proportional to (—T——)§.

Example 3.2.2 Let (Mn,g(t),f(t)) be a complete gradient shrinking soliton in
canonical form with bounded curvature. Then it follows directly from equation (2.7)

that the flow is of type A, in fact of type I.

19

3.3 Reduced distance based at singular time

In this section, we state and prove the main

Theorem 3.3.1 Let (Mn,g(t)) be a Riccifiow on [0,T) of type A. Also let p E M"

and t,- /' T. Then there exists a locally Lipschitz function
113,71: M” x (0,T) —r IR,

which is a subsequential limit

1 CPOC(M"X(0,T))
PM ’ lp,T

 

and which satisfies the differential inequality analogous to (2.18), i.e. for all
(9.5) E M” X (0,T)

8
-.—— p,T(q1{) - Ag(01p,T(Qv 0 +|Vg(21p,r(qs alga) - Rg(t7(61) + '2‘(—Tn__'t—) Z 0 (3-1)

holds in the sense of distributions. Equivalently,
D;(QUP,T(q7 5) S 01 (3'2)

where

i'p,T(q. t) :2 (47r(T — D)_2e_lP~T(q’{).

Remark 3.3.2 This theorem has been independently obtained in [23] under the type

I assumption ( and k— noncollapsedness) using very similar techniques.

Theorem 3.3.1 allows us to make the following
Definition 3.3.3 Under the assumptions of Theorem 3.3.1 we define
[1)],an X (0, T) —> R

to be a reduced distance based at singular time (p,T) in the Ricci flow
(Mn.g(t))-

20

We now give the proof of Theorem 3.3.1.

Proof. To simplify notation let 12' := lp,tz.(q,t.) and L,- := Lp,t2.(q, t). The proof will
be in 3 steps.

1. First, we will derive a basic uniform bound on 11- on compact subsets
K 2 K1 x [a, b] C M n x (0,T). By definition of l,- it suffices to show such a bound
for L,- on K. Let 77 : [0,1] ——i M be a g(O)—geodesic with 77(0) 2 q and 77(1) 2 p.

Fix It 6 (b, T). As depicted in Figure 3.2, consider the curve

a”) he [bk]

 

7(t) == k“
p t E (k, til'
Mn
77
i ‘fi i_ i i l i?
0 a t b k tz T
Figure 3.2: Construction of 7(t) for a uniform bound on Lp,t,-(Q: 0.
Since |n’(s)|§(0) = c for a constant c, the uniform equivalence of the metrics

along the Ricci flow on [0.1:] yields a constant D such that ln’(s)l§(t) S D. The

type A assumption implies that there exist constants C > O and r E [1, %) such that

21

|R| < W If Li< q,D Z O, we get the following estimate:

 

 

ti
Imam s (f (fa—'tuumgf +Rg(.)(v<t>>)dt| (3.3)
T
_ _ kg”— _Dzn n’I:(——:—- )|:()dt+C/_ 82—03111:
D\/(T 20 _,.__
—— mitt—.57” --E-

If [1,-(qt) < O, we let 7(t) be an L—minimizing L—geodesic such that
Li(q, t) = (3(7). Then we get the same bound:

ti
Lam = ]. \/t.-—"i(Ii(t)I§(,,+R(mama
2 —/ ,/t,-— tg(|R t))|dt
2 —E.

Thus, we have uniform bounds in i on any given compact subset K.

2. Next we derive uniform derivative bounds for L,- on compact subsets K of space-

time as above. We first prove

Lemma 3.3.4 Under the assumptions of Theorem 3.3.1 let 7,-(t) be an

[I— minimizing £— geodesics from (q, B to (p, ti), where (q, t) E K. Then there exists

a constant C independent ofi such that for all t E [t_, k]

IWW) 2

Proof. Denote by Vi(t) = ,/t.i — My“). Using the .C—geodesic equation (2.17) we

compute

22

2 _
gall/i“ )l g(t) —
2
W

 

—2Ric(l/;;(t), mt» +

: _2Rz'c(V,-(t), V¢(t))

 

2
+ ,___ti

= 2Ric(V(t)
Cl
(T-tl'r

02

|/\

2

 

Inmgm+

(QN/ti -- tRng ()t (Vz(t
m+¢rtuvgm

iit+§m—av“0

(T - tir—

—2Ric(V,-(t), vim) + 2<V.-,,(t)1i(t). Vimlt

(Vnm

V2“), mt)»

Rg(t)1Vi(t)>t
g() V(t M)

éll/iallgu)?

(3.4)

where in the last inequality, since t 2 t > 0, we used Shi’s derivative estimates [32]

(see also Appendix A) combined with the type A assumption to bound V9(t)Rg(t).

Note that 01,02 are constants depending on the type A constant C, n and t, but

are independent of i.

Before integrating this ordinary differential inequality to conclude the proof of the

lemma, we need to get uniform bounds on each Vi(t) for some t in a compact set of

time. By definition of L, and using again the type A assumption we estimate

ti 1 2

l/\

5(72') +

ti
con—A «a—Hgmowna

2C

3
————-—-T2_T.
3 _

2r

Now the integral mean value theorem yields the existence of t,- E [t, k], such that

 

 

. 2 A : k_1__t_/_k
g(ti)

|/\

1

|/\

23

1 [ti
_f t

Kim”) + 3— 2r

gT:.M w>PW

 

. 2

3
20 T2“ ),

 

or equivalently

 

 

. ti 4} 20 3
V. t' 2 g = _____ . 2‘7'
I 2( 2”9031) k —t (C(71) + 3 — 2rT )
VT 2C 3_
< — E 2 T =:
— k—b( + 3—27‘T ) F,

since by choice of 7i the bound (3.3) holds for C(71).

J

W.l.o.g. we can assume that lVilt)lg(t) 2 1 and estimate (3.4) for t E [a, k] to get

d 2 Cl C2 2 2
— v, g 2 V- = C v,- .
dtl (0'90) (T — k)? + (T _ k),._% I 2might) 3| (t)lg(t)

 

Integrating this, we conclude that for all t E [5, k]
mung“) g FeC3(‘—‘i) g F€C3T =; G. (3.5)

This proves the Lemma. C]

To get the gradient bounds for Li recall that from (2.15)

v9<t>L.(q.£) = —2\/t.~ Jam.

\Nith Lemma 3.3.4 we obtain for (q, 0 E K
Iv9(‘)L.-<q,nlg@ = 2M.- — wmgm s 2E. (3.6)

For the time derivative bound for Li: we compute

gin-(m = disz-(m—<v9(t‘>L.-(q.t7,~r'.-<t‘)>z

= _./—t, _ {at-(mat) + Rg(t-)(7(i))) + 2w.- — {Ii/(W309

2 72-1—44 Mt, -- manlgm — \/tz' - {Rgfihml-

Using the type A assumption and Lemma 3.3.4, we get the time derivative bound for

((1.0 E K

_£_, (3.7)

k — (T— b)r—:2

 

lgngqu)! S

:10

24

Recall that by Remark 2.3.5 each 12- is locally Lipschitz on M n x (0,tz-). The
above bounds show that lz- are in fact uniformly locally bounded and Lipschitz on

M n x (0, T). Hence, there exists a locally Lipschitz function
[p,T : A!" x (O, T) —» IR

such that a subsequence, denoted 1,- again, converges to a [p,T in CZOOC(Mn x (0, T)).

This proves the first part of the theorem.

3. To prove that the differential inequality (3.1) holds in the sense of distributions,

we first note that l,- E WIL’C2(M"’ x (O, T )) and the bounds derived above imply

'1' 1 < C.
2|W10’C2(M"X(O,T))

W.l.o.g. we can assume that
[i —‘ [p,T (3.8)

weakly 1n W10, 2(Mnx(0, T)) This implies for all (V, 10) EW 2(Mnx (0,T),Rn+1)

lloc

0 (9
[T [N V+atllwd10l )dtH/T [M V9“) lpT' V+at lpTI/Jd’UOlg (Md

In particular, if for nonnegative (b 6 C3; t(M"x x(O,T)), we let 1,0 = —¢ and

V = V9(t)cf), we get the distributional convergence

/0T -/TM V9“) ngqb—Ql- tbdvol g(Odt

—> g(t) . g(t) _fi ,

Comparing with the distributional formulation (2.22), we see that we now only need

to Show that for all nonnegative gt) E CESAR!" x (0, T))

T T
[0 [M WHOM-lg“) aidvolg(t)dt —> A; [M lVg(t)lp,:r|§(t) ¢dL’Olg(t)dt.
25

It suffices to show for each t E (0, T) and nonnegative (:5 6 3733,41)! 7‘)

t '2 i “ —) gt 2 ' ..

