Ricci Flow on Poincare' and Thurston's Geometrization Conjecture

Home , Ricci flow

Ricci flow on Poincare' and Thurston's Geometrization Conjecture

A Dissertation, Presented

Hassan Jolany

The Graduate School

in Partial Fulfillment of the

Requirements

for the Degree of

Master of Philosophy In

Mathematics

Supervisor:M.R.Darafsheh

University of Tehran

2010 Contents chapter1: Introduction………………………………………………………………..1 chapter2:

Ricci flow…………………………………………………………………9

chapter3:

overview on Poincare conjecture………………………………………..28

chapter4:

maximum principle…………………………….………………………...51

chapter5:

Li-Yau-Perelman estimate…………………………..……………….…..69

chapter6:

Two functional ĐQ ġ of Perelman………………..………………..78

chapter7:

Reduced volume and reduced length………………………….……….102

chapter8: k-non collapsing estimate ……………………………………….…..129

Biography ………………………………………………………….141

INTRODUCTION V{tÑàxÜD

Chapter 1 [PRILIMINARIES]

Chapter1

Preliminaries

We will sweep through the basics of Riemannian geometry in this chapter, with a focus on the concepts that will be important for the Ricci flow later. I will quickly review the basic notions of infinitesimal Riemannian geometry, and in particular defining the Riemann, Ricci, and scalar curvature of a Riemannian manifold. This is a review only, in particular omitting any leisurely discussion of examples or motivation for Riemannian geometry. For more details see [1-8]

Connection A definition frequently used by differential geometries goes like this.

Let ʚ͇ʛ Ɣ ʜ͙͚͙͇ͪ͗ͨͣͦͣ͘͢͝͠ʝ, ϙʚ̿ʛ Ɣ ʜ͍͙͚ͣͣͨͧ͗ͨͣͧͣ̿͜͢͡͝ʝ

Definition1.1. A connection on a vector bundle ̿ is a map ǧS ʚ͇ʛ Ɛϙʚ̿ʛ Ŵϙʚ̿ʛ

Which satisfies the following axioms (whereǧΌƔǧʚ͒Q ʛ)

ǧΌʚ͚ ƍ ʛ Ɣ ʚ͚͒ʛƍ͚ǧΌƍǧΌ

!ͮƔ͚ǧΌƍǧ΍

Here f is a real-valued function if ̿ is a real vector bundle, A complex valued function if ̿ is a complex vector bundle. Given a connection ǧ in the sense of definition (1.1), we can define a map

ǳ ͘ ȏϙʚ̿ʛ Ŵϙʚ͎ ͇̂̿ʛ ƔϙƳ͂ͣ͡ʚ͎͇Q ̿ʛƷ by

͘ʚʛʚ͒ʛ ƔǧΌ

Then ͘ satisfies .

Chapter 1 [PRILIMINARIES]

Definition1.2. A connection on a vector bundle ̿ is a map

ǳ ͘ ȏϙʚ̿ʛ Ŵϙʚ͎ ͇̂̿ʛ

Which satisfies the following axiom

͘ʚ͚ ƍ ʛ Ɣ ʚ͚͘ʛ̂ ƍ ͚͘ƍ͘

Definition1.3 In more frequently used in gauge theory .The simplest example of a connection occurs on the bundle ̿Ɣ͇Ɛ͌(, the trivial vector bundle of rank m over M. A section of this bundle ͥ ͦ Ɣƶ ƺ S͇ Ŵ͌( ˅ (

We can use the exterior derivative to define the “trivial” flat connection ͘ on : ͥ ͥ͘ ͦ ͦ ͘ ƶ ƺ Ɣ ƶ͘ ƺ ˅ ˅ ( ͘(

More generally, given an ͡Ɛ͡ matrix

ͥ ͥ !ͥ ˆ!( !Ɣʬ ˅ˆ˅ˆ˅ʭ ( ( !ͥ ˆ!(

We can define a connection ͘ by

ͥ ͥ ͥ ͘ !ͥ ˆ!ͥ ͦ ͦ͘ ͥ ( ͦ ͘ ƶ ƺ Ɣ ƶ ƺ ƍʬ ˅ˆ˅ˆ˅ʭ ƶ ƺ ˅ ˅ !( ˆ!( ˅ ( ͘( ͥ ( (

We can write this last equation in a more abbreviated fashion:͘Ɣ͘ƍ!Q Matrix multiplication being understand in the last term.

If ͚Ǩ*ʚ͐ʛ (͚ is a section of bundle͐) is such that ǧ͚ƔR for all vector field , we say that say that f is parallel to the connectionǧ .

A connection on the tangent bundle ͎͇ is known as an affine connection.

Remark1.1.Suppose &1 be a parallel transport map i.e.&1S͐3 Ŵ͐3ͮ1 to each infinitesimal tangent vector ͪǨ͐ and for fixed ͬ . We have

Chapter 1 [PRILIMINARIES]

ͯͥ &/ʚ3ʛ ʠ͚Ƴͬ ƍ ͨ͒ʚͬʛƷʡ Ǝ͚ʚͬʛ ǧ͚ʚͬʛ Ɣ*'+ /Ŵͤ ͨ

Let ǧ be a connection on ͇ . We say that this connection is torsion-free if we have the pleasent identity

ǧPǧQ͚ƔǧQǧP͚

For all scalar field͚ Ǩ ̽Ϧʚ͇ʛ.

Riemannian metrics and Levi-Civita connection

Definition1.4. Let ͇ be an n-dimensional manifold. A Riemannian metric ͛ on ͇ is a smooth section of ͎ǳ͇͎̂ǳ͇ defining a positive definite symmetric bilinear form on ͥ ) ͎+͇ for each ͤǨ͇. In local coordinates ʚͬ QUQͬ ʛ, one has a natural local basis i $ % ʜ"ͥQUQ")ʝ for ͎͇, where "$ Ɣ . The metric tensor ͛Ɣ͛$%ͬ͘ ̂ͬ͘ is represented i3Ĝ by a smooth matrix –valued function ͛$% Ɣ͛Ƴ"$Q"%Ʒ . The pair ʚ͇Q ͛ʛ is Riemannian $% manifold .We denote by Ƴ͛ Ʒ the inverse of the matrix ʚ͛$%ʛ

Definition1.5. The Levi-Civita connection ǧS ͎͇ Ɛ̽ Ϧʚ͎͇ʛ Ŵ̽Ϧʚ͎͇ʛ is the unique connection on ͎͇ that is compatible with the metric and torsion free.

͒Ƴ͛ʚ͓Q ͔ʛƷƔ͛ʚǧ͓Q ͔ʛ ƍ͛ʚ͓Q ǧ͔ʛ

ǧ͓Ǝǧ͒Ɣʞ͒Q͓ʟ

Where ʞ͒Q ͓ʟ͚Ȭ͒Ƴ͓ʚ͚ʛƷƎ͓Ƴ͒ʚ͚ʛƷ defines the Lie bracket acting on functions.

From this one can show by taking a linear combination of the above equations with permutations of the vector field ͒Q ͓ and ͔that

T͛ʚǧ͓Q ͔ʛ Ɣ͒Ƴ͛ʚ͓Q ͔ʛƷƍ͓Ƴ͛ʚ͒Q ͔ʛƷƎ͔Ƴ͛ʚ͒Q ͓ʛƷƍ͛ʚʞ͒Q ͓ʟQ͔ʛ Ǝ͛ʚʞ͒Q ͔ʟQ͓ʛ Ǝ ͛ʚʞ͓Q ͔ʟQ͒ʛ (1.1)

) ƣͬ$Ƨ ͏ ͇) Let $Ͱͥ be a local coordinate system defined in an open set in . The christoffel i & i ͏ š SƔϙ symbols are defined in by i3ĝ $% i3Ğ .Here and through the thesis; we follow šīĜ the Einstein summation convention of summing over repeated indices. By (1.1) and ʢ i Q i ʣ ƔR i3Ĝ i3ĝ we see that they are given by S " " " ϙ& Ɣ ͛&' ƴ ͛ ƍ ͛ Ǝ ͛ Ƹ $% T "ͬ$ %' "ͬ% $' "ͬ' $%

Chapter 1 [PRILIMINARIES]

$ % Also for Ɣ͕"$ ,͒ Ɣ ͖ "% we have a formula for ǧ͓ in local coordinates

$ % $ % & ǧ͓Ɣ͖"$Ƴ͕ Ʒ"% ƍ͖͕ ϙ$% "&

Let f be a smooth real-valued function on M .We define the Hessian of ͚, denoted ͙͂ͧͧʚ͚ʛQ as follows:

͙͂ͧͧʚ͚ʛʚ͒Q ͓ʛ Ɣ͒Ƴ͓ʚ͚ʛƷƎǧ͓ʚ͚ʛ

Note that the Hessian is a contra variant, symmetric two-tensor i.e., for vector fields ͒Q ͓Qwe have ͙͂ͧͧʚ͚ʛʚ͒Q ͓ʛ Ɣ͙͂ͧͧʚ͚ʛʚ͓Q ͒ʛ and ͙͂ͧͧʚ͚ʛʚ&͒Q ͓ʛ Ɣ& ͙͂ͧͧʚ͚ʛʚ͒Q ͓ʛ

For all smooth function &, . Other formulas for the Hessian are

ͦ ͙͂ͧͧʚ͚ʛʚ͒Q ͓ʛ Ɣ ˛ʚ͚ʛQ͓˜ ƔƳʚ͚ʛƷƔ ͚ʚ͒Q ͓ʛ

Also, in local coordinate we have

& ͙͂ͧͧʚ͚ʛ$% Ɣ"$"%͚Ǝʚ"&͚ʛϙ$%

The Laplacian ;͚ is defined as the Hessian .That is to say

$% ;͚ʚͤʛ Ɣ ȕ ͛ ͙͂ͧͧʚ͚ʛƳ"$Q"%Ʒ $%

Also , if ʜ͒$ʝ is an orthogonal basis for ͎͇ then

;͚ʚͤʛ Ɣ ȕ ͙͂ͧͧʚ͚ʛʚ͒$Q͒$ʛ $

Curvature tensor

Let ʚ͇Q ͛ʛ be a Riemannian manifold and ǧ the Riemannian connection, the curvature tensor is a ʚSQUʛ-tensor defined by

ͦ ͦ ͌ʚ͒Q ͓ʛ͔ƔQ͔ƎQ͔

Ɣ ͔Ǝ ƎʞQʟ͔ƔʞQʟ͔ƎʞQʟ͔

On vector fields Q͓Q͔ .

Chapter 1 [PRILIMINARIES]

Using metric ͛ we can change this to a ʚRQVʛ Ǝtensor as follows ͌ʚ͒Q ͓Q ͔Q ͑ʛ Ɣ͛ʚ͌ʚ͒Q ͓ʛ͔Q ͑ʛ

The Riemannian curvature tensor ͌ʚ͒Q ͓Q ͔Q ͑ʛ satisfies the following properties

1) ͌ is skew-symmetric in the first two and last two entries ͌ʚ͒Q ͓Q ͔Q ͑ʛ ƔƎ͌ʚ͓Q ͒Q ͔Q ͑ʛ Ɣ͌ʚ͓Q ͒Q ͑Q ͔ʛ

2) ͌ʚ͒Q ͓Q ͔Q ͑ʛ Ɣ͌ʚ͔Q ͑Q ͒Q ͓ʛ 3) ͌ satisfies a cyclic permutation property called Bianchi’s first : ͌ʚ͒Q ͓ʛ͔ƍ͌ʚ͔Q ͒ʛ͓ƍ͌ʚ͓Q ͔ʛ͒ƔR 4) ǧ͌ satisfies a cyclic permutation property called Bianchi’s second identity ʚǧ͌ʛʚ͒Q ͓ʛ͑ƍʚǧ͌ʛʚ͓Q ͔ʛ͑ƍʚǧ͌ʛʚ͔Q ͒ʛ͑ƔR

In local coordinates, the Riemann curvature tensor can be represented as

͌Ƴ"$Q"%Q"&Q"'ƷƔ͌$%&'

Definition1.6. The sectional curvature of a 2-plance ͊ɗ͎+͇ is defined as ͅʚ͊ʛ Ɣ ͌ʚ͒Q ͓Q ͒Q ͓ʛ where ʜ͒Q ͓ʝ is an orthogonal basis of ͊ . We say that ʚ͇Q ͛ʛ has positive sectional curvature for every 2-plane .

A Riemannian manifold is said to have constant sectional curvature if ͅʚ͊ʛ is the same for all ͤǨ͇ and all 2-planes ͊ɛ͎+͇.

Definition1.7. The Riemannian manifold ʚ͇Q ͛ʛ is said to be an Einstein manifold with Einstein constant if ͗͛͝ Ɣ ͛ .

Definition1.8. Using the metric, one can replace the Riemannian curvature tensor ͌ by a symmetric bilinear form ͌͡ on ͎͇ͦ . in local sections of͎͇ͦ .the formula for͌͡ is

$% &' ͌͡ʚQ ʛ Ɣ͌$%&'

We call ͌͡ the curvature operator. We say ʚ͇Q ͛ʛ has positive curvature operator if ͌͡ʚQ ʛ ƚR for any Ǩ͎͇ͦ .

In local coordinate the Ricci curvature tensor is defined by

&' ͌͗͝ʚ͒Q ͓ʛ Ɣ͛ ͌ʚ͒Q "&Q͓Q"'ʛ

Cleary ͌͗͝ is a symmetric bilinear form on ͎͇,given in local coordinates by

$ % ͌͗͝ Ɣ ͌͗͝$%ͬ͘ ̂ͬ͘

Chapter 1 [PRILIMINARIES]

Where ͌͗͝$% Ɣ͌͗͝Ƴ"$Q"%Ʒ.the scalar curvature is defined by

$% ͌Ɣ͌" Ɣͨͦ"͌͗͝ Ɣ ͛ ͌͗͝$%

Definition1.9 Let ; denote the laplacian (or laplace-Beltrami operator) acting on functions, which is globally defined as the divergence of the gradient and given in local coordinates by: "ͦ " ;Ȭ ͪ͘͝ Ɣ ͛$% Ɣ͛$% ʦ Ǝϙ& ʧ $ % "ͬ$"ͬ% $% "ͬ&

There are other equivalent ways to define + such as

) +͚ Ɣ ȕ ͙ ʚ͙ ͚ʛ ƎƳǧ ͙ Ʒ͚ Ĕ Ͱͥ

) Where ʜ͙ʝͰͥis an orthogonal frame.

Integration by parts

A basic tool is integration by parts recall that stokes theorem says that

Theorem1.1.If is an ʚ͢ƎSʛ Ǝform on a compact differentiable manifold ͇) with (possibly empty) boundary "͇, then

ǹ͘ Ɣǹ i

The divergence theorem says

Theorem1.2. Let ʚ͇Q ͛ʛ be a compact Riemannian manifold. If ͒ is a 1-form, then

ǹͪ͘͝ʚ͒ʛ͘ Ɣǹ ˛͒Q ͪ˜͘ ġ iġ

Here ͪ is the unit outward normal,͘ denotes the volume form of ͛ and ͘ is the volume form of the boundary "͇) .Also we have following properties

Ȅ ;ͩ͘ ƔR 1) On a closed manifold , ġ 2) (Green) on a compact manifold

ǹ ʚͩ;ͪʛ͘ Ɣǹ ʚͪ;ͩʛ͘ ġ ġ 3) If ͚ is a function and ͒ is a 1-form ,then

Chapter 1 [PRILIMINARIES]

ǹ͚ͪ͘͝ʚ͒ʛ͘ Ɣ Ǝ ǹ ˛ǧ͚Q ͒˜͘ ƍ ǹ ͚˛͒Q ͪ˜͘ ġ ġ iġ

Geodesics

v S ̓ Ŵ ͇ ͦʗ Ɣ R We can now define the acceleration of a curve by the formula /v ,in local coordinate this becomes

ͦ͘ & ͘ $ ͘ % ͦʗ Ɣ " ƍ ϙ&" ͨͦ͘ & ͨ͘ ͨ͘ $% &

A ̽Ϧcurve S ̓ Ŵ ͇ is called a geodesic if ͦʗ ƔR .this means a smooth curve S ̓ Ŵ ͇ is called a geodesic if ǧRʖ ʖ ƔR.

Projective linear groups

The group ͍͊͆ͦʚ͌ʛ is the group of automorphisms of the hyperbolic plane ͂ and the elements of this group are transformation of the form ͕ͮ ƍ ͖ ͮŴ ͗ͮ ƍ ͘

͕Q ͖Q ͗Q ͘ Ǩ ͌ and ͕͘ Ǝ ͖͗ Ɣ S

This group also be made into a topological space by identifying the transformation with the point ʚ͕Q ͖Q ͗Q ͘ʛ Ǩ͌ͨ in the subspace ʜʚ͕Q ͖Q ͗Q ͘ʛ Ǩ͌ͨ ȏ͕͘Ǝ͖͗ƔSʝ and in fact ͦ ͥ one show that ͍͊͆ͦʚ͌ʛ is homeomorphic to ͌ Ɛ͍ and thus being a 3-dimensional manifold, moreover the group multiplication and taking of inverses are continuous with the topology on ͍͊͆ͦʚ͌ʛ and so this is a topological Group .

Lensspace

In the 3-manifold case, a picturesque description of a lens space is that of a space resulting from gluing two solid tori together by a homeomorphism of their boundaries. Of course, to be consistent, we should exclude the 3-sphere and͍ͦ Ɛ͍ͥ , both of which can be obtained as just described.

There is a complete classification of three-dimensional lens spaces.

Three-dimensional lens spaces arise as quotients of͍ͧ ɗĿͦ by the action of the group ͫR ʠ ʡ that is generated by elements of the form Rͫ,

Chapter 1 [PRILIMINARIES]

Seifertfiberspace A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus.

A standard fibered torus corresponding to a pair of coprime integers ʚ͕Q ͖ʛ with ͕Ƙ͖ is the surface bundle of the automorphism of a disk given by rotation by an angle of T͖ ͕ (with the natural fibering by circles). If ͕ƔS the middle fiber is called ordinary, while if ͕ƘS the middle fiber is called exceptional. A compact Seifert fiber space has only a finite number of exceptional fibers.

RICCI FLOW V{tÑàxÜE

Chapter 2 [ RICCI FLOW

Chapter 2

Ricci flow

Flows on Riemannian manifold

At first we introduce flows ͨŴʚ͇ʚͨʛQ͛ʚͨʛʛ on Riemannian manifoldsʚ͇Q ͛ʛ, which are recipes for describing smooth deformations of such manifolds over time, and derive the basic first variation formulae for how various structures on such manifolds change by such flows.

In this section we get a one-parameter family of such manifolds ͨ Ŵ ʚ͇ʚͨʛQ͛ʚͨʛʛ ,parametrized by a “time” parameter ͨ in the usual manner the time derivatives ͛ʖʚͨʛ Ɣ ͛ʚͨʛ c is ͘ ͛ʚͨƍͨ͘ʛ Ǝ͛ʚͨʛ ͛ʚͨʛ Ɣ*'+ ͘ /Ŵͤ ͨ͘

͛ʚͨʛ Ė ͛ and also the analogue of the time derivative c is then the Lie derivative iħ .

Definition 2.1.(Ricci flow)A one-parameter family of metrics ͛ʚͨʛ on a smooth manifold ͇ for all time ͨ in an interval̓ is said to obey Ricci flow if we have ͘ ͛ʚͨʛ ƔƎT͌͗͝ʚͨʛ ͘

Note that this equation makes tensorial sense since ͛ and ͌͗͝ are both symmetric rank 2 tensor. The factor of 2 here is just a notational convenience and is not terribly important, but the minus signƎis crucial.