(3-8 implies the weak L2 convergence of \/q—>V9(t)[. _i fingh ’ and hence
' p,T
fM |V9(t)1p,T|§(t)odvolgut) < limian/I Ivy (0) 2'2“) ¢dU0lt g(t )-

We now show the other direction
lim sup / V9“) [light )czidvolt </ n79“) lg( nédvol
i—»oo M I 9( ) M p T 9( )
using an argument similar to Lemma 9.21 in [22]. We rewrite
- t , 2 _ g t 2
1151132" /M (W )lz'gm “7 ( ”pf-Plan) ¢dv0lg<t>

= limsup(/M(v9(t)(z,- — 1M), ¢V9(t)lp,T)t dvolgm (3.9)

i—aoo

+/M(V9( )(z —z,, T,) av“ )z,)tdvozg(,)).

We can approximate avgmzfl by V]- E Ccpt(’l~1,ll€") and conclude by weak L2
convergence of V9(t)l,-_ that the first integral goes to zero as 2' —> 00. For the second
integral, we first use the CPOC— convergence of l,- —> [p,T to get a sequence 65 \ 0,

such that on supp(¢) we have
lp,T — 12' + 62' > 0.
Then the second integral above equals
limsup / WWW,- — sz — €i):¢Vg(t)li)-tdv0lg(t).
i—->oo M ,
We multiply Perelman’s differential inequality (2.20) for l,- by (,6 and write it in the

distributional sense for a nonnegative w E CcthlI"). '

1.0 l, —
/M<V9(t )(./ vm) v.9“ hm dvolgu) g A! g(lvgfthilgu) _R’g(t) _ t :)dv019(t)°

2'

 

26

By approximation in W112, we can take w = lp,T — l,- + 62' 2 O to be only locally

Lipschitz, i.e. conclude

A 1(v9“) ((1,- — 11,1 — 50¢), vgmzm dvolgm

 

 

(RT—Q+%W t 2 h‘”
S /M p 2 (WM )li|g(t) _ Rg(t) _ t )dvolgm.

2

Since the right-hand integrand is bounded on supp(q§) and
[p,T—li'i'éz' —>0
uniformly, we obtain

lim sup /M(V9(t) ((z, — 1pm — e,)¢),v9(t)z,)t d’uolgm g 0.

i—ioo

Now, inserting the characteristic function X sup“ <25)’ we can rewrite
t t
/M(Vg( )(Ui - [p,T — e,)¢),V9( )li)tdv019(t)
t
= /M(Vg(t)(lz‘ — l119,1!“ — 60¢, Vg( )li>t (10019“)
t t
= /M(Vg(t)(li — w). ¢Vg(t)li>t .1..on
1 t 2 t 2
+ [M §<lz _ lp,T _ €Z)(ivg( )legU) + leupp(¢)Vg( )lllg(t)) dv0[g(t)
As before, the last integral tends to zero because of the uniform convergence of
[p,T — [2' + 62' —* 0.
This implies that the second term in (3.9) satisfies
lirn sup /M(v9(‘)(z,- — zfl), ¢V9(t)l,-)t dvolgm g 0
z—+oo
and finishes the proof. [:1
The proof of Theorem 3.3.1 implies that the analogues of the differential

(in)equalities (2.20) and (2.21) hold for a reduced distance based at singular time

as well. We summarize this in the the following

27

Corollary 3.3.5 For lpflw as in Theorem 3.3.1 and (of) E M" x (0,T), the

following three (in)equalities hold:

-0217p’T(q’D—A9({)IP,T(q’D+lvg®lp,T(2<Lt—)lg®-%(q+) TIE—0- 2 0, (3.10)

 

or equivalently , D;({)11paT(q,D S O. (3.11)
l (q, t.) _ n
t 2 .T
—|V90lp,T(q,t_)|g(D + R9030” + p T _t. + 2A gmlpT(q,t) g 0. (3.12)

lpT< (q7fl

_t :0. (3.13)

8
—2a—,_zp,T<q.o + lVg<E>lp,T(q,f)|§®-R g(mq q+)

3.4 Reduced volume monotonicity based at
singular time

Definition 3.4.1 Let (Mn,g(t)) be a Ricci flow on [0. T) of type A. Let p E M",
t,- /' T, and 110 T andv pT as in Theorem 3. 3. I. Then we define a reduced volume

based at singular time (17.7") by

171,103 :2 ./M up,T(q, deolg®(q) = [M ((47r(T — t_))_ge—vaT(q’£)duolg(t-)(q).

Remark 3.4.2 The finiteness of 171,105) for any Ricci flow (Mn,g(t)) of type A
and any fixed 5 E (0, T) follows from Fatou’s lemma and the finiteness of Perelman’s

reduced volume as follows:
~ .71 -—lim . l . ,t
vpmm = /M (are — 3) 2e VT “2‘" ldvozgma)

= /M.:i;nT(<<4w<T-i>> £6 ., ”WW
3 Ian/1511 171.1,. (a

51
g 1.

Now we state the analogue of Perelman’s monotonicity (Corollary 2.4.3) for the

reduced volume based at singular time. Compare also Figure 3.3.

28

Corollary 3.4.3 (Monotonicity based at singular time)

Under the assumptions as in Definition 3.4.1 we have
fl)%%1®2a
(iii) If Vp,T(t—l) = Vp,T(t—2) fOT‘ 0 < 51 < {2 < T, then (ll'ln,g(t),lp,T(-,t)) is a

gradient shrinking soliton in canonical form.

Remark 3.4.4 Similar results have also been obtained in [23] under the type I

assumption (and n—noncollapsedness).

,. f”,- (t for some gradient
[.l .
13— shrinking soliton

—n— --——-- w» ,i, — -—-r»77 ,r_ 7--—»—————————————— —— - ———¢

 

 

 

O
“l.“—
H—-----------

Figure 3.3: Reduced volume monotonicity based at singular time (p, T) for some
Ricci fiow (M",g(t)) on [0,T).

29

Proof. (i) If A!" is compact and lp,T is smooth the proof follows directly from (3.2)

in Theorem 3.3.1:

.7 8
8
= M (51? p,T + Avpfl" — R)dvolg({) (3.14)
—_ * I
— M —D vpflwdvolgm
Z 0.

In the general case, we need to justify the differentiation under the integral and

the adding in of the Laplacian term. For both arguments, we need to bound

—1
[Me PsTllpfldvolg® < oo (3.15)

for fixed time t E (0, T). From Remark 3.4.2, we know that

/M e-ZPdeuolgm < 00, and hence [M e—lvaTldvolgm < 00.

Since the proof of the finiteness of Perelman’s reduced volume Vpitz‘ relies on com-
parison of the reduced distance lpfl with the square of the distance function, we also
get

1 1
[M e-QlPdevolgm < 00, and hence /M e_?|lP,T|duolg(g) < 00.

Now let N := {q E M | lp,T(q,t) Z 0}. Then

—l —l —|l |
e P»T l duol = f e p,T l dvol + f e 191 l duol ,

1
where the second term is finite since §|lp3~| S efillpflfl. The first integral is bounded
because the type A curvature bound yields an upper bound on llp,Tl on N by

dropping the energy part in the C—functional. This proves (3.15).

The proof of Lemma 3.3.4 implies that in fact there exist constants C1 and C2

(depending on t) such that
lvgajlpfldga) S Clllp,Tl + C2.

30

(Note that Lp,T gets bounded on compact sets by the constant E in that proof.)
This implies

—l
/Me P’T|V9(Dlp,T|§®duolg(g) < oo. (3.16)

We lastly show that also

—l (9
[M e Plegflledvolgm < oo. (3.17)

The differential equality (3.13)

can be used to bound lgflp,Tl' Then (3.17) follows from (3.15) and (3.16) and the
bounded curvature assumption.

Now we can justify what corresponds to the adding in of the Laplacian term in
(3.14) in the distributional setting. As in (2.22), Theorem 3.3.1 implies that for fixed

t E (0, T) and for any nonnegative a) E 25AM")

Tl

a , '
/M ((717 P1T+lVJ(0’P,T'§(t‘)‘Rg(t)+ m) ¢+V9(5)lp,T-Vg®¢) dvolgm 2 0.