Chapter 2 [ RICCI FLOW

Let us give a quick introduction of what the Ricci flow equation means .In the harmonic coordinate ʚͬͥQͬͦQUQͬ)ʛ about ͤ, that is to say coordinates where ;ͬ$ ƔR for all ͝ ,we have " " S ͌͗͝ Ɣ͌͗͝ƴ Q ƸƔƎ ;͛ ƍ͋ ʚ͛ͯͥQ"͛ʛ $% "ͬ$ "ͬ% T $% $%

Where ͋ is a quadratic form in͛ͯͥ and "͛ .See lemma (3.32) on page (92) of [9]

So in this coordinates, the Ricci flow equation is actually a heat equation for the Riemannian metric "͛ Ɣ;͛ƍT͋ʚ͛ͯͥQ"͛ʛ "ͨ

Special solutions of flows

i" ƔƎ͌͗͝ The Ricci flow i/ introduced by Hamilton is a degenerate parabolic evolution system on metrics.

Theorem2.1 (Hamilton [9]) Let Ƴ͇Q ͛$%ʚͬʛƷ be a compact Riemannian manifold .Then ͎ƘR i" ƔƎT͌͗͝ ͛ ʚͬQ Rʛ Ɣ there exists a constant Such that the Ricci flow i/ , with $% ͛$%ʚͬʛ ,admits a unique smooth solution ͛$%ʚͬQ ͨʛ for all ͬǨ͇ and ͨǨʞRQ͎ʛ

Example 1(Einstein metric)

Recall that a Riemannian metric ͛$% is Einstein if ͌$% Ɣ͛$% for some for some constant .Now since the initial metric is Einstein, we have

͌$%ʚͬQ Rʛ Ɣ͛$%ʚͬQ Rʛ

ͦ and some ƘR .Let ͛$%ʚͬQ ͨʛ Ɣ ʚͨʛ͛$%ʚͬQ Rʛ.

Now from the definition of the Ricci tensor, one sees that

͌$%ʚͬQ ͨʛ Ɣ͌$%ʚͬQ Rʛ Ɣ͛$%ʚͬQ Rʛ

i" Ĝĝ ƔƎT͌͗͝ Thus the equation i/ corresponds to

ͦ " ʠ ʚͨʛ͛$%ʚͬQ Rʛʡ ƔƎT͛ ʚͬQ Rʛ "ͨ $%

i` ƔƎZ This gives the ODE i/ ` .

Chapter 2 [ RICCI FLOW

ͦ Whose solution is given by ʚͨʛ ƔSƎTͨ .Thus the evolving metric ͛$%ʚͬQ ͨʛ shrinks Ŵ͎Ɣ ͥ homothetically to a point as ͦZ .

By contrast, if the initial metric is an Einstein metric of negative scalar curvature ,The metric will expand homothetically for all times .Indeed if ͌$%ʚͬQ Rʛ ƔƎ͛$%ʚͬQ Rʛwith ƘR ͛ ʚͬQ ͨʛ Ɣͦʚͨʛ͛ ʚͬQ Rʛ ʚͨʛ ` Ɣ Z and $% $% ,the satisfies the ODE / ` . With the ͦ ͦ solution ʚͨʛ ƔSƍTͨ , hence the evolving metric ͛$%ʚͬQ ͨʛ Ɣ ʚͨʛ͛$%ʚͬQ Rʛ exists ͯͥ and expands homothetically formal times, and the curvature fall back to zero like / . lemma2.1 (see [2]).Let͒Ǩ͎+͇ be a unit vector. Suppose that ͒ is contained in some orthonormal basis for ͎+͇. ͌͗ʚ͒Q ͒ʛ is then the sum of the sectional curvatures of planes spanned by͒ and other elements of the basis.

Example 2 on an n-dimensional sphere of radius ͦ (where͢ Ƙ S), the metric is given by ͛Ɣͦͦ͛ ͛ ͥ where is the metric on the unit sphere.The sectional curvatures are all -v . Thus for any unit vector ͪ,the result of lemma (2.1) tells us that ͌͗ʚͪQ ͪʛ Ɣ ʚ)ͯͥʛ ͌͗ Ɣ ʚ)ͯͥʛ ͛ Ɣ ʚ͢ Ǝ Sʛ͛ -v .Therefore -v

So the Ricci flow equation becomes an ODE " ͛ƔƎT͌͗ "ͨ

ʂ i ʚͦͦ͛ʛ Ɣ ƎTʚ͢ Ǝ Sʛ͛ i/

ʂ i ʚͦͦʛ Ɣ ƎTʚ͢ Ǝ Sʛ i/

ͦ We have the solution ʚͨʛ Ɣ ǭ͌ͤ Ǝ Tʚ͢ Ǝ Sʛͨ , where͌ͤ is the initial radius of the v sphere.ThemanifoldshrinkstoapointasŴ t . ͦʚ)ͯͥʛ

Similarly,forhyperbolicnspace͂)(where͢Ƙ1),TheRicciflowreducestotheODE

" ʚͦͦʛ ƔTʚ͢ƎSʛ "ͨ

ͦ Which has the solution ͦʚͨʛ Ɣ ǭ͌ͤ ƍTʚ͢ƎSʛͨ .So the solution expands out to infinity .

Chapter 2 [ RICCI FLOW

Example3 (Ricci Soliton) A Ricci soliton is a Ricci flowʚ͇Q ͛ʚͨʛʛ , RƙͨƗ͎ƙ7

,with the property that for each ͨǨʞRQ͎ʛ there is a diffeomorphism /S͇ Ŵ͇ and a ǳ constant ʚͨʛ such that ʚͨʛ/ ͛ʚRʛ Ɣ͛ʚͨʛT That is to say , in a Ricci soliton all the Riemannian manifold ʚ͇Q ͛ʚͨʛʛ are isomorphic up to a scale factor that is allowed to ɑ vary with ͨ .The soliton is said to be shrinking if ʚͨʛ ƗR for allͨ (note thatͤ Ɣ͘͝ andʚRʛ ƔS )

ǳ Taking the derivative of the equation͛ʚͨʛ Ɣʚͨʛ/ ͛ʚRʛ and evaluating at ͨƔR yields

" "ʚͨʛ " ͛ʚͨʛ Ɣ ǳ͛ʚRʛ ƍʚͨʛ ǳ͛ʚRʛ "ͨ "ͨ / "ͨ /

ƎT͌͗Ƴ͛ʚRʛƷƔĺʚRʛ͛ʚRʛ ƍ Ė1͛ʚRʛ

ͪƔ eħ ĺʚRʛ ƔT Where / .Let us set . Now because on a Riemannian manifold

ʚ͇Q ͛ʛ,we have ʚĖ͛ʛ$% Ɣ$͒% ƍ%͒$

We get ƎT͌$% ƔT͛$% ƍ$͒% ƍ%͒$

As a special case we can consider the case that ͪ is the gradient vector field of some ) scalar function ͚ on ͇ , i.e. ͪ$ Ɣ$͚, the equation then become ͌$% ƍ͛$% ƍ$ǧ%͚Ɣ R such solutions are known as gradient Ricci solutions .

The gradient Ricci solitons play a role in motivating the definition Perelman’s Đand ġƎfunctionals.

Proposition2.1(see[10]) suppose we have a complete Riemannian manifold ʚ͇Q ͛ʚRʛʛ ,a smooth function ͚S ͇ Ŵ ͌ ,and a constant ƘR such that Ǝ͌͗͝Ƴ͛ʚRʛƷƔ ͙͂ͧͧʚ͚ʛ Ǝ͛ʚRʛ then there is ͎ƘR and a gradient shrinking soliton ʚ͇Q ͛ʚͨʛʛ defined for RƙͨƗ͎.

In 1989, Shi generalized the theorem of Hamilton about short-time existence and uniqueness theorem for the Ricci flow to complete non-compact manifolds with bounded curvature .

Chapter 2 [ RICCI FLOW

Theorem2.2(Shi[11]) Let Ƴ͇Q ͛$%ʚͬʛƷ be a complete non compact Riemannian manifold of dimension ͢ with bounded curvature .Then there exists a constant ͎ƘR such that the initial value problem

" ͛ ʚͬQ ͨʛ ƔƎT͌ ʚͬQ ͨʛ͇ͣ͢ ʰ"ͨ $% $% ͛$%ʚͬQ Rʛ Ɣ͛$%ʚͬʛ͇ͣ͢

Admits a smooth solution͛$%ʚͬQ ͨʛ,ͨǨʞRQ ͎ʟ, with bounded curvature

Recently Chen and Zhu proved the following uniqueness Theorem.

Theorem2.3 (Chen-Zhu [11]) Let Ƴ͇Q ͛$%ʚͬʛƷ be a complete non-compact Riemannian manifold of dimension n with bounded curvature. Let ͛$%ʚͬQ ͨʛ and ͛$%ʚͬQ ͨʛbe two solutions to the Ricci flow on ͇ƐʞRQ͎ʟ with ͛$%ʚͬʛ as the initial data and with bounded curvatures. Then Let ͛$%ʚͬQ ͨʛ Ɣ͛$%ʚͬQ ͨʛ for ʚͬQͨʛǨ͇ƐʞRQ͎ʟ.

DELATIONS Now we specialize to some specific flows ʚ͇Q ͛ʚͨʛʛ of a Riemannian metric on a fixed backward manifold M. The simplest such flow (besides the trivial flow ͛ʚͨʛ Ɣ͛ʚRʛ,of course) is that of a dilation

͛ʚͨʛSƔ̻ʚͨʛ͛ʚRʛ

Where ̻ʚͨʛ ƘR is a positive scalar with ̻ʚRʛ ƔS .The flow here is given by ͛ʖʚͨʛ Ɣ ͕ʚͨʛ͛ʚͨʛʚTTSʛ

ʖ ͕ʚͨʛ Ȭ ʚ/ʛ Ɣ ̻ͣ͛͠ʚͨʛ ̻ where ʚ/ʛ / is the logarithmic derivative of (or equivalently , / ̻ʚͨʛ Ɣ͙ͬͤʠȄ ͕ʚͨĺʛͨ͘ĺʡ ͤ )

In this case our variation formulas become very simple

͘ ͛PQ ƔƎ͕͛PQ ͨ͘

Chapter 2 [ RICCI FLOW

ʖ σ *PQ ƔR

ʖ ͌̓͗PQ ƔR

͌ʖ ƔƎ͕͌

͘ S ͘ʚͬQ ͭʛ Ɣ ͕͘ʚͬQ ͭʛ ͨ͘ T

͘ Ɣ ͕͘ / ͦ

Modifying Ricci flow

If ͛ʚͨʛ solves Ricci flow and we set ͛ʚͧʛ Ɣ̻ʚͧʛ͛ʚͨʚͧʛʛ for some reparameterized time ͧƔͧʚͨʛ and some scalar ̻Ɣ̻ʚͧʛ ƘR, Then the Ricci curvature here is ͌̓͗Ȳ ʚͧʛ Ɣ͌͗͝ƳͨʚͧʛƷ .We then see from the chain rule that ͛Ȭ obeys the equation

͘ ͨ͘ ͛Ȭʚͧʛ ƔƎT̻ʚͧʛ ͌̓͗Ȳ ʚͧʛ ƍ ͕ʚͧʛ͛Ȭʚͧʛ ͧ͘ ͧ͘

Where ͕ is the logarithmic derivative of ̻ .If we normalize the time reparameterization / Ɣ ͥ ͛Ȭ by requiring . ʚ.ʛ ,we thus see that obeys normalized Ricci flow

͘ ͛Ȭ ƔƎT͌̓͗Ȳ ƍ͕͛Ȭʚͧʛ ͧ͘

͛ʚͨʛ ƔƎT͌͗͝ʚͨʛ Which can be viewed as a combination of (2.1) and / .

Definition2.2 (Bounded curvature solution) we say that a solution ͛ʚͨʛ , ͨǨ̓ of the Ricci flow has bounded curvature if on every compact subinterval ʞ͕Q ͖ʟ ɗ ̓ the Riemann curvature tensor is bounded. In particular we do not assume the curvature bound is uniform in time on non-compact time intervals.

Timeevolvingmetrics: A normal coordinates about the point p are defined by

ͥ ͦ ) a) 1ʚͨʛ Ɣ ʚͨͪ Qͨͪ QUQͨͪ ʛ is a geodesic

Chapter 2 [ RICCI FLOW

b) ͛$%ʚͤʛ Ɣ $% & c) *$%ʚͤʛ ƔR,"$͛%&ʚͤʛ ƔR

Suppose that ͛$%ʚͨʛ is a time-dependent Riemannian metric and

͘ ͛ ʚͨʛ Ɣ͜ ʚͨʛ ͨ͘ $% $%

Then the various geometric quantities evolve according to the following equations:

1) Metric inverse " ͛$%ʚͨʛ ƔƎ͜$% ƔƎ͛$&͛%'͜ "ͨ &' 2) Christoffel symbols

" S ϙ& Ɣ ͛&'Ƴ ͜ ƍ͜ Ǝ͜ Ʒ "ͨ $% T $ %' % $' ' $%

3) Riemann curvature tensor " S ͜ ƍ ͜ Ǝ ͜ ' '+ $ % &+ $ & %+ $ + %& ͌$%& Ɣ ͛ Ƥ ƨ "ͨ T Ǝ%$͜&+ ƍ%&͜$+ Ǝ%+͜$& 4) Ricci tensor " S ͌ Ɣ ͛+,Ƴ ͜ ƍ ͜ Ǝ ͜ Ǝ ͜ Ʒ "ͨ $% T + $ %+ , % $+ , + $% $ % ,+ 5) Scalar curvature " ͌ƔƎ;͂ƍ+,͜ Ǝ͜+,͌ "ͨ +, +, +, Where ͂Ɣ͛ ͜+, 6) Volume element " ͂ ͘ Ɣ ͘ "ͨ T 7) Volume of manifold " ͂ ǹ͘Ɣǹ ͘ "ͨ T 8) Total scalar curvature on a closed manifold ͇ " S $% ǹ͌͘Ɣǹƴ͌͂ Ǝ ͜ ͌$%Ƹ͘ "ͨ T

Chapter 2 [ RICCI FLOW

We only proof 2), 5) and 6) Proof2) We know ͘ S ϙ& Ɣ ͛&'Ƴ" ͛ ƍ"͛ Ǝ"͛ Ʒ ͨ͘ $% T $ %' % $' ' $% So we get S S " ϙ& Ɣ ʚ" ͛&'ʛƳ" ͛ ƍ"͛ Ǝ"͛ Ʒƍ ͛&'Ƴ" " ͛ ƍ"" ͛ Ǝ"" ͛ Ʒ / $% T / $ %' % $' ' $% T $ / %' % / $' ' / $% Now we work in normal coordinates about a point ͤ .So according to properties b)and c) of normal coordinates we get "$͛%& ƔR at ,"$̻Ɣǧ$̻ at ͤ for any tensor̻. Hence S " ϙ&ʚͤʛ Ɣ ͛&'Ƴ ͜ ƍ͜ Ǝ͜ Ʒʚͤʛ / $% T $ %' % $' ' $% Now although the Christoffel symbols are not the coordinates of a tensor quantity, their derivative is. (This is true because the difference between the christoffel symbols of two connections is a tensor .Thus, by taking a fixed point connection with christoffel Ȳ& & & Ȳ& symbolsϙrs , we have "/ϙ$% Ɣ"/Ƴϙ$% Ǝϙrs Ʒand the Right hand side is clearly a tensor) Hence both side of this equation are the coordinates of tonsorial quantities, so it does not matter what coordinates we evaluate them. in particular ,the equation is true for any coordinates , not just normal coordinates ,and about any point ͤ. Proof 5) from 4) and 1) we get $% $% $% "/͌Ɣ"/Ƴ͛ ͌$%ƷƔ"/Ƴ͛ Ʒ͌$% ƍ͛ Ƴ"/͌$%Ʒ

ƔƎ͜$%͌ ƍ͛$% ƴͥ ͛+,Ƴ ͜ ƍ ͜ Ǝ ͜ Ǝ $% ͦ , $ %+ , % $+ , + $%

$%͜,+ʛƸ

+ , +, ƔƎ;͂ƍ ͜+, Ǝ͜ ͌+,

$% (Note that ǧ͛ Ɣ Rand;Ɣ ͛ $%). Proof 6) we know

ͥ ͦ ) ͘ Ɣ ǯ͙ͨ͛͘$%ͬ͘ Ȁͬ͘ Ȁͬ͘

And ͘ ̻͘ ͙̻ͨ͘ Ɣ ʚ̻ͯͥʛ$% ʦ $%ʧ ͙̻ͨ͘ ͨ͘ ͨ͘

Chapter 2 [ RICCI FLOW

Now by chain rule formula we obtain ͥ ͦ ) ͥ $% ͥ ͦ ) "/͘ Ɣ "/ǭ͙ͨ͛͘$%ͬ͘ Ȁͬ͘ Ȁͬ͘ Ɣ ͛ ͜$%͙ͨ͛ͬ͘͘ Ȁͬ͘ Ȁͬ͘ Ɣ ͘ ͦǭ /"Ĝĝ ͦ $% Where͂Ɣ͛ ͜$% .

Evolutionequationsforderivativesofcurvature

i" Ĝĝ Ɣ The Ricci flow is an evolution equation on the metric. The evolution equation i/ ƎT͌$% for the metric implies a heat equation for the Riemannian Curvature ͌$%&'which we now derive.

Theorem2.4 (Hamilton[10]) under the Ricci flow, the curvature tensor satisfies the evaluation equation " ͌ Ɣ;͌ ƍTƳ̼ Ǝ̼ Ǝ̼ ƍ̼ Ʒ "ͨ $%&' $%&' $%&' $%'& $'%& $&%'

+, Ǝ͛ Ƴ͌+%&'͌,$ ƍ͌$+&'͌,% ƍ͌$%+'͌,& ƍ͌$%&+͌,'Ʒ

+- ,. Where ̼$%&' Ɣ͛ ͛ ͌+$,%͌-&.' and ; is the Laplacian with respect to the evolving metric

Proof choose ʜͬͥQͬͦQUQͬ(ʝ to be a normal coordinate system at a fixed point .At this point, we compute

" S ϙ# Ɣ ͛#( ʠ ʚƎT͌ ʛ ƍƳƎT͌ ƷƎ ƳƎT͌ Ʒʡ "ͨ %' T % '( ' %( ( %'

" " " " " ͌# Ɣ ƴ ϙ#ƸƎ ƴ ϙ#Ƹ "ͨ $%' "ͬ$ "ͨ %' "ͬ% "ͨ $'

" " "͛ ͌ Ɣ͛ ͌# ƍ #& ͌# "ͨ $%&' #& "ͨ $%' "ͨ $%'

Combining these identities we get

" S ͌ Ɣ͛ Ƭƴ ʢ͛#( ʠ ʚƎT͌ ʛ ƍƳƎT͌ ƷƎ ƳƎT͌ ƷʡʣƸư "ͨ $%&' #& T $ % '( ' %( ( %'

S Ǝƴ ƫ͛#(Ƴ ʚƎT͌ ʛ ƍʚƎT͌ ʛ Ǝ ʚƎT͌ ʛƷƯƸ Ǝ T͌ ͌# T % $ '( ' $( ( $' #& $%'

Chapter 2 [ RICCI FLOW

+, +, Ɣ$&͌%' Ǝ$'͌%& Ǝ%&͌$' ƍ%'͌$& Ǝ͌$%'+͛ ͌,& Ǝ͌$%&+͛ ͌,' +, ƎT͌$%+'͛ ͌,& +, Ɣ$&͌%' Ǝ$'͌%& Ǝ%&͌$' ƍ%'͌$& Ǝ͛ Ƴ͌$%&+͌,' ƍ͌$%+'͌,&ƷT

Note that we have used of this formula

'( $%ͪ& Ǝ%$ͪ& Ɣ͌$%&'͛ ͪ(Q

Now according to Simon’s identity in extrinsic geometry,

;͌$%&' ƍTƳ̼$%&' Ǝ̼$%'& Ǝ̼$'%& ƍ̼$&%'ƷƔ$&͌%' Ǝ$'͌%& Ǝ%&͌$' ƍ%'͌$& ƍ +, ͛ Ƴ͌+%&'͌,$ ƍ͌$+&'͌,%Ʒ

So we obtain

+, +, ;͌$%&' Ɣ͛ +$͌,%&' Ǝ͛ +%͌,$&'

+, Ɣ$&͌%' Ǝ$'͌%& ƎƳ̼$%&' Ǝ̼$%'& Ǝ̼$'%& ƍ̼$&%'Ʒƍ͛ ͌+%&'͌,$Ǝ%&͌$' +, ƍ%'͌$& ƍƳ̼%$&' Ǝ̼%$'& Ǝ̼%'$& ƍ̼%&$'ƷƎ͛ ͌+$&'͌,% +, ƍ͛ Ƴ͌+%&'͌,$ ƍ͌$+&'͌,%ƷƎTƳ̼$%&' Ǝ̼$%'& Ǝ̼$'%& ƍ̼$&%'Ʒ

As desired, where in the last step we used the symmetries

̼$%&' Ɣ̼&'$% Ɣ̼%$'&

So proof is complete.

Corollary2.1. (see [11]) The Ricci curvature satisfies the evolution equation

" ͌ Ɣ;͌ ƍT͛+-͛,.͌ ͌ ƎT͛+,͌ ͌ "ͨ $& $& +$,& -. +$ ,&

Lemma2.2 (Hamilton[11]) The scalar curvature satisfies the evolution equation

Chapter 2 [ RICCI FLOW

"͌ Ɣ;͌ƍT ͌͗͝ ͦ "ͨ

Proof

"͌ "͌ "͛ Ɣ͛$& $& ƍ ʦƎ͛$+ +, ͛,&ʧ ͌ "ͨ "ͨ "ͨ $&

$& +- ,. +, $+ ,& Ɣ͛ Ƴ;͌$& ƍT͛ ͛ ͌+$,&͌-. ƎT͛ ͌+$͌,&Ʒƍ͌+,͌$&͛ ͛ Ɣ;͌ƍT ͌͗͝ ͦ Lemma2.3. The volume form ͪͣ͘͠ʚͬQ ͨʛ satisfy the following evolution equation under Ricci flow ͘ ͪͣ͘͠ʚͬQ ͨʛ ƔƎ͌ʚͬQ ͨʛͪͣ͘͠ʚͬQ ͨʛ ͨ͘ ͚S ͇ Ŵ ͌ i ;͚ Ɣ ; i! ƍ and if is a time dependent function, then under the Ricci flow i/ i/ T˛͌͗͝Q ͙͂ͧͧʚ͚ʛ˜

Harmonic map

Let S ʚ͇)Q͛ʛ Ŵ ʚ͈(Q͛ʛ , the map Laplacian of ͚ is defined by P Q R "͚ "͚ Ƴ; ͚Ʒ Ɣ; ʚ͚Rʛ ƍ͛$% ʠϙʚ͜ʛR ͚ͣʡ "Q# " PQ "ͬ$ "ͬ% v Ŋ Ŋ ň ŉ Ɣ͛$% ʠ i ! Ǝ*Ι i! ƍ ʠϙʚ͜ʛR ͚ͣʡ i! i! ʡ i3Ĝi3ĝ ΗΘ i3Ğ PQ i3Ĝ i3ĝ (2.2) Where ͚RSƔͭR͚ͣ and ƣͬ$Ƨ and ʜͭPʝ are coordinates on ͇ and ͈, respectively. Note Ϧ ǳ ǳ that;"Q#͚Ǩ̽ ʚ͚ ͎͈ʛ, where ͚ ʚ͎͈ʛ Ŵ͇ Is the pullback vector bundle of ͎͈ by

R R ͚.In (2.2 ) Ƴ;"͚Ʒ denotes the Laplacian with respect to ͛ of the function ͚ . As a special case, if ͇Ɣ͈ and ͚ is the identity map and we choose the ͬ and ͭ coordinates to be the same , then

& $% & & Ƴ;"Q#͘͝Ʒ Ɣ͛ ƳƎϙʚ͛ʛ$% ƍϙʚ͜ʛ$%Ʒ The derivative ͚͘ of a map ͚S ͇) Ŵ͈( is a section of the vector bundle Ɣ ͎ǳ͇͚̂ǳʚ͎͈ʛ . On ̿ is a national metric and compatible connection "Q# defined by the (dual of the)Riemannian metric ͛ and associated Levi-Civita connection on ͎ǳ͇ and the pullback by ͚ on the metric ͜ and its associated Levi-Civita connection on ͎͈.

Chapter 2 [ RICCI FLOW

"Q# ǳ ǳ ǳ So ͚͘ is a section of the bundle ͎ ͇͎̂ ͇͚̂ ʚ͎͈ʛ. The map Laplacian is the trace with respect to ͛ of "Q#͚͘ "Q# ;"Q#͚Ɣͨͦ"ʚ ͚͘ʛ

) ( A map ͚S ʚ͇ Q͛ʛ Ŵ ʚ͈ Q͜ʛ is called a harmonic map if ;"Q#͚ƔR . In the case where ͈Ɣ͌ a harmonic map is the same as a harmonic function.

Ricci flow on almost flat manifolds:

A compact Riemannian manifold ͇)is called Ǝflat if, its curvature is bounded in terms of the diameter as follows ͅ ƙ T͘ʚ͇ʛͯͦQ

Where ͅ denotes the sectional curvature and͘ʚ͇ʛ the diameter of ͇. Here we show summary result of Ricci flow that act on almost flat manifolds.

Theorem2.5. (Ricci flow on almost flat manifolds[12])

In any dimension n there exists an ʚ͢ʛ ƘR such that for any ƙ ʚ͢ʛ an Ǝflat Riemannian manifold ʚ͇)Q͛ʛ has the following properties : i) The solution ͛ʚͨʛto the Ricci flow i" ƔƎTͦ͗͝ Q͛ʚRʛ Ɣ ͛ʚTTUʛ i/ " exists for all ͨǨʞRQ 7ʟ ii) Along the flow () one has ͦ *'+ ͅ " T͘ ʚ͇Q ͛/ʛ ƔR /ŴϦ ħ iii) ͛ʚͨʛ converges to a flat metric along Ricci flow (2.3), if and only if the fundamental group of ͇ is almost abelian (abelian up to a subgroup of finite index).

In the case n=2 the normalized Ricci flow looks like "͛ Ɣ ʚͧ͗ʚ͛ʛ Ǝͅʛ͛ʚTTVʛ "ͨ

Where ͧ͗ʚ͛ʛ is average sectional curvature

Uniformization Theorem2.6 (see [12])

Chapter 2 [ RICCI FLOW

On a closed Riemannian manifold ʚ͇ͦQ͛ʛwith ʚ͇ʛ (Euler characteristic of͇ͦ) ,The normalized Ricci flow (2.4) with ͛ʚRʛ Ɣ ͛ͤ has a unique solution for all time , moreover , as ͨŴ7 , the metrics ͛ʚͨʛ converge uniformly in any ̽& Ǝnorm to a flat metric ͛Ϧ

The normalized Ricci flow In case of 3-manifolds with positive Ricci curvature and higher dimensional manifolds with positive curvature operator, works by Hamilton, Huisken , Bohm and wilking showed that the normalized Ricci flow will evolve the metric to one with constant curvature .One way to avoid collapse is to add the condition that volume by preserved along the evaluation .To preserve the volume we need to change a little bit the equation for the Ricci flow.

We know

ͥ ͦ ) ͘ Ɣ ǯ͙ͨ͛͘$%ͬ͘ Tͬ͘ Uͬ͘

Let us define the average scalar curvature Ȅ ͌͘ ͦƔ Ȅ ͘ Let ͛Ȭʚͨʛ Ɣ ʚͨʛ͛ʚͨʛ with ʚRʛ ƔS. Let us choose ʚͨʛ so that the volume of the manifold with respect to ͛Ȭ is constant. So ͪͣ͠Ƴ͛ȬʚͨʛƷ Ɣ ͪͣ͠Ƴ͛ȬʚRʛƷ

ġ But we know if ͛Ȭ Ɣ͗͛ then ͘ȬƔ͗v͘ therefore ) ʚͨʛͦͪͣ͠Ƴ͛ʚͨʛƷ Ɣ ͪͣ͠Ƴ͛ʚRʛƷ

So we obtain ͦ ͯ) Ȅ ͘ʚͨʛ ʚͨʛ Ɣʬ ġ ʭ Ȅ ͘ʚRʛ ġ

i Ȅ ͘ ƔƎȄ ͌͘ But according to relation i/ ġ ġ we get. ͦ ͯ)ͯͥ ͘ ͘ T Ȅ ͘ʚͨʛ Ȅ ġ ͘ʚͨʛ ʚͨʛ Ɣ ʬ ġ ʭ ͨ͘ ͨ͘ Ǝ͢ Ȅ ͘ʚRʛ Ȅ ͘ʚRʛ ġ ġ

Chapter 2 [ RICCI FLOW

T ʚͨʛ Ȅ ͌͘ʚͨʛ Tͦ Ɣ ġ Ɣ ʚͨʛ ͢ Ȅ ͘ʚͨʛ Ȅ ͘ʚRʛ ͢ Ƶ ġ ƹ ġ Ȅ ͘ʚRʛ ġ The normalized average scalar curvature ,ͦ1,is defined by Ȅ ͌Ȱ͘ ͦ ͦ1 Ɣ ġ Ɣ Ƴ̼͙͕͙͚͗ͩͧ͛͝Ȭ Ɣ͎͙͗͛͌͜͢Ȱ Ɣ͗ͯͥ͌Ʒ Ȅ ͘ ʚͨʛ ġ Where͌Ȱ Ȭ͌ʚ͛Ȭʛ. So ͦ ʚͨʛ Ɣ ͦ-1 Ƴ ʚͨʛƷ / ) So we get " " ͘ ͛ȬƔƴ ͛Ƹ ʚͨʛ ƍ͛ ʚͨʛ "ͨ "ͨ ͨ͘

Tͦ1 ͦ Ɣ ƎT ʚͨʛ͌͗Ƴ͛ʚͨʛƷƍ Ƴ ʚͨʛƷ ͛ ͢ Tͦ1 Ɣ ʚͨʛ ƴƎT͌͗ʚ͛Ȭʚͨʛʛ ƍ ͛ȬƸ ͢ We define a rescaling of time to get rid of the ʚͨʛ terms in this evolution equation: / Ɣǹ ʚͩʛͩ͘ ͤ i ͛Ȭ ƔƎT͌͗Ȳ ƍ ͦ-1 ͛Ȭ ͛Ȭʚͨʛ Soi/ ) , that called the normalized Ricci flow. Therefore we checked that ͛Ȭʚͨʛ is a solution to the normalized Ricci flow. Note that i ͛ ƔƎT͌ ƍ ͦ ͦ͛ i/ $% $% ) $% that called unnormalized Ricci flow, may not have solution even for short time [Hamilton].

LaplacianspectrumunderRicciflow

Let ʚ͇)Q͛ʛ be a Riemannian manifold, we know $% ;͚ Ɣ ͛ $%͚ "͚ͦ "͚ $% & Ɣ ͛ ʦ Ǝϙ$% ʧ "ͬ$"ͬ% "ͬ& By an eigenvalue of ; we mean there exists a non-zero function ͚ such that ;͚ ƍ ͚ Ɣ R

Chapter 2 [ RICCI FLOW

The set of all eigenvalues is called the spectrum of the operator it is well-known that the spectrum of the laplacian –Beltrami operator is discrete and non-negative .So we can write for the Laplacian Eigen values is the following ways

RƔͤ Ɨͥ ƙͦ ƙͧ ƙˆ Note that if ͇) is closed ,Then assuming ͚ȹR we have Ȅ ǧ͚ ͦ͘ Ɣ ġ Ȅ ͚ͦ͘ ġ Here we assume that if we have a smoothly varying one-parameter family of metrics ʚͨʛ

, each Laplacian eigenvalue Pʚ͛ʚͨʛʛ will also vary smoothly.

Lemma2.4 On a closed manifold we have ͢ƎS ǹ͌͗ʚǧ͚Q ǧ͚ʛ͘ ƙ ǹ ʚ;͚ʛͦ͘ ġ ͢ ġ Theorem2.7 (Lichrerowicz) suppose f is an eigen function of the Laplacian with Eigen value: ;͚ ƍ ͚ Ɣ R If͌͗ ƚ ʚ͢ƎSʛ͛͟, where ͟ƘR is a constant, then ƚ͢͟ Proof: By applying lemma 2.4, we have

ʚ͢ƎSʛ͟ǹ ǧ͚ ͦ͘ ƙ ǹ ͌͗ʚǧ͚Q ǧ͚ʛ͘ ƙ ġ ġ ͢ƎS ͢ƎS ƙ ǹ ʚ;͚ʛͦ͘ Ɣ ͦ ǹ͚ͦ͘ ͢ ġ ͢ ġ

Ȅ ǧ! v [ Ɣ Ćġ and because v , so proof is complete. ȄĆġ ! [ Theorem2.8. Let ͇) be a closed manifold and ͛ʚͨʛ be a smooth one-parameter family of metrics which vary along the direction of a symmetric 2-tensor ͪ$%ʚͨʛ,i.e. "͛ $% Ɣͪ "ͨ $%

Then the Ǝth Laplacian Eigen value P evolves according to the following equation

Chapter 2 [ RICCI FLOW

͘ S P $ % P ͦ ͦ ƔƎǹ ͪ$% ͚P ͚P ͘ Ǝ ǹ͚ͪP ͘ ƍ ǹͪ ͚P ͘ ͨ͘ ġ T ġ T ġ $% Where ͪƔ͛ ͪ$% and ͚ʚͬQ ͨʛ is a time- dependent eigen function of PƳ͛ʚͨʛƷ with

Ȅ ͚ ʚͬQ ͨʛͦ ͘ ƔS ͨ ġ P "ʚ/ʛ for all . Ȅ ͚ͦ͘ Ɣ S Proof since ġ , we have

Ɣǹ ǧ͚ ͦ ͘ ġ i ǧ͚ ͦ at first we compute i/ . " " " " ǧ͚ ͦ Ɣ Ƴ͛$% ͚ ͚Ʒ Ɣ ƴ ͛$%Ƹ͚ ͚ƍT͛$% ƴ ͚Ƹ ͚ "ͨ "ͨ $ % "ͨ $ % $ "ͨ % " "͚ ƔƎ͛$&͛%' ƴ ͛ Ƹ͚ ͚ƍT͛$% ƴ Ƹ͚ "ͨ &' $ % $ "ͨ % ƔƎͪ &͚'͚ƍT͛$% ʠi!ʡ ͚ &' $ i/ % We also have " ͪ ͘ Ɣ ͘ "ͨ T Ȅ ͚ͦ͘ Ɣ S Moreover ,since ġ , take / on both side gives "͚ ͪ RƔǹ T͚ ƍ ͚ͦ͘ ġ "ͨ T So "͚ ƎS Tǹ ͚ ͘ Ɣ ǹ͚ͪͦ͘ʚTTWʛ ġ "ͨ T ġ Z Now we compute the /. ͘ " Ɣǹ ʚ ǧ͚ ͦ͘ʛ ͨ͘ ġ "ͨ " ͪ Ɣǹ ǧ͚ ͦ ƍ ǧ͚ ͦ͘ ġ "ͨ T " S $ % $% ͦ ƔƎǹ ͪ$% ͚ ͚͘ ƍ Tǹ ͛ $ ƴ ͚Ƹ %͚͘ ƍ ǹͪ ǧ͚ ͘ ġ ġ "ͨ T ġ ƔƎȄ ͪ $͚%͚͘ Ǝ T Ȅ ʠ i ͚ʡ ;͚͘ ƍ ͥ Ȅ ͪ ǧ͚ ͦ͘ ġ $% ġ i/ ͦ ġ

Chapter 2 [ RICCI FLOW

ƔƎȄ ͪ $͚%͚͘ ƍ T Ȅ ͚ i ͚͘ ƍ ͥ Ȅ ͪ ǧ͚ ͦ͘ ġ $% ġ i/ ͦ ġ by plugging (2.5) in last equation, proof will be complete. So we obtain the following lemma directly

i" Ĝĝ ƔƎT͌ Ƴ͛ʚͨʛƷ Lemma2.5. If i/ $%, Then the evolution of P is given by ͘ P $ % ͦ ͦ ƔTǹ ͌$% ͚P ͚P͘ ƍ P ǹ͚͌P ͘ Ǝ ǹ ͌ ǧ͚P ͘ ͨ͘ ġ ġ ġ ͚ ʚͬQ ͨʛ Ƴ͛ʚͨʛƷ Ȅ ͚ ʚͬQ ͨʛͦ͘ Ɣ S Where P is an Eigen function of P with ġ P for all ͨǨʞRQ͎ʛ.

Cigar Soliton

In Hamilton’s program for the Ricci flow on 3-manifold, via dimension reduction, the cigar soliton is a potential singularity model. Definition2.3.Hamilton’s cigar soliton is the complete Riemannian surface ͦ Ƴ͌ Q͛? Ʒwhere ͬͦ͘ ƍͭͦ͘ ͛ Ɣ ? Sƍͬͦ ƍͭͦ Where ͬͦ Ȭͬ̂ͬ͘͘ . As a solution to the Ricci flow ,its time-dependent version is: ͬͦ͘ ƍͭͦ͘ ͛ ʚͨʛ Ɣ ? ͙ͨ/ ƍͬͦ ƍͭͦ

v v ͬȭ Ɣ͙ͯͦ/ͬ ͭȭ Ɣ͙ͯͦ/ͭ ͛ Ɣ 3ȭ ͮ 4ȭ From the change of variables and , we see that ? ͥͮ3ȭ vͮ4ȭ v is isometric to ͛? Ɣ͛?ʚRʛ. That is, if we define the 1- parameter group diffeomorphisms ͦ ͦ ͯͦ/ ͯͦ/ ǳ /S͌ Ŵ͌ by /ʚͬQ ͭʛ Ɣ ʚ͙ ͬQ ͙ ͭʛ, Then ͛?ʚͨʛ Ɣ/ ͛?ʚRʛ. So ͛?ʚͨʛ is a Ricci soliton .Thus ,for each two times ͨͥQͨͦ Ǩ ʚƎ7Q 7ʛ,͛ʚͨͥʛ is isometric to ͛ʚͨͦʛ. In polar coordinates, we may rewrite to cigar metric as ͦͦ͘ ƍͦͦͦ͘ ͛ Ɣ ? Sƍͦͦ If we define the new radial distance variable

Chapter 2 [ RICCI FLOW

ͧS Ɣ ͕ͦ͗ͧͦ͢͜͝ Ɣ ͣ͛͠ ʠͦƍǭSƍͦͦʡ

Then we may rewrite ͛? as ͦ ͦ ͦ ͛? Ɣͧ͘ ƍ͕ͨ͢͜ ͧ͘ Also we can write ͇ ͇ ͯͥ ͦ͘ ͛ ƔƴSƎ Ƹͦ͘ ƍƴSƎ Ƹ ? Vͦ Ɣ ʚ͙ͯͦ5 ƍSʛͯͥʚͮͦ͘ ƍͦ͘ʛ Ɣ͇͗ͣͧͦͧ͜ ͮƔͥ *-%ʚ - ƎSʛ ͦƔǭͬͦ ƍͭͦ Where and ͦ (where ) ͥ ͌ ͥ And Ǩ͍ ʚSʛ Ɣ ƟT͔ where ͍ ʚSʛ denotes the circle of radius 1. Corollary2.2 (uniqueness of the cigar) if ʚ͇ͦQ͛ʚͨʛʛis a complete gradient Ricci soliton with positive curvature, then ʚ͇ͦQ͛ʚͨʛʛ is the cigar soliton.