Note that

V9(5)(e—lp,T) = —e—lP,TV9®lp,T,

so using the bounds for the integrals proved above and approximating (b = e-lva by

dj 6 CgtM/I"), we conclude

0 n —l. T
«/M ( — ‘37—le _ R9“) + 2(T _ file p’ (10019“) 2 0.

Tl
Up to the factor of (47r(T — 5)) —2 this is the integral we obtain when differentiating
the integrand in the reduced volume based at (p, T). To conclude (i), we only have
to justify the differentiation under the integral sign. We will do so using the standard

dominated convergence argument. We bound, at t E (0,T), the reduced volume

31

integrand difference quotient D(q,t, h), rewritten in integral form, by an integrable

function. Note that

 

- 1 h a _
Dl‘l» ti h) = E./() (_ alpfl'an t + S) — Rg(t—+3) (q)
_ n _ - dvol - (q)
n _ _ — J l ,T(q,t+8) g(t+s)
+2(T — f— 3)) (477(T t 8)) it p duolgmm)

(compare e.g. [6] for the argument in Perelman’s case). Now from Remark 3.4.2 and
the bound (3.17) as well as the exponential growth bound of the volume form under

Ricci flow, which makes the quotient

dvolg({+s) (q)
dvolgm (q)

 

bounded near 5 (where we have bounded curvature), we conclude that D(q,f, h) is

dominated by an integrable function for 17. near 0.
(ii) This follows from (i) and Remark 3.4.2.

(iii) From the initial computation in (i), which we have justified for noncompact M n

and in the distributional setting, we get

- - {2 d - t‘2
0: t —v t =/ —_v 135:] f 43*.) dvol dt'.
p,T( 2) p,T( 1) [1 dt p,T(d {1 M p,T g(t)

From (3.2) we conclude that in the sense of distributions
D’k’b‘p’T E 0. (3.18)

and hence by parabolic regularity lp,T is smooth. Recall equation (3.13)

5 t 2
—2—lp,T + |V9<2lp,T|g(t-) — R9“) +

lp,T
at — _

Subtracting 2 - (3.18) from this, we get

._ , t 2 __
pr .— ((T — DQAQUDIILT -|V901p,T|g(t~) + Rg({)) + [p,T — n) vp,T : 0.

32

Hence,

0 E ngu'nT = _2(:r — f) thgm + v9(t)v9(t)1p,T_2 g(t)2 upfl.’

—1—29 90

where the last equality follows from a formal computation as in Proposition 9.1 [26].

Because of ”p,T > 0, we conclude
. t t
13sz + vgovgoifl—Q ——t.)-_g g(t).. — 0. (3.19)
Plugging the trace of (3.19) into (3.18), we obtain

01
7%? = ivgmlflmlgm. (3.20)

By Remark 2.2.4, this implies that (M 7”, g(t), 1p,T( - ,t)) is a gradient shrinking soliton

in canonical form. C]

Remark 3.4.5 We will discuss some issues arising with the inverse of the statement

in Corollary 3.4.3 (iii) in Section 3. 6.

3.5 Examples

3.5.1 Einstein solutions

For Einstein solutions (Mn,g(t)) on (0,T) with 129(0) > O as in Example 2.2.1, it

follows from an explicit computation (see [6]) that for (p, to) E M n x (0, T)

arctan $95)}.

d2( (p,q
lp,t0(‘110 : g(1_ _ (3 21)
t -t
TQ-TO —7t-p_—f)(/ jarctanflqtl— “to

we can easily conclude the following: if t,- /' T, then

C

 

 

l n
pati —" E

33

uniformly on M n x [0, T). In particular, any reduced distance based at any (p, T) is
given by

Tl

lp,T(CI1 5) = 5 (322)

Then the differential inequalities in Corollary 3.3.5 become equalities for the constant
function lp,T, because of equation (2.5). In particular, 171,1(0 E const. Let us

compute its value using (2.3):

p,T(t) = (4vr<T—f))‘3e‘3vw<g(t‘>>
= (47reT)—%Vol(go)

R n
= (%)2Vol(go) (3.23)

For the standard shrinking round n-sphere (8", 957;). we therefore get

n(n — 1)
2nen

 

)%V0l(gsn) = (n2;el)%Vol(gSn).

Vail-(a = (

In the following proposition, we show that the condition lp,T E % characterizes

Einstein solutions uniquely.

Proposition 3.5.1 Let (MR,g(t)) be a maximal type A Ricci flow on [0,T),
p E M n and lpflw a reduced distance based at singular time (p, T). Then we have
the following:

c
T—t

 

(i) If [p,T E c on M" x (0,T), then Hg“) = , c _<_ , and Mn is compact.

MI:

(ii) [p,T E E if and only if (Mn,g(t)) is a (compact) Einstein solution with

Proof. (i) It follows directly from (3.13) that R9“) = TC——t’ and combined with (3.1)
that c _<_ n. The compactness of M in follows from the finiteness of Vp,T by Remark

3.4.2.

34

(ii) We have seen for Einstein solutions [p,T E 3. On the other hand, if lp,T E 13-,
then it follows from (i) that (3.1) becomes an equality, and from the proof of
Corollary 3.4.3 that the corresponding reduced volume Vp,T is constant. We con-
clude (Mn, g(t)) is a gradient shrinking soliton with potential function 3, i.e.

1

—2(T _ t)g(t). Cl

Ricgfi) =

3.5.2 Gradient shrinking solitons

Recall from equation (2.11) that the potential f (t) of a gradient shrinking soliton in
canonical form (M n, g(t), f (1‘)) satisfies the differential equality

8f n

77 _ 9‘2“” + 'Vg(t)f(t)'§(o ‘ Ram + if?) = 0.

Since by Theorem 3.3.1 the reduced distance based at singular time lp,T for some
p E M 7" satisfies the same relation as an inequality, it is natural to attempt to relate
f(t) and lp!T(-,t).

Before we proceed, we discuss a normalization of the soliton potential, which pre-
serves its canonical form. Let (M n, g(t), f (t)) be a complete gradient shrinking soli-

ton in canonical form on (—oo, T) with bounded curvature. Recall equation (2.14):

Rg(.)+1v9<‘>f(t>I§(,) — £9; = C(t)

We can use this equation to normalize the soliton potential f (t), a thought which
was triggered in discussions with Xiaodong Wang. To keep the soliton in canonical

form, however, we only normalize such that
T—1 2
Rg(T_1)+iV9( MT — 1>Ig(T_1) — f(T — 1) = 0. (3.24)

and replace the original soliton potential f (t) by the gradient shrinking soliton in
canonical form obtained from f (T — 1), as discussed in Remark 2.2.4. From here on,

we will assume a given gradient shrinking soliton is normalized by (3.24).

35

Remark 3.5.2 We note the gradient shrinking solitons in canonical form
(Mn.g(t),lp,T(~.t)) arising in the equality case of the reduced volume based at
singular time (P, T) are in fact normalized for all t E (0,T), i.e. satisfy

_M=o

Rg(t) + Ivg(t)lp,T( ' 10'3“) T —t

This follows directly by plugging (3.20) into (3.13).
We will next show that for compact gradient shrinking solitons f = lpflw holds.

Theorem 3.5.3 Let (1)1”,g(t),f(t)) be a compact normalized gradient shrinking
soliton in canonical form on (—oo,T), and let p E M. If lpf is a reduced distance

based at singular time (p, T), then it is independent of p and

lp,T( ' , t) = f(t)-
Remark 3.5.4 While the theorem is mentioned in the literature [10], [3], we are
not aware of any published proof. There are however results [6] using an alternative
definition of a reduced distance based at singular time l, which we will briefly discuss
in Section 6.1. The results can be improved to obtain f = l, rather than f = l + 0,

when using the above normalization. Our proof utilizes techniques from Section 4.1.

Proof. We know from (2.7) that

g(t) = (T - t)¢2‘9(T — 1). (3-25)
In particular g(t) is of type I. For p E M 7‘, let [p,T and Vp,T a reduced distance
and corresponding reduced volume based at singular time, respectively. We refer to

Section 4.1 for more details of the following argument: For tj /' T, consider the

rescaled metrics pointed at p and compute using (3.25):

 

at) = Tltjg<r+(T—tj>t>
= T i tj (T —— (T+ (T — tj)t)) ¢}+(T_tj)tg(T -1)

 

= _t¢T+(T—tj)tg(T ‘ 1)-

36

With the definition of Cheeger-Gromov convergence, this implies that the rescaling
limit (.Mgo,goo(t),p30) is compact. In particular, we can assume Mgo = M" and
P00 = P-

Next, note that gj(—1) = 65?]. g(T — 1), i.e.