Ricci soliton Definition2.4. Let ʚ͇Q ͛ʛ be a Riemannian manifold .A Ricci soliton structure on ͇ is a smooth vector field ͒ satisfying the Ricci equation S ͌͗͝ ƍ ͆ ͛Ɣ͛ T For some constant Ǩ͌ . Here Ric =Ricci curvature of M

͆=Lie derivative in the direction ͒ (a sort of directional derivative in the direction of ͒for tensor fields. Built using the local 1-parameter group of diffeomorphisms generated by ) We say the Ricci soliton is Contractive or shrinking if ƘR Steady if ƔR Expansive if ƗR

If ͒Ɣǧ͚ for some smooth function ͚S ͇ Ŵ ͌ .we say that ʚ͇Q ͛Q ǧ͚ʛis a gradient Ricci soliton with potential f. In this situation, the soliton equation reads ͌͗͝ ƍ ͙͂ͧͧʚ͚ʛ Ɣ ͛

Chapter 2 [ RICCI FLOW

Ricci solitons are a generalization of Einstein manifolds .they give rise to self similar solutions of the Ricci flow and arise as the blow up of some of the singularities of the Hamilton Ricci flow.

OVERVIEW ON POINCARE CONJECTURE V{tÑàxÜF

Chapter 2 overview on Poincare conjecture

Chapter 3

The Ricci flow approach to Poincare conjecture

We will spend this section giving a high-level overview of Perelman's Ricci flow –based proof of the Poincare conjecture ,and in particular how that conjecture is reduced to verifying a number of (highly non-trivial) facts about Ricci flow .Our exposition is based on [13],[14],[15],[16],[17] and [18]

We start with a question.

''If ͇ is a closed UƎmanifold with trivial fundamental group, then is M diffeomorphic to͍ͧ.

The Poincare conjecture is that the answer to this question is ''yes''. In 1980's Thurston developed another approach to 3-manifolds with Riemannian metrics of constant negative curvature -1 .These manifolds, which are locally isometric to Hyperbolic 3- space ,are called Hyperbolic manifold .There are fairly obvious obstructions showing that not every 3-manifold can admit such a metric . Thurston formulated a dereral conjecture that roughly says that the obvious obstructions are the only ones, Should they vanish for a particular 3-manifold, then that manifold admits such a metric .The aspect of Thurston's Geometrization conjecture that is most relevant for us is that the conjectural existence of especially nice metrics on 3-manifolds suggests a more analytic

28 Chapter 2 overview on Poincare conjecture approach to the problem of classifying 3-manifolds.Hamilton's formalized one such approach by introducing the Ricci flow on the space of Riemannian metrics .

Hamilton made significant progress on the program he initiated by establishing many crucial analytic estimates for understanding the evolving metric and its curvatures. After Perelman, on this work has claimed to surmount all of the various technical, geometric and analytic difficultics of Hamilton's program .In this way he claims to have established Thurston's Geometrization conjecture and hence the Poincare Conjecture .

Definition3.1. A homogeneous Riemannian manifold ʚ͇Q ͛ʛ is one whose group of isometries acts transitively on the manifold.

Examples of homogeneous manifolds are the round sphere ͍) ,Euclidean space ͌) and Hyperbolic space ͂) .

We say that a Riemannian manifold is modeled on a given homogeneous manifold ʚ͇Q ͛ʛ if every point of the manifold has a neighborhood isometric to an open set in ʚ͇Q ͛ʛ .Such manifolds are called locally homogeneous manifolds provided that they are complete .

Remark3.1. In dimension2 there are four simply connected homogeneous manifolds up to isometry :͍ͦQ ͌ͦQ ͂ͦ ,and ́ ,with a left invariant metric ,where ́ is the group ͌ʠ͌ǳ with the natural action of ͌ǳͣ͌͢.

Remark 3.2 .the geometry and topology of the surface are related by the Gauss –Bonnet T ʚ͇ʛ Ɣ Ȅ ͐͘͟ ʚ͇ʛ ͇ formula ,where is the Euler characteristic of ,Which is related with the genus ͛ of ͇ by ʚ͇ʛ ƔTƎT͛ ,and every oriented closed surface has a genus ͛ and can be described as a sphere with g handles glued to it ,where a handle is ̓Ɛ͍ͥ .

Theorem3.1 (uniformization in dimension 2) let ͒ be a compact surface then͒ admits a locally homogeneous metric locally isometric to one of constant curvature models above .The model will be positively curved if ʚ͒ʛ ƘR ,flat if ʚ͒ʛ ƔR ,and negatively curved or hyperbolic if ʚ͒ʛ ƗR .

29 Chapter 2 overview on Poincare conjecture The Sphere (or prime) Decomposition:

The first two main steps in the way to classification of 3-manifolds are the sphere Decomposition Theorem.

Definition 3.2.(connected sum) if a closed 3-manifold ͇ contains an embedded sphere ͦ ͦ ͍ separating ͇ into two components, we can split ͇ along this ͍ into manifolds ͇ͥ ͦ and ͇ͦ with boundary ͍ .We can then fill in these boundary spheres with 3-balls to

produce two closed manifolds ͈ͥ and ͈ͦ .One says that ͇ is the connected sum of ͈ͥ

and ͈ͦ ,and one writes ͇Ɣ͈ͥ3͈ͦ .This splitting operation is commutative and associative .

There is also a strong relationship between the topology of a connected sum and that of its components.

Remark.3.3 Leet ͇ and ͇ be connected manifolds of the same dimension

1) ͇3͇ is compact if and only if ͇ and͇ are both compact

2) ͇3͇ is Orientable if and only if ͇ and͇ are both Orientale

3)͇3͇ is simply connected if and only if ͇ and͇ are both simply connected

Remark3.4 .on rather trivial possibility for the decomposition of ͇ a connected sum is Ɣ ͇3͍ͧ .

Illustration of connected sum.

30 Chapter 2 overview on Poincare conjecture Now we give a theorem from Grushko about connected sum.

Theorem3.2 (Grushko) Let ͇ be a compact, connected 3-manifold, andͥʚ͇ʛ Ɣ́ͥ ǳ

́ͦ Then ͇Ɣ͇ͥ3͇ͦ where $ʚ͇$ʛ Ɣ́$ ʚ͝ Ɣ SQTʛ .

Definition3.3 (prime manifolds) a closed 3-manifold is called prime if every separating embedded 2-sphere bounds a 3-ball.

Or a another simple definition: a 3-manifold ͒ is said to be prime if it is not diffeomorphic to ͍ͧ and if every ͍ͦ ɗ͒ that separates ͒ into two pieces has the Property that one of the two pieces is diffeomorphic to a 3-ball.

Definition 3.4 : an embedded 2-sphere ͍ͦ ƌ͇ͧ is essential if it does not bound a ball in ͇ or is not parallel to a sphere in "͇ͧ .

An orientable 3-manifold ͇is irreducible if any embedding of the 2-sphere into ͇ extends to an embedding of the 3-ball into͇ .

One of the first theorems in the topology of three manifolds in due to Keneser in 1929.

Theorem3.3 .every closed, oriented 3-manifold admit a decomposition as a connected sum of oriented prime 3-manifolds, called its prime factors. This decomposition is unique up to the order of the prime factors (and orientation-preserving diffeomorphism of the factors).

There are a countably infinite number of prime 3-manifolds up to diffeomorphism .

There are three main classes of prime manifolds

Type I: With finite fundamental group .All the known examples are the spherical 3- w ͇Ɣ ϙ ͍͉ʚVʛ manifolds, of the form, 8 ,where is a finite subgroup of acting freely on ͧ ͍ by rotations .Thus ϙƔͥʚ͇ʛ .It is and old conjecture that spherical 3-manifolds are the only closed 3-manifolds with finite fundamental group (it is ,in fact ,Poincare conjecture ) .

31 Chapter 2 overview on Poincare conjecture Type II) with infinite cyclic fundamental group .there is only one prime 3-manifold satisfying this condition: ͍ͥ Ɛ͍ͧ .This also the only orientable 3-manifold that is prime

but not irreducible .it is also the only prime orientable 3-manifold with non-trivial ͦ .

Type III) with infinite non cyclic fundamental group .these are ͅʚQSʛ Manifolds (also called aspherical); i.e , manifolds with contractible universal cover. Any irreducible 3-

manifold͇, with ͥ infinite is a ͅʚQSʛ.

Suppose that we start with a 3-manifold ͇ which is connected and not prime.

Then we can decompose ͇Ɣ͈ͥ3͈ͦ ,where no ͈$ is a sphere .Now ,either each ͈$is irreducible ,or we can iterate this procedure. The theorem of Keneser (1929) states that this procedure always stops after a finite number of steps, yielding a manifold ͇ such that each connected component of ͇ is irreducible .

Theorem3.4 (prime Decomposition) Let ͇ be a orientable closed 3-manifold.Then ͇ admits a finite connected sum decomposition

- ͦ ͥ ͇Ɣʚͥͅ3ͦͅ U3ͅʛ3ʚ͆ͥ3͆ͦ U3͆,ʛ3ʚ3͍ͥ Ɛ͍ ʛ

The ͅ and ͆ factors here are closed and irreducible 3-manifolds.The ͅ factors have infinite fundamental group and are aspherical 3-manifold (are of type III), while the ͆ factors have finite fundamental group and have universal cover a homotopy 3-sphere (are of type I)

The Thurston Geometrization conjecture

In dimension three, every finite volume, locally homogeneous manifold is modeled on one of the eight homogeneous manifolds listed below .First; we have the constant (sectional) curvature examples:

1) ¯̌,of constant curvature +1 2) ®̌,which is flat 3) ¤̌of constant curvature -1

32 Chapter 2 overview on Poincare conjecture Next we have the homogeneous 3-manifold with product metrics:

4)¯̋ Ɛ®

5)¤̋ Ɛ®

Finite volume locally homogeneous manifolds modeled on ͍ͦ Ɛ͌ are automatically compact and either are isometric ͍ͦ Ɛ͍ͥ or ͌͊ͧ3͌͊ͧ.

Finite volume locally homogeneous manifolds modeled on ͂ͦ Ɛ͌ either are of the form ?Ɛ͍ͥ ,where ? is a finite area hyperbolic surface ,or are finitely covered by such manifolds.

Lastly, we have the homogeneous manifolds ʚ͇Q ͛ʛ where ͇ is a simply connected lie group and ͛ is a left-invariant metric.

Three-dimensional examples of this type admitting locally homogeneous examples of finite volume are:

Definition3.5. Heisenberg group. Is a group of UƐU upper triangular matrices of the S͕͗ form ƵRS͖ƹ .Elements ͕Q ͖Q ͗ can be taken from some arbitrary commutative ring. RRS

6) The unipotent group (Heisenberg group) Locally homogeneous manifolds modeled on this group are called Nil-manifolds. Note that this is the only 3-dimensional nilpotent but not abelian connected and simply connected Lie group; This explains the term Nil geometry topologically, Nil is diffeomorphic to ͌ͧ under the map

S͕͗ ͈͝͠ ǫ Ɣ ƵRS͖ƹ Ŵ ʚ͕Q ͖Q ͗ʛ Ǩ͌ͧ RRS

Under this identification, left multiplication by corresponds to the map

͆RʚͬQ ͭQ ͮʛ Ɣ ʚͬƍ͕Qͭƍ͖Qͮƍ͕ͭƍ͗ʛ

In other words, from this point of view,͌ͧ has the multiplication

33 Chapter 2 overview on Poincare conjecture

ʚͬͤQͭͤQͮͤʛʚͬQ ͭQ ͮʛ Ɣ ʚͬƍͬͤQͭƍͭͤQͮƍͮͤ ƍͬͤͭʛ

It is easy to prove that Nil is a Lie group, so it admits a metric invariant under left multiplication; we shall take ͧͦ͘ Ɣͬͦ͘ ƍͭͦ͘ ƍͮͦ͘ at ʚRQRQRʛ -the unit of the Heisenberg group and the extend ͧͦ͘ at all other points of ͒ as a left invariant metric .The result is

ͦ ͧͦ͘ Ɣͬͦ͘ ƍͭͦ͘ ƍƳͮ͘Ǝͬͭ͘ Ʒ

Next we check that ͧͦ͘ is invariant under multiplications

ͦ ǳ ͦ ǳ ͦ ǳ ͦ ǳ ǳ ǳ ͆R ʚͧ͘ ʛ Ɣ͘Ƴ͆R ͬƷ ƍ͘Ƴ͆R ͭƷ ƍ ʠ͘Ƴ͆R ͮƷ Ǝ ͆R ͬ͘Ƴ͆R ͭƷʡ

ͦ ͦ ͦ Ɣ ͘Ƴͬͣ͆RƷ ƍ͘Ƴͭͣ͆RƷ ƍ ʠ͘Ƴͮͣ͆RƷƎƳͬͣ͆RƷ͘Ƴͭͣ͆RƷʡ

ͦ Ɣ͘ʚͬƍ͕ʛͦ ƍ͘ʚͭƍ͕ʛͦ ƍƳ͘ʚͮƍ͕ͭƍ͗ʛ Ǝ ʚͬƍ͕ʛ͘ʚͭƍ͖ʛƷ

Ɣͬͦ͘ ƍͭͦ͘ ƍ ʚͮ͘ ƍ ͕ͭ͘ Ǝ ͬͭ͘ Ǝ ͕ͭ͘ʛͦ Ɣͧͦ͘

If we identify ͍ͥwith the interval ʞRQTʟwith the ends identified then a point Ǩ͍ͥacts on Nil by ċS͍ͥ Ɛ͈͝͠ Ŵ ͈͝͠ such that

ċ ƳQ ʚͬQ ͭQ ͮʛƷ Ŵ ʚͬQ ͭQ ͮʛ Where

˫ ͬ Ɣ ͬ͗ͣͧ Ǝ ͭͧ͢͝ ˮ ˮ ˮͭ Ɣ ͬͧ͢͝ ƍ ͭ͗ͣͧ ͚ʚͬʛ Ɣ ˬ S ˮ ͮ Ɣ ͮ ƍ ͧ͢͝ʚ͗ͣͧʚͬͦ Ǝͭͦʛ ƎTͧͬͭ͢͝ʛ ˮ T ˮ ˭

Where ċ is an action of ͍ͥ on Nil which is a group of automorphisms of Nil preserving the above metric.

34 Chapter 2 overview on Poincare conjecture Finite volume manifolds with this geometry are compact and orientable and have the structure of Seifert fiber space.

Also we can identify Nil with the subset ́ of Ŀͦdefined as ʜʚͩQ ͪʛ ǨĿͦS̓ͪ͡Ɣ ͩ ͦʝ and with multiplication on ́ defined by

ʚͩQ ͪʛT ʚͩQͪʛ ƔʚͩƍͩQͪƍͪ ƍTͩͩ͝Ŭʛ

An isomorphism from Nil to ́ can be given by the formula

S S S ʚͬQ ͭQ ͮʛ Ɣ ʦ ʚͬƍͭ͝ʛQͮƎ ͬͭ ƍ ͝ʚͬͦ ƍͭͦʛʧ T T V

7) Sol geometry

At first we give two points .

1) A Lie group ́ is said to be solvable if it is connected and its Lie algebra is solvable.

2) Let ́ and ͂ two Lie groups and consider a homomorphism from ́ to the abstract

group of automorphisms of ͂ ,that is ,S ́ Ŵ ̻ͩͨʚ͂ʛT The semi direct product ͂Ɛ` ́ of ͂ and ́ with respect to is the product manifold ͂Ɛ́ endowed with the Lie group structure given by

ʚ͜Q ͛ʛT ʚ͜ĺQ ͛ʛ Ɣ ʚ͜ʚ͛ʛ͜ĺQ ͛͛ʛ , ʚ͜Q ͛ʛͯͥ Ɣ ʚʚ͛ͯͥʛͯͥ͜Q͛ͯͥʛ Ǣ͜Q ͜ Ǩ͇ and Ǣ͛Q ͛ Ǩ́

Topologically, we can identify Sol with ͌ͧ so that the multiplication is given by

ͯ5t 5 ʚͬͤQͭͤQͮͤʛʚͬQ ͭQ ͮʛ Ɣ ʚͬƍ͙ ͬͤQͭƍ͙ ͭͤQͮƍͮͤʛ

Clearly ʚRQRQRʛ is identity and ͬͭ Ǝplane is a normal subgroup isomorphic to ͌ͦ .In fact ,

ʚͬQ ͭQ ͮʛͯͥT ʚ͕Q ͖Q RʛT ʚͬQ ͭQ ͮʛ Ɣ ʚƎ͙ͬͯ5QƎ͙ͭ5QƎͮʛʚ͕Q ͖Q RʛT ʚͬQ ͭQ ͮʛ

Ɣ ʚ͕Ǝ͙ͬͯ5Q͖Ǝ͙ͭ5QƎͮʛʚͬQ ͭQ ͮʛ Ɣ ʚǳQǳ QRʛ

35 Chapter 2 overview on Poincare conjecture Metrically, Sol is just ͌ͧ but endowed with the left invariant Riemannian metric which at ʚͬQ ͭQ ͮʛ is

ͧͦ͘ Ɣ͙ͦ5ͬͦ͘ ƍ͙ͯͦ5ͭͦ͘ ƍͮͦ͘ (3.1)

Another equivalent approach to the algebraic definition of Sol will be given along the proof of Thurston theorem .In this way; we can say that Sol is the Unimodular Lie group completely determined by

ʞ͙ͥQ͙ͦʟ Ɣ RQʞ͙ͦQ͙ͧʟ Ɣ͙ͥQʞ͙ͧQ͙ͥʟ ƔƎ͙ͦ

Where ʜ͙ͥQ͙ͦQ͙ͧʝ is an orthonormal basis of eigenvectors of the Lie algebra of ͒ ȸ ͍ͣ͠ ͦ .As the generators ͙ͥ and ͙ͦ are commute .Sol contains a copy of ͌ which is a normal subgroup and the quotient group is R. The group is therefore a semi direct product of / ͌ͦ ͌ ͨ Ŵ ʠ͙ R ʡ with .Therefore the transformations in this basis are of the form R͙ͯ/ .