9j(—1) '3 g(T - 1)

are isometric for all j. We can therefore identify the reduced distances defined on the

rescaled metrics by
using the diffeomorphism. After shifting time from T to O, we have

i 9:
[13,0 — p,T.

Next, we conclude by Cheeger-Gromov convergence

goo(-1) '-—‘-’9(T— 1).

which gives us the identification

where lgoo is the limit as extracted in Section 4.1 from the reduced dis-
tances based at singular time 11])0 of the rescaled metrics. By Theorem 4.1.3,
(i\1&,gx(t),l;f30(-.t)) 2’ (Mn,g(t),lp‘T(-,t)) is a normalized gradient shrinking
soliton in canonical form.

To see lp,T( - , t) = f (t), note that both potentials satisfy equation (2.8) for the

same Ricci flow. Hence, hence
v9<t>v9<t><lp,r( - ,t) — f(t)) = 0.

As observed by Xiaodong Wang, this implies lp,T( - ,t) = f(t) +c, unless the manifold

splits off a line in the direction of V9(t)(lp3T( - , t) — f(t)), which we can exclude since

37

we assumed compactness of Mn. Since both soliton potentials are normalized, we

conclude

Given a normalized complete gradient shrinking soliton in canonical form
(M n, g(t), f (t)), in general for a base point p E M" and a reduced distance lp,T
based at singular time, we cannot expect to conclude f (t) = [p,T( - ,t). The following

example (see also Figure 3.4) illustrates, that the choice of base point p is one issue.

Example 3.5.5 Let (N"_k,gN(t)), 0 < k < n — 1 be an Einstein solution on
(——oo,T) with R90 > 0. As in Example 2.2.8, we consider the normalized gradient
shrinking soliton in canonical form given by

IIlEle Tl — k
+ ),
4(T — t) 2

 

(W‘k x R’“. g(t) —-— we) +9Rk, new) =

where 6 E N714“ and :r E W“. Note that the constant 15—!“ normalizes the soliton by
equation (3.24).

We can then conclude by the obvious additivity of the L— distance for product
manifolds, as well as Example 2.4.2 and (3.22), that for p = (90,$0) and
q = (6,510) 6 MM x nk

2
l' qt = —— ————-u
P’qu' ) 2 + 4(T—t)

Hence, the equality f(t) = [p,T( - ,t) holds if and only if we choose the base point
p E Nn—k x {0}.

Using Example 2.4.2 and {3.23), the corresponding reduced volume based at

38

singular time can be directly computed:

Vflu) an_k [112k (47r(T— t))_%e_lpsT(0’x’t) dvong(,)(e) dxk

_n—k _n—k
[Nu-k (47r(T — t)) Te T dvolgN(t)

lx~xol2k
. ka(4r(T-t))-'§e_ 4ZT—ti dark (3.26)
R n—k
= (57%)TVng0) (3.27)

In particular, it is constant in t, independent of the base point p and the sequence
{ti} used to define lpflw, as well as equal to the reduced volume based at singular time
for the Einstein part (N ”’k , gN(t)) of the soliton.

We also make the observation that while f 76 lpgw, we do have

VP,T(t) = /Nn—k [Ric (47r(T —— t))_zzle_f(0’$’t) dvolgN(t)(0) dzrk, (3.28)

so in examples of this kind, we can define the reduced volume in terms of the soliton

potential.

 

A
a

----—--¢-’

 

% ' 1R"

 

Figure 3.4: Einstein solution on N71"k x Rk as an example of a normalized gradient
shrinking soliton in canonical form with f (t) 75 [p,T( - ,t) + c.

While the choice of a suitable base point p in the very special Example 3.5.5 is

obvious, that choice of base point p E M" such that f = 1191 might not even exist

39

for general complete gradient shrinking solitons, but depend on the structure of the
soliton, and in particular the critical points of f.

For complete gradient shrinking solitons with quadratic curvature decay and non-
collapsedness assumption, which converge to an incomplete metric cone which is
smooth away from the parabolic vertex (consisting of a collapsed compact set), Cao,
Hamilton and Ilmanen state in [3] that the reduced distance and volume based at
singular time are well-defined. Also, this reduced volume is constant. They use this
result to define a notion of density for such shrinking solitons, which we generalize to

type A solutions in Section 4.2.

3.6 The inverse of Corollary 3.4.3 (iii

Recall that Corollary 3.4.3 (iii) stated that if a reduced volume based at singular time
is constant, the corresponding Ricci flow is a gradient shrinking soliton in canonical
form (which is in fact normalized).

The inverse statement holds, whenever for some p E M n, we have f = [p,T' More

generally, the following is true:

Proposition 3.6.1 If (Mn,g(t),f(t)) is a gradient shrinking soliton in canonical
form on (~oo,T), and f(t) = lp,T( - ,t) + c for some p E M", c E R, then Vp,T(t)

is constant in t.

Proof. From Example 3.2.2, we get that the type A assumption is satisfied. Since

lpflw = f is smooth, the inequality (3.1) becomes the equality (2.11), i.e.

* 1 _
From the arguments in the proof of Corollary 3.4.3 (i), we conclude

—v (t) = o. C]

40

Remark 3.6.2 As discussed in Section 3.5.2, compact gradient shrinking solitons
in canonical form will satisfy the assumptions of the proposition for any p E M n,
even without being normalized. Similarly, this is the case for solitons of the form in

Ezrample 3. 5.5 with the correct choice of base point, as well as the ones in [3].

41

CHAPTER 4

Applications of monotonicity based

at singular time

One of the main applications of monotone quantities in geometric evolution equations
is the analysis of singularities. We will first show in Section 4.1 how the notion of a
reduced distance based at singular time can be used to model type I singularities by
gradient shrinking solitons. Results of that kind are very important, as they reduce
the study of singularities in the Ricci flow to the classification of gradient shrinking
solitons.

In Section 4.2, based on the reduced volume and its extension to singular time,
we will define a notion of density for any point in space-time in the Ricci flow. This
is motivated by the Gaussian density in mean curvature flow based on Huisken’s
monotonicity formula [20] [34], which can abstractly be regarded as an analogue to

the monotonicity of a reduced volume based at singular time.

42

4.1 Rescaling limits of type I Ricci flows are
gradient shrinking solitons

We will describe in this section one of the main motivations for extending the notion
of the reduced distance to singular time, as done in Chapter 3.

Let (Mn,g(t)),t E [0, T), T < 00 be a complete maximal type I Ricci flow, and let
p E M n. We rescale the flow parabolically: Let tj /‘ T and consider the sequence

of pointed 1-parameter families of Riemannian manifolds M n, g- t ,p , where
.7

 

gJ-(t) := Titjg(T+(T—tj)t). (4.1)

Then g.- t is a Ricci flow on — T ,,0 for each j. Because of the type I curvature
J 7 _ J

bound, we have at any x E M n that

lngjtting-(tflx) - (T—ti)lng(T+(T—tj)t)l9(T+(T—tj)t)("’)

C
s (T—tj)T_ (T+ (T—tjlt)

 

(4.2)

C

t.

This gives a uniform curvature bound on compact subsets of (—oo,0). Together
with Perelman’s no local collapsing theorem (which here also holds for complete M n
because of the uniform (in t) lower bound on the reduced volume as described below),
we can use the Cheeger-Gromov-Hamilton Compactness Theorem lB.O.2 to extract a
complete pointed subsequential limit Ricci flow (M30, goo(t),poo) on (—00, 0), which
is still type I. An example of the convergence of rescales around a neckpinch to a
shrinking cylinder is shown in Figure 4.1.

As ancient solutions are expected to have a very special structure, Hamilton [18]
conjectured (under a different rescaling procedure) that (Male, goo(t)) is a gradient

shrinking soliton.

43

 

 

 

 

 

 

 

i i >t
O T
M” Z X
9.0) U
4. # >t
T 0
awe) (1: P°°(j: (‘1’?
% >t
0

Figure 4.1: Parabolic rescalings (M 7‘, gj(t)) of a neckpinch (Mn, g(t)) and pointed
Cheeger-Gromov-Hamilton convergence near a singular point p to a shrinking cylinder

44

In general, Mg}, and M" are different smooth manifolds. The definition
of Cheeger-Gromov convergence implies that if M30 is compact, then in fact

Mgg = M n. It follows from [29], that Hamilton’s conjecture holds in this case:

Theorem 4.1.1 The rescaling limit (Mgo, goo(t)) obtained as described above is a

gradient shrinking soliton, if 11130 is compact.

The proof uses the monotonicity of Perelman’s entropy functional, which is only
defined on compact manifolds.

We will now describe our idea on how to use the monotonicity of a reduced volume
based at singular time to obtain a proof of the conjecture without the very restric—
tive assumption of Mgo being compact: Let lp,T be any reduced distance based at
singular time (p, T) for the Ricci flow (Mn,g(t)) on [0, T) as defined in Chapter 3.

For each (q, t) E M" x (—oo.0), consider for large enough j

zgfltq. a := we T + (T - mi). (4.3)

which is a reduced distance based at singular time (p, O) for the Ricci flow ( M n, gj (t))

on l—TT—j’ 0) because of the scaling properties of the reduced distance.

17,3106) = Vp,T(T + (T -— tj)t) (4.4)

We can conclude from (4.4) that

~ ' j—pw . ~
Vito ———* ,ll/rg. are

uniformly on compact subsets of (—00, 0). Hence by Corollary 3.4.3, I7; 0 converge

to a constant in (O. 1]. Figure 4.2 illustrates this convergence.

45

 

 

 

Figure 4.2: Convergence of the rescaled reduced volumes 173 0 based at singular time

to the constant lim I7 t .
t/T p,T()

We would like to conclude that there exists a locally Lipschitz function lg; 0 on

the limit manifold Mg) x (—oo,0), such that

0
l] CIOC I“)
p O pmao’

and which satisfies the differential (in)equalities of Corollary 3.3.5. Since its
corresponding formal reduced volume V290; 0 is constant, those are equalities for

l and we can conclude that (Mgo, goo(t), lg; 0( - , t)) is a normalized gradient

00
190090,
shrinking soliton in canonical form. To do this, we need uniform bounds on the

j

p 0, as in the proof of Theorem 3.3.1. However, those bounds

sequence of functions 1
must survive under Cheeger-Gromov convergence.
In [23], this problem has been worked on independently, and estimates have been

derived to define a “singular reduced length”, which is a reduced distance based

at singular time as we constructed in Section 3.3. Those estimates are derived by

46

comparison with a minimizing geodesic, and hence, are in terms of the Riemannian
distance away from the singular time. As such, they are preserved under Cheeger-

Gromov convergence. In our notation, they read as follows:

Theorem 4.1.2 Let (Mn,g(t)) be a complete type I Ricci flow on [0,T), and let
(p, to) E M” x [0,T). Then there exist 0 > 0 (only dependent on n and the type I

constant) such that

 

 

 

 

 

. 1 0HM) ding)
(2) 50+ tt0_t.>2—cszp,.0(q,asc<1+ :04)?
(a) WWII. i (q bl (q) < C (1+ dim”)

M ’ 9(5) - tO—t sag—4’
8 C d~(p.q)
(2”) la—{lpat0(Q~{)lg(fl(q) S to _ {(1 + \/tt—0——_—t—)2

In particular, those estimates are satisfied by any reduced distance based at
singular time [p,T by replacing to by T in Theorem 4.1.2. For the metrics gj(t)
defined in (4.1), we have the same type I bound (see (4.2)). We can therefore con-
clude from Theorem 4.1.2 that
(15(1). (1))2.

H

In the same way, the other bounds hold and are preserved under Cheeger-Gromov

 

OO
lPoo

Corollary 3.3.5 as equalities and is normalized for the same reasons as explained in

convergence. This allows us to extract 0 as we described above, which satisfies

Remark 3.5.2.
we summarize the above discussions by stating the theorem, which is stated

slightly differently in [23]:

Theorem 4.1.3 Let (Mn,g(t),p), t E [0,T), T < 00, p E M" be a complete maxi-

mal pointed type I Ricci flow. For tj / T, consider gj(t) := Til—{g(T + (T -— tj)t)
J

47

on M n x [~Tg—7, 0). Then there exists a Cheeger-Gromov-Hamilton limit Ricci flow
'3

(AvIgo,goo(t),poo) on (—oo,0) and a potential function if); 0 on M” x (0,T), such

that (M30, goo(t), [8:0 0( - , t)) is a normalized gradient shrinking soliton in canonical

form with bounded curvature.

We saw in the proof of Theorem 3.5.3, that if (M n, g(t), f (t)) is a compact normal-
ized gradient shrinking soliton in canonical form, then the limit soliton constructed

above equals the original soliton, and moreover, the soliton potentials are equal.

4.2 Definition of a density

Consider a type A Ricci flow (M 7‘, g(t)) on [0, T), and let p E M n. For a sequence
t,- /‘ T defining a reduced distance based at singular time (p,T: we have from Defi-

nition 3.4.1 that

we?) = M 3% (((47r(T — t))"i e—z,.,.,<qa) dvozgmtq).

Using the uniform lower estimate on [Pitt by the square of the distance function in

Theorem 4.1.2, we actually obtain from the dominated convergence theorem that

‘7 T09 = lim {Ci-(12.
p3 ti/T I);

which is illustrated in Figure 4.3.
We will now consider points in the closure of space—time, i.e. in M n x [0, T], to

include the singular time.

Definition 4.2.1 Let (Mn,g(t)) be a type A Ricci flow on [0,T). For any
(p, to) E M" x [0,T] and a reduced distance [Pitoi we define a density of (p, to)
in the Ricci flow (Mn,g(t)) by

s =limi7 tEO,1.
p,t0 f/to N00 ( l

48

density loss

 

 

du—
up—

0 T

Figure 4.3: Convergence of the reduced volumes Vpiim to a reduced volume based
at singular time Vp,T(t‘) and density loss.

If to < T, then 9W0 is well-defined since the reduced distance Into is unique,
and we have 6pm = 1 by Corollary 2.4.3. For to = T the density might depend
on the choice of the reduced distance functions, which in general is only defined as a
subsequential limit. If (M n, g(t), f (t)) is a compact gradient shrinking soliton, then
by Theorem 3.5.3 the density 9p,T is well-defined, in fact even independent of the
base point p. More general, for products of Einstein solutions with Euclidean space
as in Example 3.5.5, we also obtain that the density is well-defined by equation (3.28).
This provides a collection of densities E (0, 1] occuring in Ricci flows, and the values
can be computed by (3.26). In [3], Cao, Hamilton and Ilmanen define an analogous
density for gradient shrinking solitons satisfying their assumptions.

In general, for complete type I Ricci flows (Mn, g(t)) on [0,T), it follows from
Theorem 4.1.3 that for p E M n and choice of lp,T there exists a gradient shrinking

soliton in canonical form (ll/{(7310, goo(t), loo(t)), which models the singularity at p and

49

whose potential function satisfies

_n _100
61).?" = Mn(4rr(T—t)) 26 dvolgm

for any t. This is depicted in Figure 4.4. While the integral is constant in t, it is
only a formal reduced volume, i.e. 100 might not be a reduced distance. This could
even be the case when (.M”, g(t)) is a gradient shrinking soliton in canonical form if

the limit soliton is not isometric to it.

GET = /M&(47r(T — t))—3e_loodvolg(t)

for a gradient shrinking soliton
(MgO)QW(t)il°o(t))

  
     

Vp,T(t) for some
Ricci flow (M",g(t))

 

 

 

—q.—-—-—-——--------

H

0

Figure 4.4: Relation between density 612,1" for a type I Ricci flow (M n, g(t)) and
limit gradient shrinking soliton (ll/[30,g(t), l°°(t)).

To distinguish points p E M n where the curvature blows up at the singular time

T, from those where the curvature stays bounded, we make the following
Definition 4.2.2 The singular set E C M" for a maximal Ricci flow (Mn,g(t))
on [0,T) is given by

z := {q e M | Ing(,)(q)]g(,) t—‘i oo}.

50

Remark 4.2.3 The points in the singular set 2 are equivalently those, where the

metric g(t) becomes degenerate as t —+ T.
We can prove

Proposition 4.2.4 Let (Mn,g(t)) be a maximal type I Ricci flow on [0,T) with

singular set )3. If p E Mn\2, then 6p,T = 1.