The metric (1.3) is preserved by the group G of transformations of X of the form

ʚͬQ ͭQ ͮʛ Ŵ ʚ ͙ͯͬƍ͕Q ͙ͭƍ͖Qͮƍ͗ʛͣͦʚ ͙ͯͭƍ͕Q ͙ͬƍ͖Qͮƍ͗ʛ

Where ͕Q ͖Q ͗ Ǩ ő and Q Ɣ ƏST

ȳ 8)£ Ɣ ¬¯¨̋ʚőʛ The universal covering group of ͍͊͆ͦʚőʛ .This manifold can also be viewed by as the universal covering of the unit tangent bundle to ͂ͦ with its induced metric .Finite volume locally homogeneous manifolds modeled on this example are circle bundles over hyperbolic surfaces.

Theorem3.5 suppose that ͒ͧ is connected and orientable and admits a locally homogeneous Riemannian metric of finite volume. Then,͒ is diffeomorphic to the interior of a compact 3-manifold with boundary ,all of whose boundary components are Tori. Furthermore, each of these tori has fundamental group which injects into the fundamental group of ͒ .if ͒ is non-compact, then it is modeled on ͂ͧ,͂ͦ Ɛ ő or ȳ ͍͊͆ͦʚőʛ ,and hence , ͒either is a hyperbolic 3-manifold or is Seifert –fibered with hyperbolic two –dimensional orbifold base.

36 Chapter 2 overview on Poincare conjecture The Geometrization conjecture reads as follows:

Geometrization conjecture: Every closed orientable UƎmanifold ͇is a connected sum

of closed UƎmanifold ͇$ such that for every ͝ there is a finite collection of pair wise

disjoint embedded Tori ͎$% ɗ͇$ such that

ƣ͎ Ƨ ͇ ͎ ƌ ͇ 1) The Tori $% % are incompressible in $ (i.e. ͥ $% ͥ $)

2) The components of ͇$ ǲȃ% ͎$% are diffeomorphic to metric quotients with finite volume of one of the following 8 homogeneous geometries

͍ͧQ͍ͦ Ɛ͍ͥQ͂ͧQ͌ͧQ͂ͦ Ɛ͌Q ͍͊͆Ȳ ʚTQ ͌ʛQ ͈͍ͣͦͣ͝͠͠

Note that the Poincare conjecture follows easily from the geometrization conjecture

Actions on geometric manifolds

Definition3.6 Ƴʚ͒Q ϙʛ Ǝ͙ͧͨͦͩ͗ͨͩͦƷ.Let ͒ be a topological manifold and ϙ a group acting on͒ .An ʚ͒Q ϙʛ Ǝ ͙ͧͨͦͩ͗ͨͩͦ for a manifold ͇ is a maximal ʚ͒Q ϙʛ Ǝcompatible

collection of charts $S͏$ Ŵ͒ covering ͇.Two charts $Q% are ʚ͒Q ϙʛ Ǝcompatible ͯͥ ,if on each component ͐ of ͏$ Ȃ ͏% the coordinate change %ͣ$ ɳ u is the eĜ ʚʛ restriction of some element ͛Ǩϙ.

Definition 3.7(model geometry, geometric manifold) a model geometry is a smooth, simply connected manifold ͒ with a Lie- group ϙ acting transitively on ͒ such that,

1) ϙ has compact point stabilizer 2) ϙ Is maximal in the sense that it is not contained in any larger group of diffeomorphisms of ͒ with compact point stabilizer. 3) There exists at least one compact manifold with an ʚ͒Q ϙʛ Ǝ ͙ͧͨͦͩ͗ͨͩͦ if ʚ͒Q ϙʛ is a model geometry, then an ʚ͒Q ϙʛ Ǝ͙ͧͨͦͩ͗ͨͩͦ is called geometric structure and a manifold with a geometric structure is called geometric manifold.

We consider smooth actions S ́ ƙ ͇ of a finite group ́ on a smooth manifold͇.

37 Chapter 2 overview on Poincare conjecture Definition3.8 (standard action) let ʚ͒Q ϙʛ be a model geometry and ͇ an ʚ͒Q ϙʛ Ǝmanifold .We say, the action S ́ ƙ ͇ is standard, if there exists a ʚ́ʛ- invariant complete locally homogeneous metric on ͇ .

Theorem3.6 :Let ͇Ɣ͍ͦ Ɛ͍ͥ ,and S ́ ƙ ͇ a smooth finite group action .Then the action is standard.

Any locally homogeneous manifold modeled on the ͍ͧ of constant curvature +1 is a Riemannian manifold of constant positive sectional curvature.

These are of the from ͍ͧ ϙ where ϙ is a finite subgroup of ͍͉ʚVʛ acting freely on͍ͧ ,͌͊ͧ,lens space ,as well as the quotients by the symmetry groups of the exceptional regular solids .These manifolds are called spherical space-form.

Definition 3.9. A Riemannian four-manifold is said to have positive isometric curvature if for every orthonormal four-frame the curvature tensor satisfies

͌ͥͧͥͧƍ͌ͥͨͥͨ ƍ͌ͦͧͦͧ ƍ͌ͦͨͦͨ ƘT͌ͥͦͧͨ

Definition3.10. An incompressible space from͈ͧ in a four-manifold ͇ͨ is a three dimensional sub manifold diffeomorphic to ͍ͧ ϙ , (the quotient of the three sphere by a ͧ group of isometries without fixed point) such that the fundamental group ͥʚ͈ ʛ ͨ injects into ͥʚ͇ ʛ .

The space form is said to be essential unlessϙ Ɣ ʜSʝ ,or ϙƔ͔ͦ and the normal bundle is non-orientable .

Theorem3.7 (Bing-Long chen Xi-Ping Zhu) let ͇ͨ be a compact four-manifold with no

essential incompressible space-form and with a metric ͛$% of positive isotropic ͛ʚ&ʛ ʚͨʛ ͟Ɣ curvature. Then we have a finite collection of smooth solutions $% , ͨ RQSQTQ U Q ͡.to the Ricci flow ,defined on ͇& Ɛʞͨ &Qͨ&ͮͥʛ, (RƔͨ& ƗˆƗͨ(ͮͥ)with ͇ ͨ Ɣ͇ͨ ͛ʚͤʛ ʚͨ ʛ Ɣ͛ ͨŴͨ ͤ and $% ͤ $% ,which go singular as &ͮͥ ,such that the following properties hold :

38 Chapter 2 overview on Poincare conjecture i) For each ͟ Ɣ RQSQ U Q ͡ Ǝ S,the compact (possible disconnected) four-manifold ͇ ͨ ͛ʚ&ʛ ʚͨʛ & contains an open set & such that the solution $% can be smoothly

extended to ͨƔͨ&ͮͥ over& ;

͟ Ɣ RQSQ U Q ͡ Ǝ S ƴ Q ͛ʚ&ʛ ʚͨ ʛƸ ƴ͇ͨ Q ͛ʚ&ͮͥʛ ʚͨ ʛƸ ii) For each , & $% &ͮͥ and &ͮͥ $% &ͮͥ

contain compact (possible disconnected) four-dimensional sub manifolds with ͨ smooth boundary ,which are isometric and then can be denoted by ͈& , ͨ ͨ iii) For each ͟ Ɣ RQSQ U Q ͡ Ǝ S, ͇ & ǲ ͈ & consists of a finite number of disjoint ͧ ͨ ͨ ͨ ͨ ͨ pieces diffeomorphic to ͍ Ɛ̓,̼ or ͌͊ ǲ ̼ ,while ͇ &ͮͥ ǲ ͈ & consists of a finite number of disjoint pieces diffeomorphic to ̼ͨ ; ͨ iv) For ͟Ɣ͡ ,͇( is diffeomorphic to the disjoint union of a finite number of ͍ͨ ,or ͌͊ͨ ,or ͍ͧ Ɛ͍ͥ ,or ͍ͧ Ɛȭ ͍ͥ ,or ͌͊ͨ ʬ ͌͊ͨ .(note ͍ͧ Ɛȭ ͍ͥ ,is equal ͧ ͥ ͧ ͥ Ǵ with͍ Ɛ͍ ͔ͦ where͔ͦ flips ͍ antipodally and rotates ͍ by SZR )

As a direct consequence we have the following classification result of Hamilton.

Corollary (Hamilton): a compact four manifold with no essential incompressible space form and with a metric of positive isometric curvature is diffeomorphic to ͍ͨ , or ͌͊ͨ , or ͍ͧ Ɛ͍ͥ ,or ͍ͧ Ɛȭ ͍ͥ ,or a connected sum of them .

Hanilton introduced a program to study all 3-manifolds using the Ricci flow.

Short-time existence and uniqueness :if ͛ͤis a smooth metric on a compact manifold

,then there is some ƘR depending on ͛ͤ and a unique solution to the Ricci flow

equation defined for ͨǨʞRQ ʛ with ͛Ɣ͛ͤ. Curvature characterization of singularity formulation :if the solution exists on the time interval ʞRQ ͎ʛ but does not extend to any strictly larger time interval ,Then there is a point ͬ in the manifold for which curvature tensor ͌͡ʚͬQ ͨʛ of the metric ͛ʚͨʛ is unbounded as ͨ approaches ͎. Hamilton was discovered quite early that the Ricci flow may develop singularities even in the case of a sphere if the Ricci curvature is not positive .an example is the so-called

39 Chapter 2 overview on Poincare conjecture neck pinch singularity .Perelman’s work describes what happens to the Ricci flow nears a singularity and also how to Perform the surgery .The new flow is called Ricci flow with surgery. Theorem 3.8(Hamilton) Let ͒ͧbe a compact connected 3-manifold with non-negative Ricci curvature .Then one of the following happens: 1) The Ricci curvature becomes strictly positive for all ͨƘR sufficiently small. In this case, the Ricci flow develops a singularity in finite time.

As the singularity develops ,the diameter of the manifolds goes to zero .Rescaling the evolving family of metrics so that their diameters are one leads to a family of metrics converging smoothly to a metric of constant positive curvature .In particular ,the manifold is diffeomorphic to a spherical space-form (note that if ͎Ɨ7 and the curvature becomes unbounded as ͨ tends to T ,we say the maximal solution develops singularities as ͨ tends to ͎ and͎ is the singular time )

2) There is a finite cover of the Riemannian manifold which ,with the induced metric ,is a metric product of a compact surface of positive curvature and ͍ͥ.this remains true for all the Riemannian metrics in the Ricci flow .The Ricci flow develops a singularity in finite time ,and the manifold in question is diffeomorphic to͍ͦ Ɛ͍ͥ ,or ͌͊ͧ ʬ ͌͊ͧ (the case of͌͊ͧ ʬ ͌͊ͧ is interesting in that it is apparently the only non prime 3- manifold which admit a geometric structure) 3) The metric is flat and the evolution equation is constant. In this case, of course, the manifold is covered by͎ͧ.

Theorem3.9 (Hamilton) if the Ricci flow with initial conditions (͇,͛ͤ) be a connected, compact and Ricci flow exists for all ͨǨʞRQ7ʛ and also the normalized curvature . ͌͡ʚͬQ ͨʛ is bounded as ͨŴƍ7.Then ͇ satisfies Thurston’s Geometrization conjecture .

Perelman’sclaims

Regions of High curvature in the flow :in order to do surgery we need to understand regions of the flow where the scalar curvature is large .of course ,since it is possible to

40 Chapter 2 overview on Poincare conjecture rescale the metric and time in any flow ,we must normalize in some fashion to have an invariant notion of large curvature .Thus ,we arrange that our flow has normalized initial conditions in the sense that at ͨƔR the absolute value of the Riemannian curvature at each point is at most one, and the volume of any metric ball of radius one is at least half that of the unit ball in ͌ͧ.

From now on we implicity assume that all flows under consideration have normalized initial conditions.

RƗ Ɨͥ ͟ For all the following definitions we fix ͦ .Set equal to the greatest integer less than or equal to ͯͥ .in particular,͟ ƚ T.

Definition3.11 suppose that we have a fixed metric ͛ͤon a manifold ͇ and an open sub

ʢuʣ manifold ͒ɗ͇ .We say that another metric ͛ on ͒ is within of ͛ͤ in the ̽ Ō - ͟ Ɣ ʢͥʣ Topology if, setting T we have

& ͦ ͧͩͤ Ƶ ͛ʚͬʛ Ǝ͛ ʚͬʛ ͦ ƍ ȕɳ' ͛ʚͬʛɳ ƹ Ɨ ͦ ͤ "t "t 3Ǩ "t 'Ͱͥ

' ͛ Where the covariant derivative "t is the Levi-Civita connection of ͤand norms are the

point wise ͛ͤ Ǝnorms on

͍͎ͭͦ͡ǳ͇ ɬ ͎Ƀǳɇ͇U͎ɇɄɇɇǳ͇Ʌ 'ͯ/$( .

Definition3.12 Letʚ͈Q ͛ʛbe a Riemannian manifold and ͬǨ͈ a point .Then an -neck structure on ʚ͈Q ͛ʛ centered at ͬ consists of a diffeomorphism

S ͍ͦ ƐʚƎ ͯͥQ ͯͥʛ Ŵ ͈

ʢuʣ With ͬǨʚ͍ͦ Ɛ ʜRʝʛ ,such that the metric ͌ʚͬʛǳ͛ is within in the ̽ Ō Ǝ ͎ͣͤͣͣ͛ͭ͠ of the product of the usual Euclidean metric on the open interval with the ͥ ͍ͦ metric of constant Gaussian curvature ͦ on .

41 Chapter 2 overview on Poincare conjecture

Now we give a claim from Perelman.

Claim ( Perelman) there is ͦƘR such that the following holds .Let ʚ͇Q ͛ʚͨʛʛ be a Ricci flow with normalized initial conditions defined for RƙͨƗ͎ with ͇ a closed ,orientable 3-manifold.Then , for any point ʚͬQ ͨʛ in the flow with ͌ʚͬQ ͨʛ ƚ ͦͯͦ one of the following holds

1) The components of ͇/ containing ʚͬQ ͨʛ is diffeomorphic to a spherical space-form.

2) ʚͬQ ͨʛ is the center of an Ǝ͙͗͢͟ in ͇/ ͧ 3) ʚͬQ ͨʛ is contained in a sub manifold of ͇/ diffeomorphic to ̾ with every point in the

boundary being the center of an Ǝneck in͇/ .

Note that: such a region is called an Ǝ͕͗ͤ

Definition3.13 supposes that as ͨŴ͎ the Ricci flow becomes singular .there is an

extension of the above theorem to this time as well. There is an open subset ɗ͇ consisting of all the points where the Riemannian Curvature tensor remains bounded .and according to papers of Shi and Hamilton, There is a limiting metric on this open

subset as ͨŴ͎ .we denote itʚQ͛ʚ͎ʛʛ.

Definition3.14 in doing surgery there are three parameters that are fixed .The first is the coarse control parameter ƘR ,sufficiently small, a universal constant fixed once and for all .The other two parameters are non-increasing functions of ͨ limiting to Zero asͨ Ŵ 7 .They are ʚͨʛ ƘR,The fine control parameter, and ͜ʚͨʛ,The scale parameter .we also fix an auxiliary parameter ʚͨʛ Ɣ ʚͨʛ .

42 Chapter 2 overview on Poincare conjecture

Definition 3.15 we fix ͦƘR and a surgery parameter ƘR .set ƔͦT .Let ʚʛ be the subset of points ʚͬQ ͎ʛ for which ͌ʚͬQ ͎ʛ ƙ ͯͦ

ͦ 1) An ƎTube in is a sub manifold diffeomorphic to͍ Ɛ̓ such that each point is the

center of an Ǝneck in .

2) An ƎCircuit in is a component of which is a closed manifold and each one of its points is the center of an Ǝneck .It is diffeomorphic to ͍ͦ Ɛ͍ͥ. ͦ 3) an Ǝhorn is a closed subset ͂ɗ diffeomorphic to ͍ ƐʞRQSʛ with boundary

contained in ʚʛ such that every point of ͂ is the center of an Ǝneck in .of course ,the scalar curvature goes to infinity at the other end of H. ͦ 4) A double Ǝhorn is a component of diffeomorphic to ͍ ƐʚRQSʛ such that every

point is the center of an Ǝneck in .The scalar curvature goes to infinity at each end of this component . ͧ 5) A capped Ǝhorn is a component of diffeomorphic to ͨ͢͝ʚ̾ ʛ such that each is either the center of an Ǝneck or is contained in an Ǝcap .The scalar curvature goes to infinity near the end of a capped Ǝhorn

43 Chapter 2 overview on Poincare conjecture

Corollary At time ͎ every ͬǨ Ǝʚʛ,is contained in one of the following :

1) A component of containing ͬ is diffeomorphic to a quotient of a sphere , 2) An Ǝcircuit diffeomorphic to ͍ͦ Ɛ͍ͥ,

3) An ƎTube with boundary components in ʚʛ

4) an ƎCap with boundary in ʚʛ

5) an Ǝhorn with boundary in ʚʛ 6) a capped Ǝhorn 7) a double Ǝhorn

Perelman’s functional ŀ̡̮̤ő

One of perelmans claims is that, he proved that Ricci flow is a gradient flow up to diffeomorphism .Of course note that Ricci flow is not gradient flow .for explain the trick of Perelman for it we at first define a function that called Đ-functional

44 Chapter 2 overview on Poincare conjecture

Đʚ͛Q ͚ʛ Ɣǹ ʚ͌ƍ ͚ ͦʛ͙ͯ! ͐ͣ͘͠T

Where ͛ is a Riemannian metric on M and f is a smooth function .

Restrict this function to the subspace of ė Ɛ̽Ϧʚ͇ʛ given by ʚ͛Q ͚ʛ such that the volume form͙ͯ!͐ͣ͘͠ is constant, equal to a fixed one denoted by ͘͡.The gradient flow equation forĐ is then

i" ƔƎTʚ͌͗͝ ƍ ͚̾ͦʛ i! ƔƎ;͚Ǝ͌ i/ , i/ (3.1)

Now the idea is to deform a solution ʚ͛ʚͨʛQ ͚ʚͨʛʛ of (3.1) to kill the term ͚̾ͦ and obtain a solution ͛Ȭʚͨʛ of Ricci flow .Now we recall a classical result of Riemannian Ɣ͚ ʚǳ͛ʛ Ɣ geometry .if is a family of diffeomorphisms such that . . " , then . . T͚̾ͦ Ɣ͚ .Given a solution of (3.1) one can define diffeomorphisms /such that / ǳ ȱ at time ͨ .Then͛ȭ ʚͨʛ Ɣ/ ͛ʚͨʛ and͚ʚͨʛ Ɣ͚ʚͨʛͣ/ solve the system :

ɺ ͦ i"Ȭ ƔƎT͌͗͝ i! ƔƎ;͚ɺ Ǝ͌ƍɳ͚ɺɳ i/ , i/

The first equation gives a solution to the Ricci flow .Moreover, Đʚ͛ʚͨʛQ ͚ʚͨʛʛ Ɣ ĐƳ͛ȬʚͨʛQ ͚ɺʚͨʛƷ.

ElliptizationandHyperbolizationconjecture

Definition3.16 A Seifert fibration is a partition by circles (locally a product) except for a finite number of singular fibres, that have the following local model .We consider a cylinder, the product ̾ͦ ƐʞRQSʟ of a disc with an interval, and glue ̾ͦ Ɛ ʜRʝ with ̾ͦ Ɛ ʜSʝ by a rotation of finite order .The fibration by horizontal intervals induces a partition by circles of the solid torus,that is a fibration except for the singular fibre ʜRʝ Ɛ͍ͥ,which is shorter .