Proof. Give p E M"\E, we rescale g(t) for a sequence tj / T by

gjm := Titjgm (T —- mt).

 

The bounded curvature of g(t) near p for all t implies the Cheeger-Gromov-Hamilton
limit flow is

(NI&.9m(t)) ’5 (Rnignn).

which by Theorem 4.1.3 is a gradient shrinking soliton in canonical form on fiat 112”.
The soliton equation (2.8) immediately implies this is the Gaussian soliton, and hence,

6

p,T = 1 by the discussions above. Cl

We generally expect that the structure of a singularity at p E 2 determines a

“density loss”, so that 6p; < 1 as shown in Figure 4.3,

51

 

CHAPTER 5

Reduced distance and
monotonicity based at a

submanifold

5. 1 Motivation

Our main reason to study a reduced distance based at a point (p, T) at singular time
in Chapter 3 was the application to singularity analysis in Chapter 4. In particular,
the case where p E E, the singular set of the Ricci flow (M 7’, g(t)), is of interest.
In this chapter, we are motivated by the case where the singular set E is a lower
dimensional submanifold of M n_ In general, not much is known about the size of the
singular set for a Ricci flow. The considerations here were triggered by the example
of a neckpinch as shown in Figure 5.1. We will extend the notions of the reduced
distance and volume from being based at a point p E M n to being based at any
submanifold S C M n.

First, we briefly discuss the neckpinch in Figure 5.1. In dimension n = 2 the

picture is actually false, since the curvature in the neck is negative. For n 2 3 the

52

Mn

Figure 5.1: Formation of a neckpinch.

existence of symmetric neckpinches has been rigorously shown in [1]. The authors
identify the n-sphere Sn minus two points with (—1,1) x S"—1 and consider the

evolution of a metric of the form
9 = amide + $602994. (5.1)

where as before 9371-1 denotes the standard round metric on S"—1 (see Figure 5.2.)

311— 1

    

Sn—l Sift—1

i i > t

O T

Figure 5.2: A symmetric neckpinch.

For a neckpinch the singular set 2 is of lower dimension than the manifold M 7".
Motivated by this, we will study .C—geodesics, which are minimizing the .C—length

to any given k—dimensional submanifold S of M n, where k < n and (Mn, g(t)) is

53

a Ricci flow on [0, T). We show that the monotonicity of the corresponding reduced

volume still holds, and will also extend it to singular time.

Remark 5.1.] Besides the symmetric neckpinch, there are other examples where
similar pinching behavior occurs on compact lower dimensional submanifolds of non-

compact manifolds, see e.g. [33/ and [13].

5.2 Reduced distance based at a submanifold

In this section, we study the reduced distance based at a lower dimensional sub-

manifold S of M”.

Definition 5.2.1 For a Ricci flow (Mn,g(t)) on [0,T) and a compact (not neces-
sarily connected) embedded k— dimensional submanifold S C M", 0 S k < n, we

define for given to E (0,T) and any (q,t) E M" x (0,t0)

(i) the L-distance from (q,t) to (S, to) by
L ,t := inf L ,t ,
s,.,<q 7 p€S{ mu 3}

(ii) the reduced distance based at (S, to) by
l ( L) ._ LS,t0(Qat7
S,t0 q, ° 2W)
(iii) and

_n —z ,t
vs.to(q.t) == (4WD - t)) 96 S"0(q 3-

Remark 5.2.2

(i) The compactness of S implies that the infimum in (i) is in fact assumed for
some p E S, i.e. L5,t0(q,t—) is given by the L—length of an £— minimizing

.C—geodesic from (q, t) to (p, to) for some p E S as shown in Figure 5.3.

54

 

 

eI-I-
(.5
O

'8

Figure 5.3: The .C—distance from (q,t) to (S, to) assumed by an .C-minimizing
L—geodesic 7 for some Ricci fiow (Mn,g(t)) with submanifold S C M".

(ii) It follows trivially from Definition 2.3.3 that

l ,t = inf l ,t
S,t0(q 3 p€S{ p,t0(q 7}
and
vs,t0(q. 1f) = sup{vp,t0(q. 0}-
p68
To justify the picture in Figure 5.3, we first state and prove the following
Lemma 5.2.3 We make the assumptions as in Definition 5.2.]. If 7 : [t,t0] —>

M n with 7(0 = q and 7(t0) = p E S is an .C— minimizing L— geodesic such that

[3(7) = L5,t0(q, t) then 7 is orthogonal to S at t = t0.

Proof. By the assumption [2(7) :2 L s,t0(q, t), we consider the following variational
problem: Let

CD I (—c,e) x [1: tOl _. M"

be a (Cl) variation with

¢(S,t0) E S,
its?) = q.
and (MW) = alt).

55

Then the first variation formula (2.15) for [I at the [I—geodesic 7 implies

d
0 = we) —

“a; £<¢<s¢>>=<2 to—tv<t>,Y<t>>g(t) go, (5.2)

 

320
where

W) = gauges),
”D = 0.

and Y(t0) E T'y(t0)S'

Recall that by Proposition 2.3.2, the initial vector

w 2: lim t0 — t7(l) E T7(t0)M

t—9t0

exists. Hence, we conclude from (5.2) that

0 = tlim ( t0 '- t7(t), Y(t)>g(t) = (w, Y(t0)>g(t0)'

—+t0
This holds for all Y(t0) E T'y(t0)S (arising from any variation (b as above) and implies

that w E (T7(tO)S)i, the normal space to S at 7(t0) in (Mn,g(t0)). Cl

Remark 5.2.4 The case k = 0 is included in the above proof; S is then a finite

collection of points in M n.

The main result of this section is that the three differential (in)equalities for [P.to

in Theorem 2.3.4 also hold for lS,t0 :

Theorem 5.2.5 Under the assumptions of Definition 5.2.], let to E (0,T). Then
for all (q,fl E M" x (O,t0) the following three (in)equalities hold:
n

2(to - D 2
(5.3)

3

56

or equivalently , D;(Di'5,t0(q, D S 0. (5.4)

lS,t0 (q: B _ n
to — f

 

—|v9®15,,0(q, mgm + 12mm) + + 2Ag({)lg,t0(q,t) g 0. (5.5)

[3,150 (q, f) _

t0 _ f 0. (5.6)

6
“25-15mm, a + Ivgmzmq. Olga) — 3905(0) +

Remark 5.2.6 Similarly to the reduced distance [PJO’ the reduced distance based at
a submanifold 13,150 is locally Lipschitz on M n x (0,130), and in fact smooth away
from a closed set of Lebesgue measure zero for all t. Hence, the (in)equalities hold in

the weak sense, or in particular in the sense of distributions. Compare Remark 2.3.5.
The proof of Theorem 5.2.5 will follow from the next two propositions.

Proposition 5.2.7 Under the assumptions of Definition 5. 2.1, let to E (0, T). Then

for all q,t) E M” x (0,t ) we have
0
_3
(2') Ivzs,.,<q, 0:2 = —Rg(a<q) + Fly-15mm. t‘) - (to — 2?) MM),

(a) 5353mm = ~Rg(a(q> + flames — in — a Name,

where
t
Km. t‘) == h 0(to — 0% Hunt) dt,

3 I .
H(X,t) := aRglt) + ERQU) - 2(VR,X) + 2R2C(X,X),

and 7(t) is as in Lemma 5.2. 3.

Remark 5.2.8 The equalities in the proposition were observed for lp,t0 by Perelman.
H (X ,t) is Hamilton’s Harnack expression, which is nonnegative for all vector fields
X along any complete Ricci fiow (Mn,g(t)) with bounded nonnegative curvature
operator. We will not need any properties of this expression, as will become clear in

the proof of Theorem 5.2.5.

57

Proof. It follows from Lemma 5.2.3 and the first variation formula (2.15) that

with qt‘>=—2 to— (5.

Hence,

IVLs,.0(q. $112 = 400-75) 171512 = —4<to—DR,777 +4<to—£)<R,(77 +1712?» (57)

We also get

0 d .

= —(/t0-t (|7(i)|2 +R g((qt') ))+2(/t0—t t|7(t)|2
= —2\/to—t‘R,(77<q>+ to—M<,(.7<<n)+l7®12>. (58)

we can use Perelman’s [26] computation along the .C—minimizing L—geodesic 7(t)

as in Lemma5 ..2 3, which in our context reads

(to—MR, so am (£11) —K(7,a+§Ls,.O<q.t‘>. (5.9)

Plugging (5.9) into (5.7) and (5.8), we obtain
2_
IVLs,t0(q q.=t7| will? g({)+\/tO—t( 2 L,(qSt0 (qt)- K( N?)