45 Chapter 2 overview on Poincare conjecture

We know a manifold said to be hyperbolic if its interior has a complete metric of constant curvature -1 .it is important to notice that manifolds with finite fundamental group cannot be hyperbolic.

Hyperbolization conjecture: Let ͇ͧ be a prime, closed three manifold, with ͧ ͧ ͧ ͥʚ͇ ʛinfinite and such that every subgroup ͔ ɪ ͔Ɨͥʚ͇ ʛ comes from "ʚ͇ ʛ .Then ͇ͧ is hyperbolic.

Also a 3-manifold is said to be elliptic if it admits a metric of constant curvature +1.

ͧ ͧ ͧ If ͇ is a closed three manifold and ͥʚ͇ ʛ is finite, then equipping ͇ with an elliptic metric is equivalent to admit a Seifert fibration, using the fact both families elliptic manifolds and Seifert fibered ones are classified.

ͧ ͧ Elliptization conjecture: let ͇ be a prime, closed three-manifold, with ͥʚ͇ ʛ finite .Then ͇ͧ is elliptic.

Now we explain some partial result about Geometrization before Perelman

Definition 3.17 A 3-manifold is called a graph manifold if it is a union of Seifert manifolds along the boundary, consisting of Tori.

Definition3.18 on orientable 3-manifold is called sufficiently large if it contains an property embedded incompressible surface.

The next result is Thurston’s Geometrization for sufficiently large manifolds, that was one of the main evidences to support this conjecture

A surface ̀ͦ ɗ͇ͧ is called properly embedded if it is embedded and "̀ Ɣ ̀Ȃ"͇ͧ

46 Chapter 2 overview on Poincare conjecture Theorem3.10 (Thurston) a sufficiently large three manifold satisfies the Geometrization conjecture.

The following theorem is involves convergence groups

ͧ ͧ Theorem3.11 let ͇ be a compact irreducible 3- manifold .if ͥʚ͇ ʛ has an infinite cyclic normal sub-group, then ͇ͧ is Seifert fibered

ConsequencesofGeometrization:

Some consequences of Geometrization in 3-dimensional topology are listed in this section:

Theorem3.12 (Borel conjecture in dimension three) if two aspherical compact three manifolds are homotopically equivalent, then they are homeomorphic .

Theorem3.13: compact aspherical three manifolds are classified by its fundamental group.

Theorem3.14 (Gromov-Lawson) if a compact three manifold admits a metric of non- negative scalar curvature then it is either flat or a connected sum of elliptic manifolds and ͍ͦ Ɛ͍ͥ or its quotients.

HamiltonclaimsonGeometrizationconjecture

In the three dimensional case, the curvature operator (acting on 2-forms) diagonalizes

ͥ RR Ƶ R ͦ R ƹ RR ͧ

͇ PĜ So that the $ are functions on .Then the ͦ are sectional curvatures,

47 Chapter 2 overview on Poincare conjecture ƍ ͦ ͧ RR ˟ T ˢ ͥ ƍ ͦ ˠ R R ˣ ˠ T ˣ ƍ RRͥ ͧ ˡ T ˤ

And ͌Ɣ ͥ ƍ ͦ ƍ ͧ

The evolution equations for the $ are

ɑ ͦ ͥ Ɣ; ͥ ƍ ͥ ƍ ͦ ͧ

ɑ ͦ ͦ Ɣ; ͦ ƍ ͦ ƍ ͧ ͥ

ɑ ͦ ͧ Ɣ; ͧ ƍ ͧ ƍ ͥ ͦ

Hamilton proved, in dimension three,͌͗͝ ƚ R ,͌͗͝ Ƙ R ,͍͙͗ ƚ R and ,͍͙͗ Ƙ R are conditions invariant under the Ricci flow .

Maximum principles where also used by Hamilton to control $ in the following Theorem.

Theorem 3.15(Hamilton) if a compact three manifold ͇ͧ admits a metric with ͌͗͝ Ƙ R ,then the Ricci flow, after rescaling ,converges to a metric with constant positive sectional curvature .In Particular ͇ͧ is elliptic .

Also Hamilton developed a strong maximum principle for tensors, and used it to show that if ͌͗͝ ƚ R ,Then one of the three possibilities happen

1) the metric is flat 2) ͌͗͝ Ƙ R at ͨƘR , hence ͇ is elliptic ͧ 3) The metric is locally a product ͛Ɣ͛ͥ ɪ ͬ͘ .In this case the manifold is diffeomorphic to ͍ͦ Ɛ͍ͥ or it is a quotient ͌͊ͧ ʬ ͌͊ͧ .

Hence we conclude the following Theorem from Hamilton

Theorem3.16 a closed manifold with a Riemannian metric with ͌͗͝ ƚ R satisfies Thurston’s Geometrization conjecture.

48 Chapter 2 overview on Poincare conjecture Hamilton –Ivey estimates are another example of cleaver application of maximum principles for tensors in dimension three .Let S ʞƎSQ ƍ7ʛ ŴʞSQ7ʛ be the inverse map of ͬŴͣ͛ͬ͠Ǝͬ .Then we have the following Theorem

Theorem3.17 (Hamilton-Ivey pinching) The inequalities

͌ƚƎS and ͥQ ͦQ ͧ ƚƎʚ͌ʛ

Are invariant under the Ricci flow.

Hamilton’sworkonformationofsingularitiesandtheZoomtechnique

The Ricci Tensor is invariant by homoteties ,Thus the rescaled metric still satisfies the Ricci flow equation .Now according to geometric tools we consider the Hamilton’s work on singularities .One of geometric tools is zoom ,or parabolic rescaling .The idea is to dilate the metric and the time .

Definition3.19 (parabolic rescaling) Let ͛ʚͨʛ be a Ricci flow on ͇ƐʞRQ ͎ʛ , ͬͤ Ǩ

͇,ͨͤ ǨʞRQ͎ʛ Such that ͌ʚͬQ ͨʛ ƙ ͤ ȸ ͌ʚͬͤQͨͤʛ for all ͬǨ͇ and ͨƙͨͤ. Then

ͨ ͛ͤʚͨʛ Ɣͤ͛ƴͨͤ ƍ Ƹ ͤ

Is a Ricci flow on ʞƎͨͤͤQ ʚ͎Ǝͨͤʛͤʛ and ͌ͤʚͬQ ͨʛ ƙ S for every ͬǨ͇ and ͨƙR

.This is called a parabolic rescaling of ͛ʚͨʛ at ʚͬͤQͨͤʛ .Perelman showed if The Ricci flow on a closed three manifold ͇ encounters a singularity then an entire connected component disappear or there are nearby 2-Spheres on which to do surgery .To attack this ,Hamilton initiated a blowup analysis for Ricci flow .It is known that singularities arise from curvature blowups .That is ,if a Ricci flow solution exists on a maximal time

interval ʞRQ ͎ʛ ,with ͎Ɨ7 ,then ͠͝͡/Ŵu ͍ͩͤ3Ǩ ͙͌͝͡ʚͬQ ͨʛ Ɣ7, where Riem denotes the sectional curvature .

Assume we have a singularity developing at time ͎ under the Ricci flow ʚ͇Q ͛ʛ.we take

a sequence ͬ$ Ǩ͇ and ͨ$ Ŵ͎ so that ͌ʚͬ$ʛ Ɣ͕ͬ͡ ͌ at time ͨ$ ,and we Parabolic

rescale to have ͌ʚͬ$ʛ ƔS .we also move the time by a translation ,So that the initialͨ$ becomes zero .In order to analyze the singularity ,The idea is to look at the limit of

49 Chapter 2 overview on Poincare conjecture

pointed Ricci flows (with base pointͬ$ at time 0, after the translation ).There is a compactness theorem for pointed Ricci flows ,proved we have a positive lower bound on the injectivity radius of the base point .if we had This lower bound ,Then there would be a convergent subsequence to a flow ,and Hamilton work would yield that the limit has the following properties :

1) It is an ancient solution ,i.e. defined on time ʚƎ7Q Rʟ 2) The metric is complete 3) The sectional curvature is non-negative ,because we rescale by R and apply Hamilton- Ivey Pinching

4) ͌ƚR,because ͌($) is non decreasing .

Now suppose that ͎Ɨ7 and is maximal and therefore ͧͩͤ ͙͌͝͡ʚͬQ ͨʛ Ŵ7 as

ͨŴ͎.We can pick a sequence of points ʚͬ&Qͨ&ʛ such that ͨ& Ŵ͎ and & ȸ

͌ʚͬ&Qͨ&ʛƚ͌ʚͬQͨʛfor any ͬǨ͇and ͨƗͨ& .Now consider the sequence ͛&ʚͨʛ of

parabolic rescaling at ʚͬ&Qͨ&ʛ .it is defined at least on intervals ʞƎͨ&&QRʟ ,where ,thanks to the Hamilton-Ivey pinching ,we have uniform bounds on the sectional

curvature .Note that Ǝͨ&& ŴƎ7 .and by additional hypotheses we can take a limit and get a Ricci flow on (Ǝ7Q Rʟ.note that using theorem of M.gromov we can define pointed convergence of the Ricci flow .

Let ʚ͕Q ͖ʛ be an interval such that Ǝ7ƙRƙ͖ƙƍ7 .Let ʚ͇&Q͛&ʚͨʛQͬ&ʛ be a

sequence of pointed Ricci flows onʚ͕Q ͖ʛ with ͬ& Ǩ͇& .One says that the sequence converges in the pointed topology to the pointed Ricci flowʚ͇Q ͛ʚͨʛQͬʛ, if there exist,

for any ͟,embeddings Đ& from ̼ʚͬQ RQ ͟ʛ into ͇& ,taking ͬ to ͬ& ,such that the pull- ǳ Ϧ back metrics Đ&͛&ʚͨʛ converges to ͛ʚͨʛ in the ̽ Ǝ Topology uniformly on any compact set of ͇Ɛʚ͕Q͖ʛ.

MAXIMUM PRINCIPLE V{tÑàxÜG

Chapter 4 [MAXIMUM PRINCIPLE]

Chapter 4

Maximum principle

In this section we will look at some basic PDE techniques and apply them to the Ricci flow to obtain some important results about preservation and pinching of Curvature quantities. The important fact is that the curvatures satisfy certain Reaction - diffusion equations which can be studied with the maximum principle. The essence of parabolic maximum principles is to compare the solution of a heat- type PDE by an ODE obtained by dropping the Laplacian and gradient term. The Solution to the ODE, which is easier to obtain, will act as a barrier to the PDE solution. .Our exposition is based on [20-32] .

) Recall that if a smooth function ͩȏ͏Ŵ͌ where ͏ɛ͌ has a local minimum at ͬͤ in the interior of͏, Then

"ͩ ʚͬ ʛ ƔR "ͬ$ ͤ

"ͦͩ ʚͬ ʛ ƚR "ͬ$"ͬ% ͤ

Chapter 4 [MAXIMUM PRINCIPLE]

Where the second statement is that the Hessian is nonnegative definite (has all nonnegative eigenvalues). The same is true on a Riemannian manifold, replacing regular derivatives with covariant derivatives.

Lemma4.1 Let ʚ͇Q ͛ʛ be a Riemanian manifold and ͩ ȏ ͇ Ŵ ͌ be a smooth (or at ͦ least ̽ ) function that has a local minimum at ͬͤ Ǩ ͇. Then

$ͩʚͬͤʛ ƔR

$%ͩʚͬͤʛ ƚR

$% ;ͩʚͬͤʛ Ɣ͛ ʚͬͤʛ$%ͩʚͬͤʛ ƚR

ͩƔ i0 Proof. In a coordinate patch, the first statement is clear, since $ i3Ĝ the second statement is that the Hessian is positive definite, Recall that in coordinates, the Hessian is

"ͦͩ "ͩ ͩƔ Ǝϙ& $ % "ͬ$"ͬ% $% "ͬ&

But at a minimum, the second term is zero and the positive definiteness follows from the case in͌). The last statement is true since both ͛ and the Hessian are positive definite.

Note: there is a similar statement for maxima.

Lemma4.2. Let ʚ͇Q ͛ʚͨʛʛ be a smooth family of compact Riemannian manifolds forͨ Ǩ ʞRQ ͎ʟ. ͙ͨͩ͠ ȏ ʞRQ ͎ʟƐ͇ Ŵ ͌ be a ̽ͦ function such that

ͩʚRQ ͬʛ ƚ R

For allͬ Ǩ ͇. Then exactly one of the following is true

1. ͩʚͨQ ͬʛ ƚ R for all ʚͨQ ͬʛ Ǩ ʞRQ ͎ʟ Ɛ͇ ,or

2. there exists a ʚͨͤQͬͤʛǨ ʚRQ͎ʟƐ͇such that all of the following are true: a) ͩʚͨͤQͬͤʛ ƗR b) $ͩʚͨͤQͬͤʛƔ R ; ͩʚͨ Qͬ ʛƚR c) "ʚ/tʛ ͤ ͤ

Chapter 4 [MAXIMUM PRINCIPLE]

i0 ʚͨ Qͬ ʛƙR d) i/ ͤ ͤ

Proof. Certainly both cannot hold. Now suppose 1) fails. Then there must exists

ʚͨͤQͬͤʛsuch that ͩʚͨͤQͬͤʛ ƗR. We may move this to the minimum point, at which all of the first three must hold. If we take this to be the first time that such a point occurs, the last must hold as well.

Theorem4.1 (pichotomy) Let ʚ͇Q ͛ʚͨʛʛ be a smooth family of compact Riemannian Manifolds for ͨǨʞRQ͎ʟTLet ͩQ ͪ Ǩ ʞRQ ͎ʟƐ͇ Ŵ ͌ by ̽ͦ functions such that

ͩʚRQ ͬʛ ƚ ͪʚRQ ͬʛ For allͬ Ǩ ͇. also let ̻ Ǩ ͌. Then exactly one of the following is true: 1. ͩʚͨQ ͬʛ ƚ ͪʚͨQ ͬʛ for allʚͨQͬʛǨʞRQ͎ʟƐ͇, or

2. there exists a ʚͨͤQͬͤʛǨʚRQ͎ʛɗʚRQ͎ʟƐ͇ sush that all of the following are true a) ͩʚͨͤQͬͤʛ ƗͪʚͨͤQͬͤʛ b) $ͩʚͨͤQͬͤʛ Ɣ$ͪʚͨͤQͬͤʛ ; ͩʚͨ Qͬ ʛƚ; ͪʚͨ Qͬ ʛ c) "ʚ/tʛ ͤ ͤ "ʚ/tʛ ͤ ͤ i0 ʚͨ Qͬ ʛ ƙ i1 ʚͨ Qͬ ʛ ƍ̻ʞͩʚͨ Qͬ ʛ Ǝͪʚͨ Qͬ ʛʟ d) i/ ͤ ͤ i/ ͤ ͤ ͤ ͤ ͤ ͤ Proof .replace ͩ with ͙ͯ/ʚͩ Ǝ ͪʛ.

Note: For elliptic equations on a manifold, the facts we use are that if a function͚ ȏ

͇ Ŵ ͌ attains its minimum at a point ͬͤ Ǩ͇, then

͚ʚͬͤʛ ƔR and ;͚ʚͬͤʛ ƚR

For equations of parabolic type, a simple version says the following. Theorem4.2 (weak maximum principle for super solutions of the heat equation). Let ͛ʚͨʛ be a family of metrics on a close manifold ͇) and let ͩȏ͇) ƐʞRQ͎ʛ Ŵ ͌ satisfy

Chapter 4 [MAXIMUM PRINCIPLE]

"ͩ ƚ; ͩT "ͨ "ʚ/ʛ

Then if ͩƚ͗ at ͨ Ɣ R for some ͗ Ǩ ͌, then ͩƚ͗ for all ͨƚR.

Proof. The idea is simply that given a timeͨͤ ƚR, if the spatial minimum of ͩ is attained at a point ͬͤ Ǩ͇, then "ͩ ʚͨ Qͬ ʛ ƚ; ͩʚͨ Qͬ ʛ ƚR "ͨ ͤ ͤ "ʚ/ʛ ͤ ͤ so that the minimum should be non decreasing. Note that at ʚͨͤQͬͤʛwe actually have

ʚ$%ͩʛ ƚ R. More rigorously, we proceed as follows. Given any Ƙ R, define

ͩTS͇ƐʞRQ͎ʛŴ͌ ,

ͩT Ɣͩƍ ʚSƍͨʛ since ͩ ƚ ͗ at ͨƔR, we have ͩT Ƙ͗ at ͨ Ɣ R. Now suppose for some ƘR we have

ͩT ƙ͗ somewhere in ͇ƐʚRQ͎ʛ. Then since ͇ is closed, there existsʚͨͥQͬͥʛ such that

ͩTʚͨͥQͬͥʛ Ɣ͗ andͩTʚͨͥQͬͥʛ Ƙ͗ for all ͬǨ͇ and ͨǨ͇ and ͨ ǨʞRQ ͨͥʛT we then have atʚͨͥQͬͥʛ

"ͩ Rƚ T ƚ; ͩ ƍ ƘR "ͨ "ʚ/ʛ T

Which is a contradiction. Hence ͩT Ƙ͗ on ͇ƐʞRQ͎ʛ for all Ƙ R and by taking the limit as Ŵ R we get ͩ ƚ ͗ on ƐʞRQ ͎ʛ .

Let us work on a closed manifold ͇ with a Riemannian metric ͛ʚͨʛ that varies with time .in this section we will consider PDE s of the form

i0 Ɣ; ͩƍ˛͒ʚͨʛQǧͩ˜ ƍ ̀ʚͩʛ i/ "ʚ/ʛ (4.1) where ͩȏ͇ƐʞRQ͎ʛŴ͌ is a time - dependent real - valued function on ͇, ͒ʚͨʛ is a time - dependent vector field on ͇ and ̀ ȏ͌Ŵ͌. we will see many PDEs of this broad type - they consist of a Laplacian term ;"ʚ/ʛͩ and the reaction terms ˾͒ʚͨʛQ ͩ˿ ƍ ̀ʚͩʛT we call such PDEs heat - type equations, because of the analogy with the heat equation(4.1)

Chapter 4 [MAXIMUM PRINCIPLE]

Lemma4.3 let ʚ͇Q ͛ʚͨʛʛ be a closed manifold with a time - dependent Riemannian metric ͛ʚͨʛT suppose that ͩȏ͇ƐʞRQ͎ʛŴ͌ is initially non-positive (i.e. ͩʚͬQ Rʛ ƙR for all ͬ Ǩ ͇) and that is satisfies the differential inequality "ͩ ƙ; ͩƍ˛͒ʚͨʛQǧͩ˜ "ͨ "ʚ/ʛ at all points ʚͬQ ͨʛ Ǩ ͇ ƐʞRQ ͎ʛ where ͩʚͬQ ͨʛƘ R. Then ͩʚͬQ ͨʛ ƙ Rfor all ͬ Ǩ ͇ and ͨ ǨʞRQ ͎ʛT

Proof. By applying ͩT ƔͩƎ ʚSƍͨʛ in proof of previous theorem, proof will be complete.

The following theorem essentially tells us that out upper bound grows no faster that we would expect from the reaction term ̀ʚͩʛ in (4.1)

The Scalar maximum principle

Proposition4.1 Let ʚ͇Q ͛ʚͨʛʛ be a closed manifold with a time - dependent Riemannian metric ͛ʚͨʛT suppose that ͩȏ͇ƐʞRQ͎ʛŴ͌ satisfies "ͩ ƙ; ͩƍ˛͒ʚͨʛQǧͩ˜ ƍ ̀ʚͩʛ "ͨ "ʚ/ʛ

ͩʚͬQ Rʛ ƙ ͗ for all ͬǨ͇

For some constant͗, where ͒ʚͨʛ is a time - dependent vector field on ͇ and ̀ ȏ͌Ŵ ͌ is locally Lipschitz. Suppose that&S͌ Ŵ ͌is the solution of the associated ODE. Which is formed by neglecting the Laplacian and gradient terms:

͘& Ɣ ̀ʚ&ʛ ͨ͘

&ʚRʛ Ɣ͗

Then ͩʚͬQ ͨʛ ƙ &ʚͨʛ for all ͬǨ͇ ͕ͨ͘͢ Ǩ ʞRQ ͎ʛ such that &ʚͨʛ exists.

Proof. Let us set ͪ Ɣ ͩ Ǝ & we know that ͪʚͬQ Rʛƙ R for all ͬ Ǩ ͇, and we desire to show that ͪʚͬQ ͨʛƙ R for all ʚͬQ ͨʛ Ǩ ͇ ƐʞRQ ͎ʛT to do this, we fix an arbitrary ǨʞRQ ͎ʛ and show that ͪƙRon ʞRQ ʟQ for any Ǩ ʞRQ ͎ʛ.

Chapter 4 [MAXIMUM PRINCIPLE]

First note that

"ͩ "ʚͩ Ǝ &ʛ Ɣ "ͨ "ͨ

ƙ Ϛͩ ƍ˾͒Q ͩ˿ƍ̀ʚͩʛƎ ̀ʚ&ʛ

ƔϚͪƍ˾͒Q ͪ˿ƍʚ̀ʚͪʛ Ǝ ̀ʚ&ʛʛ (4.2)

Note that ͩ Ɣ ͪ and Ϛͩ Ɣ Ϛͪ because &depends only on ͨ. we now want to deal with the last term on RHS.

Because͇ƐʞRQ ʟ is compact, there exists a constant c (dependent on) such that ͩʚͬQ ͨʛ ƙ ͗ and &ʚͨʛ ƙ ͗ on ͇ ƐʞRQ ʟ . Because ̀ is locally Lipschitz and the interval ʞƎ͗Q ͗ʟ is compact, there exists ͗ͥ (also dependent on ) such that ̀ʚͬʛƎ

̀ʚͭʛ ƙ ͗ͥ ͬ Ǝ ͭ for all ͬQ ͭ ǨʞƎ͗Q ͗ʟTTherefore, because ͩQ & ǨʞƎ͗Q ͗ʟ,

̀ʚͩʛƎ ̀ʚ&ʛ ƙ ͗ͥ ͩ Ǝ & Ɣ ͗ͥ ͪ on ͇ƐʞRQʟ. Plugging this into the evolution equation (4.2) forͪ , we obtain

"ͪ ƙ;ͪƍ˛͒Q ǧͪ˜ ƍ͗ ͪ "ͨ ͥ

Now let Ɣ͙ͯu/ͪ . Then we have

"! ƙ͙ͯu/ʚ+ͪ ƍ ˛͒Q ǧͪ˜ ƍ͗ ͪ Ǝ͗ ͪʛ "ͨ ͥ ͥ

Ɣ+!ƍ˛͒Q ǧ!˜ ƍ͗ͥʚ ! Ǝ!ʛ (4.3)

We are going to apply previous lemma to the function!. Because ͪʚͬQ Rʛƙ R for allͬ Ǩ ͇, we have !ʚͬQ Rʛƙ R for all ͬ Ǩ ͇. Furthermore, if ! Ƙ R then ! Ɣ !, so the differential inequality (4.3) gives us

"! ƙ;!ƍ˛͒Q ǧ!˜ "ͨ at any point ʚͬQ ͨʛ Ǩ ͇ ƐʞRQ ʟ such that!ʚͬQ ͨʛ ƚ R, Hence, by previous lemma!ʚͬQ ͨʛ ƙ R for all ʚͬQ ͨʛ Ǩ ͇ ƐʞRQ ʟ. It follows that ͪʚͬQ ͨʛƙ R, and hence

Chapter 4 [MAXIMUM PRINCIPLE]

ͪʚͬQ ͨʛ ƙ &ʚͨʛ for all ʚͬQ ͨʛ Ǩ͇ƐʞRQ ʟQ for any Ǩ ʞRQ ͎ʟT Therefore ͩʚͬQ ͨʛ ƙ &ʚͨʛ for any ʚͬQ ͨʛ Ǩ ͇ ƐʞRQ ͎ʛ . So proof is complete.

The maximum principle on non-compact manifolds

We start with following general results of Karp - Li and Ni – Tam

ͥ ) Definition4.1 we say that ͩǨ͂'*ʚ͇ ƐʞRQ ͎ʟʛ is a weak sub-solution of the heat i ʚ Ǝ ;ʛͩʚͬQ ͨʛ Ɣ R ̽ & equation i/ if for every non-negative function with compact support in ͇) ƐʚRQ ͎ʛ we have "& ǹǹ ƴͩ Ǝ ǧͩǧ&Ƹ ͘ʚͬʛͨ͘ ƚ R ͤ ġ "ͨ

Theorem 4.3(see [11-30]) Assume that the curvature of ʚ͇)Q͛ʚͨʛʛ, ͨ ǨʞRQ ͎ʛQ are uniformly bounded. if ͩ is a weak sub-solution of the heat equation on ͇) ƐʞRQ ͎ʟ with ͩʚRQ Rʛƙ R and if

ͦ ͦ ǹǹ ͙ͬͤʠƎ ͘"ʚͤʛʚͬQ ͣʛͩͮʚͬQ ͨʛʡ ͘"ʚ/ʛʚͬʛͨ͘ Ɨ 7 ͤ ġ ) For some ƘR, then ͩƙR on ͇ ƐʞRQ ͎ʟ (ͩͮ Ȭ͕ͬ͡ʜRQ ͩʝand dʚͬQ ͣʛ denote the distance function of ͬ to a fixed pointͣ Ǩ ͇)) Theorem4.4(see[31]) Let ʚ͇)Q ͛ʚͨʛʛ,ͨǨʞRQ͎ʛ, be a complete solution to the Ricci flow with bounded curvature and let ͤ, be a ʚͤQ ͥʛ Ǝtensor with ʚ ʚ3Q+ʛͮͥʛ ͤʚͬʛ "ʚͤʛ ƙ͙ For some̻Ɨ7. Let ̿+Q, Ȭ ʚ̂+͎ǳ͇ʛ̂ʚ̂,͎ǳ͇ʛ and suppose that +Q, +Q, ̀/S̿ Ŵ̿ Is a fiber –wise linear map with ɳ̀Ƴ ʚͬʛƷɳ "ʚ/ʛ Ĺ̀/ĹϦ Ɣ 13. Ɨ7 Qʚ3ʛǨģQĤ ʚͬʛ "ʚ/ʛ Then there exists Ɨ7 and a solution ʚͨʛQͨ ǨʞRQ ͎ʛQof " Ɣ; ƍ̀ ʚ ʛ "ͨ "ʚ/ʛ /

Chapter 4 [MAXIMUM PRINCIPLE]

Qʚ ʚ3Q+ʛͮͥʛ With ʚRʛ Ɣ ͤ and "ʚ/ʛ ƙ͙ . This solution is unique among all solutions ʚ ʚ3Q+ʛͮͥʛ with "ʚ/ʛ ƙ͙ for all ͗Ɨ7. Li - Yau proved the uniqueness of solutions bounded from below under a certain lower bound assumption on the Ricci curvature. Theorem4.5(see [31]) Assume that the curvature and their first derivatives of ʚ͇)Q͛ʚͨʛʛ, ͨ ǨʞRQ ͎ʛ, are uniformly bounded. For any ͕ƘR and ̻ƘR, there exists a positive function &ʚͬQ ͨʛ and ͖ƘR such that " ƴ Ǝ+Ƹ&ƚ̻& "ͨ On ͇) ƐʞRQ ͎ʛ and ͙ͬͤʚ͕T ͘ʚͣQ ͬʛʛƙ&ʚͬQͨʛƙ͙ͬͤʚ͖T ͘ʚͣQ ͬʛʛ

The maximum Principle for tensors

Let͇be an n - dimensional complete manifold. Consider a family of smooth metrics

͛$%ʚͨʛevolving by the Ricci flow with uniformly bounded curvature for ͨ ǨʞRQ ͎ʟ with

͎ Ɨ 7. Denote by ͘/ʚͬQ ͭʛthe distance between two ͤͣͨͧͬ͢͝Q ͭ Ǩ ͇ with respect to the metric͛$%ʚͨʛ. Now we give a lemma that will be useful in next theorems Lemma4.4 There exists a smooth function ͚ on ͇ such that ͚ƚS everywhere, ͚ʚͬʛ Ŵ

ƍ7 as ͤ͘ʚͬQ ͬͤʛ Ŵƍ7 (for some fixedͬͤ Ǩ͇ ) ǧ͚ ƙ͗ ǧ͚ͦ ƙ͗ "Ĝĝʚ/ʛ and "Ĝĝʚ/ʛ On ͇) ƐʞRQ ͎ʟ for some positive constant ͗. Here we give one of applications of weak maximum principle .

Proposition4.2 if the scalar curvature ͌ of the solution ͛$%ʚͨʛ ,Rƙͨƙ͎, to the Ricci flow is non- negative at ͨƔR ,Then it remains so on Rƙͨƙ͎ . Proof Let f be the function constructed in previous lemma, recall

"͌ Ɣ;͌ƍT ͌͗͝ ͦ "ͨ For any small constant ƘR and large constant̻ Ƙ R, we have

Chapter 4 [MAXIMUM PRINCIPLE]

" "͌ ʚ͌ƍ ͙/͚ʛ Ɣ ƍ ̻͙/͚ "ͨ "ͨ Ɣ;ʚ͌ƍ ͙/͚ʛ ƍT ͌͗͝ ͦ ƍ ͙/ʚ̻͚ Ǝ ;͚ʛ By choosing ̻ large enough. We claim that ͌ƍ ͙/͚ƘR on ͇) ƐʞRQ ͎ʟ

Suppose not, then there exists a first time ͨͤ ƘR and a point ͬͤ Ǩ͇ such that / ʚ͌ƍ ͙ ͚ʛʚͬͤQͨͤʛ ƔR / ǧʚ͌ƍ ͙ ͚ʛʚͬͤQͨͤʛ ƔR / ;ʚ͌ƍ ͙ ͚ʛʚͬͤQͨͤʛ ƚR " ʚ͌ƍ ͙/͚ʛʚͬ Qͨ ʛ ƙR "ͨ ͤ ͤ Rƚ i ʚ͌ƍ ͙/͚ʛʚͬ Qͨ ʛ Ƙ;ʚ͌ƍ ͙/͚ʛʚͬ Qͨ ʛ ƚR Then i/ ͤ ͤ ͤ ͤ So we have RƘR that is contradiction. So we have proved that ʚ͌ƍ ͙/͚ʛ ƘR on ͇ƐʞRQ͎ʟ Letting ŴR, we get ͌ƚR on ͇ƐʞRQ͎ʟ . so proof is complete.

Hamilton’s maximum principle

To introduce Hamilton’s maximum principle, let us start with some basic set-up. We assume ʚ͇Q ͛$%ʚͬQ ͨʛʛQ ͨ Ǩ ʞRQ ͎ʟQ is a smooth complete solution to the Ricci flow with bounded curvature. Let ͐ be an abstract vector bundle over ͇ with a metric͜PQ, and & $% connection Ɣϙ$% compatible with ͜. Now we may form the Laplace Ϛ Ɣ ͛ $% which acts on the sections Ǩϙʚ͐ʛ of . Suppose͇PQʚͬQ ͨʛ is a family of bounded symmetric bilinear forms on ͐satisfying the equation i ͇ Ɣ;͇ ƍͩ$ǧ ͇ ƍ͈ i/ PQ PQ $ PQ PQ (4.4) where ͩ$ʚͨʛ is a time - dependent uniform bounded vector field on the manifold M, and

͈PQ Ɣ͊ʚ͇PQQ͜PQʛ is a polynomial in ͇PQ formed by contracting products of ͇PQ with itself using the metric h = {͜PQ}. Hamilton established the following weak maximum principle:

Let ͇PQ be a bounded solution to (4.4)and suppose ͈PQ satisfies the condition that

Chapter 4 [MAXIMUM PRINCIPLE]

P Q Q ͈PQͪ ͪ ƚR whenever ͈PQͪ ƔR

ForRƙͨƙ͎. If ͇PQ ƚR atͨ Ɣ R ,then it remains so for R ƙ ͨ ƙ ͎. Hamilton also established a strong maximum principle for solutions to equation (4.4): Let ͇PQ be a $ bounded solution to (4.4) withͩ ƔR. and ͈PQ satisfies “͈PQ ƘRwhenever ͇PQ ƚ

R”. suppose ͇PQ ƚR at ͨ Ɣ R. then there exists an interval RƗͨƗ on which the rank of ͇PQ is constant and the null space of ͇PQ is invariant under parallel translation and invariant in time and also lies in the null space of ͈PQ.

The maximum principle for systems Recall that the Riemannian curvature tensor may be considered as an operator S͎ͦǳ͇Ŵ͎ͦǳ͇ . As &- '. $ % ͌͡ʚ͕ʛSƔ͛ ͛ ͌$%&'͕.-ͬ͘ ʗͬ͘ to simply the evolution equation of ͌͡ we need to introduce the notion of Li algebra square ͌͡ʬS͎ͦǳ͇Ŵ͎ͦǳ͇. First we introduce the lie algebra structure on ͎ͦǳ͇ by defining its lie bracketʞTQTʟS͎ͦǳ͇Ɛ͎ͦǳ͇Ŵ͎ͦǳ͇. For any two forms ͕͘͢ , we define &' ʞ Q ʟ$% Ɣ͛ Ƴ $& '% Ǝ $& '%Ʒ

P Q ͦ ǳ PQ Let ƣ& Q& QUƧ be a basis of ͎ ͇ ,and ̽R be the structure symbols defined by

P Q PQ R ƫ& Q& ƯƔ̽R & . We define the lie algebra square by ʚ ʬʛ RS T] ͌͡ PQSƔ̽P ̽Q ͌͡RT͌͡S] Lemma 4.5(Evolution of the curvature operator ) i ͌͡ Ɣ +͌͡ ƍ ͌ͦ͡ ƍ͌͡ʬ i/ (4.5) Where ͌ͦ͡ is the composition ͌ͣ͌͡͡ and ͌͡ is the lie algebra square. ͯͥ ͦ ) ͦ ) Let ̿3 Ɣ ʚͬʛ (S ̿ Ŵ ͇ where̿S Ɣ ͇ ̂. ͇ )be the fiber over ͬ . For each

Ǩ͇ , consider the system of ODE on ̿3 corresponding to the PDE (4.5) obtained by dropping the laplacian term : ͇Ɣ͇ͦ ƍ͇ʬ / (4.6) ͇Ǩ̿ ͈Ɛ͈ ͈Ɣ)ʚ)ͯͥʛ Ɣ͍ͣ͘͝͡ʚ͢ʛ Where 3 is a symmetric matrix, where ͦ the maximum principle for systems says the following. A set ͅ in a vector space is said to

Chapter 4 [MAXIMUM PRINCIPLE] be convex, if for any Q͓ Ǩͅ , we have ͧ͒ ƍ ʚS Ǝ ͧʛ͓ Ǩ ͅ for all ͧ Ǩ ʞRQSʟ . A subset ͅ of the vector bundle ̿ is said to be invariant under parallel translation, if for every path S ʞ͕Q ͖ʟ Ŵ ͇ and vector Ǩͅʙ̿Rʚʛ , the unique parallel section ͒ʚͧʛ Ǩ ̿Rʚ.ʛ ,ͧǨʞ͕Q͖ʟ , along ʚͧʛ with ͒ʚ͕ʛ Ɣ͒ is contained in ͅ.

Theorem4.6 (see [20-31]): Let ͛ʚͨʛ ͨǨʞRQ͎ʛ be a solution to the Ricci flow on a closed manifold ͇).Let ͅɛ̿ be a subset which is invariant under parallel translation and whose intersection ͅ3SƔͅʙ̿3 with each fiber is closed and convex . Suppose the ODE (4.6) has the property that for any ͇ʚRʛ ǨͅQ we have ͇ʚͨʛ Ǩͅ for all ǨʞRQ͎ʛ , if ͌͡ʚRʛ Ǩ ͅ, then ͌͡ʚͨʛ Ǩ ͅ for all ͨǨʞRQ͎ʛ .

Strong Maximum principle In order to give an idea of the proof of the strong maximum principle we need to make some remarks on functions which are not quite differentiable Definition4.2 let ͚ be a real valued function defined for all ͬ in an interval containing , we define

*'+ ͚ʚͬʛ ȸ *'+13.3Ŵ4 ͚ʚͬʛ Ɣ',$Sϥͤ 13. 3ͯ4 ϤS ͚ʚͬʛ Ɣ*'+SŴͤ 13. 3ͯ4 ϤS ͚ʚͬʛ 3Ŵ4

͠͝͡3Ŵ4 ͚ʚͬʛ ȸ *'+',$3Ŵ4 ͚ʚͬʛ Ɣ 13.Sϥͤ ',$ 3ͯ4 ϤS ͚ʚͬʛ Ɣ*'+SŴͤ ',$ 3ͯ4 ϤS ͚ʚͬʛ

͠͝͡3Ŵ4~ ͚ʚͬʛ ȸ *'+',$3Ŵ4 ͚ʚͬʛ Ɣ 13.Sϥͤ ',$ͤϤ3ͯ4ϤS ͚ʚͬʛ Ɣ*'+SŴͤ ',$ͤϤ3ͯ4ϤS ͚ʚͬʛ

*'+ ͚ʚͬʛ ȸ *'+13.3Ŵ4~ ͚ʚͬʛ Ɣ',$Sϥͤ 13.ͤϤ3ͯ4ϤS ͚ʚͬʛ Ɣ*'+SŴͤ 13.ͤϤ3ͯ4ϤS ͚ʚͬʛ 3Ŵ4~ Definition4.3 Let S͌ Ŵ͌ . The upper right and lower right Dini derivatives of ͚ at ͨǨ͌ are , respectively, defined by ͚ʚͨƍ͜ʛ Ǝ͚ʚͨʛ ͚̾ͮʚͨʛ Ɣ*'+13. #Ŵͤ~ ͜ ͚ʚͨƍ͜ʛ Ǝ͚ʚͨʛ ͚̾ͮʚͨʛ Ɣ*'+',$ #Ŵͤ~ ͜ Lemma4.6 Let ͚S ʞ͕Q ͖ʟ Ŵ ͌ be a Lipschitz function such that ͚ʚ͕ʛ ƙ R and ͚̾ͮʚͨʛ ƙ R when ͚ƚR for ƙͨƙ͖ , then ͚ʚ͖ʛ ƙ R ͮ Corollary 4.1 if ͚ʚ͕ʛ ƙ ͛ʚ͕ʛ and ̾ ͚ʚͨʛ ƙ ̾ͮ͛ʚͨʛ for all ͕ƙͨƙ͖ .Then ʚ͖ʛ ƙ ͛ʚ͖ʛ .