8 1
and 18L5t0(q,L)— — —2\/l0-l lR W (2",(CILSt0 CIT)- (7,5)):

and therefore

I _3
le,t0(qa{) — (t0 _ f) 2K(71{)

2 _. _
lVls,t0(q.t)| — Rgmm + to

0 I 1 _3
and dt‘lS,t0(q (la: g(jlq (1+) —lS,(qt0 (q‘i) — 5(t0_0 2K<71Ph D
To obtain the monotonicity result of the next section, we will now show that the

crucial differential inequality (5.4) holds for the reduced distance to a submanifold.

58

Proposition 5.2.9 Under the assumptions of Definition 5. 2.1, we have

ngvs,to(q,a S 0
in the weak sense.

Proof. We argue by contradiction: Assume there exists a (small) parabolic cylinder
P = U x [t2,t1) C 1171’" X (0, to), U open, such that for all 0 S d) E Cgpt(M" x (0,t0))

with support in P

ff}? v3,t0Cl¢ dvolgmdt > 0.

Inverting time (r := t0 — t) implies that —v5,to is strictly subparabolic in the weak
sense of Friedman [14], and we will apply his (strong) maximum principle several

times to derive a contradiction. By Remark 5.2.2 (ii)

”S,t0((17t_) = sup{vp,t0(q. t7}-
1265'
Let p E S, up :2 vp¢0(q,t7 and ”S := vS’tO(q,t) Then up 3 ”Slot and we know
from (2.19) that
l:l*’Up S 0
in the weak sense. Now let P := P\P and wp be a solution to
Cl*'l.Up = O In P

wplp = 7.17.

Hence,

[Ni/p ‘ wp) S 0,
and the maximum principle implies
Similarly, let w S be a solution to

CHIHS = 0 in P

wslr = vslr-

Since

D*(wp — HIS) : O
and wplp = vplp S vslp = wslp, the maximum principle implies that
wp S wS in P. (5.11)
As p E S was arbitrary, we conclude from (5.10) and (5.11) that
vs S wS in P.

Using v3|p = w SH“ and the maximum principle again, this contradicts the assump—

tion that in the weak sense in P
[3*(125 — 105) > 0,
which finishes the proof. El

Remark 5.2.10 In the above proof, we only needed the weak maximum principle,
which follows trivially from Friedman’s strong maximum principle. Moreover, by
Remarks 2.3.5 and 5.2.6, we could have assumed vp and v5 are smooth and P,

and purely worked in the smooth category.

Proof of Theorem 5.2.5. The first differential inequality is equivalent to Proposi-
tion 5.2.9 by definition of ”S,t0-

We obtain equation (5.6) by subtracting (i) ——2- (ii) in Proposition 5.2.7, since then
the term involving K (”7, t) cancels.

Finally, the inequality (5.5) follows as —2-(5.3)+(5.6). [3

Remark 5.2.11 By combining the ideas of Perelman’s original proof for the first two
equalities with the comparison principle proof for the differential equality, we avoid a
lengthy discussion of a Laplacian comparison theorem for L SatO in the setting of the

submanifold S.

60

5.3 Reduced volume monotonicity based at a
submanifold

The purpose of this section is to show the monotonicity of the reduced volume corre-
sponding to the reduced distance based at a submanifold S. Therefore, we now make

a further generalization of Perelman’s reduced volume.

Definition 5.3.1 Under the assumptions of Definition 5.2.1, we define for each

t E (O,t0) the reduced volume based at (S, to) by
VSxOlD I: [M ”3x001, 15) dwigmkfl-
By definition, we have for any p E S and t E (0, to) that
175,700) 2 Vpxofll.

To guarantee the existence of the reduced volume based at (S, to), we prove the

following
Lemma 5.3.2 I75,t0(t—) as in Definition 5.3.1 is finite for any fE (0, t0).
Proof. Fix t E (0, to). For any q E M, there exists a p E S such that

The known estimate for the reduced distance in terms of the Riemannian distance at

time to (see e.g. [36]) gives

lp,t0(q1t_) Z Cldt20(p1q) _ CZ,

where Cl > 0 and C2 > O are constants only depending on t and the curvature
bound. Using the equivalence of the metrics along the Ricci flow on [0, to], we obtain

for a possibly different constant Cl (with dependencies as above) that

15.10% t‘) Z Ciditp. q) - 02,

61

and thus
mom 2 oldest) — 02. (5.12)

The finiteness of V3206) now follows from the convergence of the integral

—d3<Sq)
e t ’ dvol (q) < 00
/M g(fi
for compact S C M”. C]

Corollary 5.3.3 (Monotonicity based at a submanifold)

Under the some assumptions as in Definition 5.3.1, we have
. (1 ~
(U Emma) 2 0.

(ii) If stOU—fl = 29,1092) for 0 < t.1 < t’2 < to. then (Mn.9(t).ls,tO(-.t)) is a

flat gradient shrinking soliton in canonical form.

Proof. This is very similar to the proof of Corollary 3.4.3 (i) and (iii), since the
differential (in)equalities in Theorem 5.2.5 hold. The technical ingredient in the
setting here is the comparison of ls,tO(q,fl with dt2(S, q) in (5.12).

To see the flatness of the gradient shrinking soliton (M n, g(t), lS,t0( . ,t)) obtained

in (ii), we first use Remark 2.2.4: For t E [to — 1, to], we have

g(t) = (to - t)¢2‘9(to -1)-

Together with the fact that the curvature of (Mn,g(t)) is bounded by C on

[to — 1, to], we obtain

3],? lng(t0_1) |g(t0_1) = ski/[p lRmabz‘guo—l) lab? 9(t0-1)
= R
sxlpl m%g(t)lzoi_fglt)
= (t0 — t) 8:11p |ng(t)lg(t)

_<_ (t0 — l.)C.
Letting t —7 t0 finishes the proof. Cl

62

5.4 Reduced distance and volume based at a
submanifold at singular time

The purpose of this section is to show that similar to Section 3.3, we can define
a reduced distance and volume based at a submanifold at singular time under the

type A curvature bound assumption.

Theorem 5.4.1 Let (Mn,g(t)) be a Ricci flow on [0,T) of type A, and S C M"
be a compact (not necessarily connected) embedded k— dimensional submanifold,

O S k < n. Let also t,- /‘ T. Then there exists a locally Lipschitz function
$1]:an X (0,T) —> IR,

which is a subsequential limit

1 cfocmnx (0,T); l
S,t2' ' S,T

 

and satisfies for all (q, t) E M n x (0,T) the following differential (in)equalities:

 

a r t 2 n
(5.13)
or equivalently for 1151((1, t) :2 (47r(T -— t))—ge—lsvflq’fl
D;(Dvs,T(q,D S O. (5.14)
l (q) i) _ n
—|Vg(tlls,:r(q. W33) + 1396701) + S’TT _ t- + QAgmlst) S 0- (5-15)
5 l (at)
452-15101. 5) + Ivgmlmq, 221370 — Hymn) + S—fi—_—_,—— = 0. (516)

Proof. This proof is now almost identical to the proof of Theorem 3.3.1. For the
uniform bound on LSJZ',’ we can use the same estimate: For p E S, (q, f) E Mnx (0, T)
and 118,15”an Z 0, we have

lLS,ti(q1t_)l S le,ti(q1£)l'

63

If L San-(‘1’?) < 0 the bound follows from the same argument as in the proof of
Theorem 3.3.1, with 7(t) being an L—minimizing .C—geodesic from (q,t) to (S, ti)-

Similarly, the derivative bounds follow by letting 7,-(t) in Lemma 3.3.4 be an
£—minimizing L— geodesic from (q,t) to (S, til With the results about lS,t0 we
have stated earlier, in particular Lemma 5.2.3 and Theorem 5.2.5, the rest of the
proof follows. Cl

With Theorem 5.4.1, we can make the following
Definition 5.4.2 Under the assumptions of Theorem 5.4.1 we define
l5] : M" x (0,T) —» IR

to be a reduced distance based at singular time (S, T) in the Ricci flow

(M ". g(t)).
Next, we define the corresponding reduced volume.

Definition 5.4.3 Under the assumptions as in Theorem 5.4.1 we define a reduced

volume based at singular time (S, T) by

Vsr® == fM Us,x(q, deolgmml

Remark 5.4.4 Contrary to Remark 3.4.2, the finiteness of 173,776) does not follow
by the same argument. However, we can transfer the estimates from Theorem 4.1.2
to the submanifold setting, in the same way as we did in the proof of Lemma 5.3.2.

We conclude that for every f, Vsz-(D subconverge to V5,T(t_), which is finite.

Now we state the monotonicity for this reduced volume.

64

Corollary 5.4.5 (Monotonicity based at a submanifold at singular time)

Under the assumptions as in Definition 5.4.3, we have
' it? t) > 0
(2) d, s,T( _ 7

(ii) If V5161) = V5162) for 0 < t_1 < t”2 < T. then (M".g(t).ls,T(-t)) is a

gradient shrinking soliton in canonical form.

Proof. This is now standard. D

5.5 Examples

We consider examples, to indicate that the structure of the submanifold S in the
Ricci flow (M 77’, g(t)) determines the behavior of the monotone quantities defined in

the previous sections.

Example 5.5.1 Consider the Ricci flow on [0,T) from Example 3.5.5 on

M” = Nn-k >< Rk, where Nn—k is an Einstein solution. Let
S = N71",c x {x0} C M”.

Then we use (3.21) to compute for t,- E [0,T), p = (190,130), as well as

q = (0,2) 6 N"—k x W“

l.,t=infl.,t
S,t,(q -) pES p,t,(q d

 

 

l3: "’ (”0'2 k
= inf 1 .(6,t)+ ——3—
soeNn-k 90‘” 4051 - 5)
— n—k 1 are an f—_ti)+|x—x0]Rk
2 t~—-t 4(ti — f)

65

In particular, we find that for any p = (60,x0) E 1W

2
lS,T(q1 a = 2 + 4(T _ a = lp,T(q, [)1

and therefore, as computed before,
~ R90 3 ~
V510 (2%”) V0 (90) V7510.

i.e. we can say that

 

R90 2
95,1" - (27m) V01(90) — 9px-

Alternatively, in Example 5.5.1, we can immediately see from Lemma 5.2.3 and
the [I—geodesic equation that [I—minimizing L-geodesics from (q,t) to (S, to) are
constant in N ”‘k . This is similarly the case for a symmetric neckpinch as discussed

in Section 5.1. In fact, the symmetries in equation (5.1) imply

lp,T = 12,2"

for a symmetric neckpinch, where p E E. For general Ricci flows, however, we do not

obtain this.

66

CHAPTER 6

Conclusions

6.1 Reduced distance based at a point at singular
time

In Chapter 3 of this dissertation, we have extended the reduced distance to base points
at singular time, and have shown that the associated reduced volume is monotone. We
can make the following alternative definition of a reduced distance based at singular

time, which we denote by l.

Definition 6.1.1 Let (M",g(t)) be Ricci flow on [0,T) of type A. For p E M”, we
define
1 . T . 2
lat“) .— We /, fT“ 1707111197,, + 1297771710» at,

where the infimum is taken over all (piecewise C1 ) curves
7 = [LT] -> M”. 7(6 = q. 7(T) = p-

The infimum is well-defined, since the functional is bounded from below because of the
type A assumption. In general, however, l (q, B will not be assumed by a minimizing

curve 2' : If. Tl -* M”. W) = €1.7(T): 11

67

Understanding the relation between l(q,t) and [131(an might help to answer
both, the question of uniqueness of lpflw, i.e. independence of the sequence {ti}
used to define it, as well as the question of the dependence of lpJ‘ on the base
point p E M". This is of particular importance in the case of gradient shrinking
solitons, where the relation between the soliton potential f and a reduced distance
[p,T based at singular time needs to be further understood (in the noncompact case).
The failure of f to be equal to lpgw seems to contain geometric information about
the structure of the soliton, in particular the relation between the critical points of
f and the choice of the base point p. Here, both the comparison geometry and the
partial differential equations points of view of the reduced distance should be helpful.
Techniques involving the reduced distance based at singular time, similar to the ones
in this dissertation, are also a main tool in [23] to classify 4-dimensional solitons of
nonnegative curvature operator.

We discussed applications to the singularity analysis using [p,T in Chapter 4.
The uniqueness of the reduced distance based at singular time would imply a well-
defined notion of the density 611.7" in Section 4.2. Regularity theorems similar to
W hite‘s local regularity theorem for the mean curvature flow [35] should then hold,
i.e. using localized quantities one should get a generalization to singular time of the
local regularity theorem in [25].

One can obtain similar results to those in this dissertation for the forward reduced
distance and volume as defined in [12]. In this case, gradient expanding solitons
replace the role of gradient shrinking solitons. We will not include further details

here.

68

6.2 Reduced distance based at a submanifold

We have generalized the reduced distance to be based at a compact k—dimensional
submanifold S, O S k < n, of a Ricci flow on M” in Chapter 5, and extended
the associated reduced volume monotonicity to singular time. We mention that
one can also develop the theory of LS—geodesics similar to Perelman’s theory of
£—geodesics. In particular, for each t’ E [t, to), the time t’ [IS-exponential map

is
.C . _L n
Sexpt/ . (TS) ——> M ,

(P, U) H 7p,v(t,)7

where p E S C M", v E (TpS)J' E’ Rn—k and 719,77 is the C—geodesic with
7p,v(t0) = p and initial vector v at to. One can then study the [IS-cut locus and
[IS-Jacobi fields to obtain the expected results.
Similarly to the reduced volume based at a point at singular time, we expect
{Ii/Hf], Vp,T({),
if finite, to carry information about how the submanifold S evolves under the Ricci
flow on M n. We can then define a density of a submanifold in the flow. When S = Z},
the singular set, this limit is related to the structure of the singularity as is the case
for the reduced volume based at a point in the singular set at singular time. If the
singular set does not collapse completely (unlike a neckpinch), this quantity should
provide insight which the reduced volume based at a point cannot provide. Finally, we
expect that one can weaken the smoothness assumptions on the set S, in particular

to include more general (compact) singular sets.

69

APPENDIX

70

Appendix A

Shi’s derivative estimates

For the convenience of the reader, we give the following statement of Shi’s local

derivative estimates [32], as it can be found for example in [7]:

Theorem A.O.1 For any a,K,r > O and n,m E N, there exists a constant
C = C(a,K,r,n,m) such that if M" is a manifold, p E M", and g(t) is a Ricci

flow on U x [0,T], 0 < T < 1%, where Bg(0)(p, r) C U open, and if
[ng(t)(x,t)|g(t) S K for all x E U and t E [0,T],
then

C(a, K, r, n, m)
77?.
t2

 

lvamg(t) (ya t) [g(t) S

for all y E Bg(o)(p, 5) and t E (0,T].

71

Appendix B

Cheeger-Gromov—Hamilton

Compactness Theorem

For statement and proof of the following compactness theorem, see for example [7]. It
is Hamilton’s adaptation of the well-known Cheeger-Gromov Compactness Theorem

from Riemannian geometry to the Ricci flow.

Theorem B.0.2 (Cheeger-Gromov-Hamilton Compactness) Let
{(i’l’Iz-T‘,gi(t),xi)},t E (a,w), be a sequence of complete pointed solutions to the

Ricci flow and to E (a,w). Assume that

(i) llegiltlllgiU) S C on Mi" x (a,w) for some constant C independent ofi

and
(ii) injg,(t0)($il Z (5 > 0 for some constant 6 independent of i.

Then there exists a subsequence of {(Mzin, gi(t),xi)}, which converges to a complete

pointed solution {(ll/Igo,goo(t),xoo)},t E (oz,w), to the Ricci flow and

lRm(QOO(t))lgoo(t) S C on M30 x (a,w).

72

The convergence is in the Cp(goo(t0))-topology for each p Z 0, i.e. there exists a
sequence of open sets U,- C 11ng with x00 E U7, U,- C 7+1 and UiUz- = 11130, and
there exists a sequence of difleomorphisms F,- : U,- —-+ Mil with V,- := Fi(Uz-) and

F.7(xoo) = x,- such that for every compact K C Mg, and I C (a,w)
Fz-*g,'(t) —> goo(t) in the Cp(K x 1,900) topology .

The injectivity radius bound needed to extract the limit flow generally follows from

Perelman’s “No Local Collapsing Theorem.”

73

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74

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