Chapter 4 [MAXIMUM PRINCIPLE]

Theorem4.7 (Hamilton) Let ͛ʚͨQ ͭʛ be a smooth function of ͨ Ǩ ͌ and ͭǨ͌&. Let & ͚ʚͨʛ Ȭͧͩͤ4Ǩ ͛ʚͨQ ͭʛ, where ͓ɛ͌ is a compact set .Then ͚ is a Lipschitz function and its upper right derivative satisfies " ͚̾ͮʚͨʛ ƙ 13. ͛ʚͨQ ͭʛ 4Ǩʚ/ʛ "ͨ Being ͓ʚͨʛ Ɣ ʜͭǨ͓S͚ʚͨʛ Ɣ͛ʚͨQͭʛʝ. Ϧ ͨ Ǩ͌ ƣͨ Ƨ ͨ Proof: choose an arbitrary ͤ and a sequence % %Ͱͥ decreasing to ͤ for which

!Ƴ/ Ʒͯ!ʚ/ ʛ *'+ ĝ t ͓ /ĝŴ/t equals the limsup . since is a compact set, the maximum is attained /ĝͯ/t

; so for each index ͞ we can take ͭ% Ǩ͓ such that ͚Ƴͨ%ƷƔ͛ʚͨ%Qͭ%ʛ, therefore ,ʜͭ$ʝ is a sequence in ͓ and , because of the compactness of ͓, there is a subsequence convergent to some ͭͤ Ǩ͓. We can assume (for simplicity of the notation) ͭ% Ŵͭͤ taking limits in

͚Ƴͨ%ƷƔ͛ʚͨ%Qͭ%ʛ and using the continuity of ͚ and ͛, we have ͚ʚͨͤʛ Ɣ͛ʚͨͤQͭͤʛ; soͭͤ Ǩ͓ʚͨͤʛ. By definition of͚, ͛ʚͨͤQͭǳʛƙ͛ʚͨͤQͭͤʛ Ǣͭǳ Ǩ͓ ; then

͚Ƴͨ%ƷƎ͚ʚͨͤʛ Ɣ͛Ƴͨ%Qͭ%ƷƎ͛ʚͨͤQͭͤʛƙ͛Ƴͨ%Qͭ%ƷƎ͛ʚͨͤQͭ%ʛ

Dividing by ͨ% Ǝͨͤ and using the mean value theorem , we obtain

͚Ƴͨ%ƷƎ͚ʚͨͤʛ ͛Ƴͨ%Qͭ%ƷƎ͛ʚͨͤQͭ%ʛ " ƙ Ɣ ͛ʚ͎%Qͭ%ʛ ͨ% Ǝͨͤ ͨ% Ǝͨͤ "ͨ

With ͨͤ Ɨ͎% Ɨͨ%. Taking limits when ͨ% Ŵͨͤ, we have

͚Ƴͨ%ƷƎ͚ʚͨͤʛ " " *'+ ƙ ͛ʚͨͤQͭ$ʛƙ 13. ͛ʚͨQ ͭʛ /ĝŴ/t ͨ% Ǝͨͤ "ͨ 4Ǩʚ/ʛ "ͨ

Since ͨͤ is arbitrary, we have proved the estimate on the upper right derivative of ͚. Moreover, since ͓ʚͨʛ is a compact set, this supremum is attained and the above inequality shows that f has bounded first derivative and so it is a Lipschitz function. Next, we state the analog result for lower right derivatives. Theorem4.8 (Hamilton) Let ͛ʚͨQ ͭʛ be a smooth function of ͨ Ǩ ͌ and ͭǨ͌&. Let & ͜ʚͨʛ Ȭ͚͢͝4Ǩ ͛ʚͨQ ͭʛ, where ͓ɛ͌ is a compact set .Then ͜ is a Lipschitz function . " ̾ͮ͜ʚͨʛ ƚ 13. ͛ʚͨQ ͭʛ 4Ǩʚ/ʛ "ͨ Being ͓ʚͨʛ Ɣ ʜͭǨ͓S͜ʚͨʛ Ɣ͛ʚͨQͭʛʝ.

Chapter 4 [MAXIMUM PRINCIPLE]

Corollary4.2 if ͚ʚ͕ʛ ƙ R and ͚̾ͮʚͨʛ ƙ͚͗ for some ͗Ǩ͌ and for ƙͨƙ͖ , then ͚ʚ͖ʛ ƙ R. Proof take Ɣ͙ͯ/͚ , then ͙ͯ.͚ʚͧʛ Ǝ͙ͯ/͚ʚͨʛ ̾ͮ͛ʚͨʛ Ɣ*'+13. .Ŵ/~ ͧƎͨ ͙ͯ.͚ʚͧʛ Ǝ͙ͯ/͚ʚͧʛ ƍ͙ͯ/͚ʚͧʛ Ǝ͙ͯ/͚ʚͨʛ Ɣ*'+13. ʦ ʧ .Ŵ/~ ͧƎͨ ͙ͯ. Ǝ͙ͯ/ ͚ʚͧʛ Ǝ͚ʚͨʛ Ɣ*'+13. ʦ͚ʚͧʛ ƍ͙ͯ/ ʧ SŴͤͤϤ.ͯ/ϤS ͧƎͨ ͧƎͨ ͙ͯ. Ǝ͙ͯ/ ͚ʚͧʛ Ǝ͚ʚͨʛ ƙ*'+ʨ 13. ʦ͚ʚͧʛ ʧ ƍ 13. ʦ͙ͯ/ ʧʩ SŴͤ ͤϤ.ͯ/ϤS ͧƎͨ ͤϤ.ͯ/ϤS ͧƎͨ ͙ͯ. Ǝ͙ͯ/ ͚ʚͧʛ Ǝ͚ʚͨʛ Ɣ*'+13. ʦ͚ʚͧʛ ʧ ƍ͙ͯ/*'+13. ʦ ʧ .Ŵ/~ ͧƎͨ .Ŵ/~ ͧƎͨ ͙ͯ. Ǝ͙ͯ/ Ɣ*'+13. ʦ͚ʚͧʛ ʧ ƍ͙ͯ/͚̾ͮʚͨʛ .Ŵ/~ ͧƎͨ So the upper limit in the first addend above is actually a limit, and the limit of a product is the product of limits. In the second addend , we can apply the hypothesis of the ͮ ʚ ʛ ͮ ʚ ʛ ʚ ʛʚ ͯ.ʛ ͯ/ ʚ ʛ ʚ ʛ ͯ/ corollary about ̾ ͚ ͨ . So ̾ ͛ ͨ ƙ͚ ͨ ͙ .Ͱ/ ƍ͙ ͚͗ ͨ ƔƎ͚ ͨ ͙͗ ƍ ͙ͯ/͚͗ʚͨʛ ƔR . As a result of applying lemma (4.6) to ͛ ,we have ͛ʚ͖ʛ ƙ R ; but ͛ʚ͖ʛ Ɣ͙͚ͯʚ͖ʛ so ͚ʚ͖ʛ ƙ R.

Let ʚ͇Q ͛ʛ be a compact Riemannian manifold and let ͚Ɣʚ͚ͥQ͚ͦQUQ͚&ʛS͇ Ŵ͌&be a system of k functions on ͇.Let ͏ɗ͌&be an open subset and let &S ͏ Ŵ ͌&be a smooth vector field on . We let ͚Q ͛ and & depend on time, also consider the nonlinear heat equation i! ƔƎ+ ͚ƍ&͚ͣ i/ Ε (4.7)

With ͚ʚRʛ Ɣ ͚ͤ, and suppose that it has a solution for some time interval Rƙͨƙ͎ . before dealing with this ,we need some definitions. & Definition4.4 we define the tangent cone ͎5ʚ͒ʛ to a closed convex set ͒ɗ͌ at a point ͮǨ"͒ as the smallest closed convex con with vertex at ͮ which contains͒. It is the

Chapter 4 [MAXIMUM PRINCIPLE] intersection of all the closed half spaces containing ͒ with ͮ on the boundary of the half space. Definition 4.5we say that a linear function ͠S͌& Ŵ͌ is a support function for ͒ɛ͌& at ͮǨ"͒ (and write ͠Ǩ͍5͒) if 1. ͠ ƔS 2.͠ʚͮʛ ƚ ͠ʚͬʛ for all ͬǨ͒ (i.e.͠ʚͮ Ǝ ͬʛ ƚ R) Remark 4.2from the view point of the support functions, we can write

ͬǨ͎5ʚ͒ʛ if and only if ͠ʚͬ Ǝ ͮʛ ƙ R for every ͠Ǩ͍5͒ ! Ɣ&͚ͣ Lemma4.7 the solution of the PDE equation / which are in the closed convex & set ͒ɛ͌ at ͨƔR will remain in ͒ if and only if &ʚͮʛ Ǩ ͎5ʚ͒ʛ for every ͮǨ"͒ .

! Ɣ&͚ͣ ͚ʚRʛ Ǩ ͒ ͒ Theorem4.9 if the solution of the ODE equation / with stays in , then the solution of the PDE (4.7) with͚ʚRʛ Ǩ ͒ stays in ͒ (suppose ͒ is a compact set) Proof first we introduce a notation

ͧʚͮʛ Ȭ͘ʚͮQ ͒ʛ Ɣͧͩͤʜ͠ʚͮƎͬʛSͬ Ǩ"͒Q͠ Ǩ͍3͒ʝ ͧʚͮʛ Ɣ͘ʚͮQ ͒ʛQ ͮ Ǩ ͌& ͚S ͇ Ɛ͌ Ŵ ͌ & i! ƔƎ+ ͚ƍ&͚ͣ Let . Given a solution of i/ Ε , we define ͧʚͨʛSƔ13.3Ǩ ͧʚ͚ʚͬQ ͨʛʛ , so, by notation we have

ͧʚͨʛ Ɣ 13.ʚ3Q,Q'ʛǨ ͠ʚ͚ʚͬQ ͨʛ Ǝ ͥʛ (4.8)

Being ͓ƔƣʚͬQ ͥQ͠ʛSͬ Ǩ͇Qͥ Ǩ"͒Q͠Ǩ͍,͒Ƨ a compact set. So previous Hamilton’s Theorem 4.10assures that " ̾ͮͧʚͨʛ ƙ 13. ͠ʚ͚ʚͬQ ͨʛ Ǝ ͥʛ ʚ3Q,Q'ʛǨ "ͨ Where ͓ʚͨʛ Ɣ ʜʚͬQ ͥQ͠ʛ Ǩ͓Sʚ͚ʚͬQ ͨʛ Ǝ ͥʛ Ɣͧʚͨʛʝfrom this definition of ͓ʚͨʛ and (4.8)

;its follows that if ʚͬQ ͥQ͠ʛ Ǩ͓ʚͨʛ , then + 6ʚ3Q,Q'ʛǨʚ/ʛ ͠ʚ͚ʚͬQ ͨʛ Ǝ ͥʛ Ɣ͠ʚ͚ʚͬQ ͨʛ Ǝ ͥʛ,Since ͠ is a linear function independent of ͨ ,we have

ͮ i i!ʚ3Q/ʛ ̾ ͧʚͨʛ ƙ 13.ʚ3Q,Q'ʛǨʚ/ʛ ͠ʚ͚ʚͬQͨʛ Ǝ ͥʛ Ɣ 13. ͠ ʠ ʡ Ɣ i/ ʚ3Q,Q'ʛǨʚ/ʛ i/

13. ƣƎ͠Ƴ;"͚ʚͬQ ͨʛƷ ƍ ͠Ƴ&ʚ͚ʚͬQ ͨʛʛƷƧ (4.9) ʚ3Q,Q'ʛǨʚ/ʛ

Chapter 4 [MAXIMUM PRINCIPLE]

͚ i! ƔƎ+ ͚ƍ&͚ͣ Note that the last equality is true because is a solution of the PDE, i/ Ε .By definition of ʚͨʛ ,͠ʚ͚ʚͬQ ͨʛʛ has its maximum at ͬ ;so

͠ƳƎϚ"͚Ʒ Ɣ ƎϚ"͠ʚ͚ʛ ƙ R (4.10) i! Ɣ&͚ͣ ͒ On the other hand, by hypothesis, the solution of the ODE i/ stays in and to easily we can prove that this means ͠ʚ&ʚͮʛʛ ƙR for every ͠Ǩ͍5͒ and Ǩ"͒ . Then, in particular, ͠ʚ&ʚͥʛʛ ƙR. So we get ͠ʚ&ʚ͚ʚͬQ ͨʛʛʛ ƙ͠ʚ&ʚ͚ʚͬQ ͨʛʛʛ Ǝ͠ʚ&ʚͥʛʛ Ɣ͠Ƴ&Ƴ͚ʚͬQ ͨʛƷƎ&ʚͥʛƷ ƙ ͠ ɳ&Ƴ͚ʚͬQ ͨʛƷƎ&ʚͥʛɳ ƙ͗ ͚ʚͬQ ͨʛ Ǝ ͥ Ɣ͗͠ʚ͚ʚͬQ ͨʛ Ǝ ͥʛ

Where c is the Lipschitz constant of & and the last equality follows by the definition of ͓ʚͨʛ .finally, by sub situation of the above inequality and (4.9)in (4.10) ,we obtain ̾ͮͧʚͨʛ ƙ͗ͧʚͨʛ , and , since ͚ʚRʛ Ǩ ͒ ,ͧʚͨʛ ƔR. Then applying lemma (4.6), we conclude that ͧʚͨʛ Ɣ 13.3Ǩ ͧʚ͚ʚͬQ ͨʛʛ ƔR for all time in which the solution is defined. But this shows that ͚ʚͬQ ͨʛ remains in ͒

Applications of maximum principle

By applying the maximum principle to various equations and inequalities governing the evolution of curvature, we will get some preliminaries control on how ͌ and ͌͡ evolve . Theorem4.11: Under the Ricci flow, the scalar curvature , satisfy "͌ T ƚ;͌ƍ ͌ͦ "ͨ ͢ ͌͗ ͦ ƚ ͥ ͌ͦ ͕ ͦ ƚ ͥ ͕͙ͨͦ͗ ʚ͕ʛͦ Proof since ) (more generally, " ) " for any 2- tensor a ) i Ɣ;͌ƍͦ ͌͗͝ ͦ i ƚ;͌ƍͦ ͌ͦ also we know that i/ ) , so we get i/ ) . Now we recall if ͛ʚͨʛ be a family of metrics on a closed manifold ͇)and suppose ͪS ͇) ƐʞRQ ͎ʛ Ŵ ͌ i ͪƚ; ͪƍ͒ʚͪʛ ͒ ʚͪʛ satisfies i/ "ʚ/ʛ / where / denotes the directional derivative of ͪ in the direction of some time-dependent vector field ͒/. If ͪƚ͗ at ͨƔR for some Ǩ͌ , then ͪƚ͗ for all ͨƚR. Now if we set ͪʚͬQ ͨʛ Ɣ͙ͯ/ͩʚͬQ ͨʛ we get the following lemma.

Chapter 4 [MAXIMUM PRINCIPLE]

Lemma4.8: suppose " ͩƚ; ͩƍ͒ʚͩʛ ƍ͗ͩ "ͨ "ʚ/ʛ / Whereͩ ƙ R. Then if ͩƚR at ͨƔR so ͩƚR for all time. Theorem4.12 we have ͯͥ S T ͌ʚͬQ ͨʛ ƚ Ƶ Ǝ ͨƹ +', ͌ʚͬQ Rʛ ͢ 3Ǩ

If +',3Ǩ ͌ʚͬQ Rʛ ƕ R. If +',3Ǩ ͌ʚͬQ Rʛ Ɣ R, then we have ͌ʚͬQ ͨʛ ƚ R . Proof we know that i ƚ;͌ƍͦ ͌ͦʚVTSSʛ i/ ) Now we apply a trick which is useful for obtaining sharp estimate. Let ʚͨʛ be a solution of the ODE " T Ɣ ͦ "ͨ ͢ This ODE is obtained by replacing ƚ by Ɣ and dropping the ; term in(4.11) . we then have " T ʚ͌Ǝʛ ƚ;ʚ͌Ǝʛ ƍ ʚ͌Ǝʛʚ͌ƍʛ "ͨ ͢ Let ʞRQ ͎ʛ be the time interval of existence of the solution of Ricci flow .For any RƗͨƗ͎ we have͌ ƍ ƙ ͗ for any constant Ɨ7 . Hence " T͗ ʚ͌Ǝʛ ƚ;ʚ͌Ǝʛ ƍ ʚ͌Ǝʛ "ͨ ͢

Whenever Ǝ ƙ R . Now we choose so thatʚRʛ Ɣ +',3Ǩ ͌ʚͬQ Rʛ ,by exercise, since ͌ƎƚR at ͨƔR, we have ͌ƎƚR on ͇ƐʞRQʟ, since Ɨ͎ is arbitrary ͌ƎƚR ͇ƐʞRQʟ ` Ɣ ,it follows that on . we also easily see that the solution to / ͯͥ ͦ ͦ ʚͨʛ Ɣʠ ͥ Ǝ ͦ ͨʡ ʚRʛ ƔR ʚͨʛ ȸ R ) is `ʚͤʛ ) unless in which case so proof is complete.

Remark4.1 in particular, if+',3Ǩ ͌ʚͬQ Rʛ Ƙ R , then the maximal time interval of ) existence ʞRQ ͎ʛ satisfies Ɨ . ͦ($)īǨĆ ʚ3Qͤʛ We mention some useful applications of the weak and strong maximum principle:

Chapter 4 [MAXIMUM PRINCIPLE]

Corollary 4.3let ͇ƐʞRQ͎ʛ be a Ricci flow on a compact n-dimensional manifold ͇ and suppose that ͍($) ƙ͍ʚTQRʛƙ͍(3 on ͇ .Then S ƙ͍ʚTQͨʛ S T Ǝ ͨ ͍($) ͢ ͥ And if ͌͗͝ ƚ R everywhere, ͍ʚT Q ͨʛ ƙ u . ͯͦ/ ČĠĔī Corollary4.4(see [32] ) let ͇ƐʞRQ͎ʟ be a Ricci flow on an n-dimensional manifold. Assume that the curvature operator ͌Ȩ is everywhere non-negative definite and ͇ʚ͎ʛ Ȟ

͈Ɛ͌,where ʚ͈Q ͛ʛ is an ͢ƎS dimensional Riemannian manifold . Then the splitting already existed before ͎ and the Ricci flow ͇ƐʞRQ͎ʟ is of product form ʚ͈Ɛ͌ʛ Ɛ ʞRQ ͎ʟ where͈ƐʞRQ͎ʟ denotes a Ricci flow on N. Corollary 4.5(see [32] ) consider a Ricci flow ͇ƐʞRQ͎ʛ on a compact manifold M .If the curvature operator at time 0 is everywhere non-negative definite ,then this property is preserved under the Ricci flow .In particular ,in dimension 3 non-negative sectional curvature is preserved .

Now we give some another applications of maximum principle. Theorem4.13(see [32] ) let ʚ͏Q ͛ʚͨʛʛ ,Rƙͨƙ͎ be a a 3-dimensional Ricci flow with non-negative sectional curvature with ͏ connected but not necessarily complete and ͎ƘR.If ͌ʚͤQ ͎ʛ ƔR for some ͤǨ͏, Then ʚ͏Q ͛ʚͨʛʛ is flat for every ͨǨʞRQ͎ʟ.

Proposition4.3 (see [32] ) Let ʚ͏Q ͛ʚͨʛʛ ,Rƙͨƙ͎ , is a 3-dimensional Ricci flow with non-negative sectional curvature ,with ͏ being connected but not necessarily complete and ͎ƘR .Suppose that ʚ͏Q ͛ʚ͎ʛʛ is isometric to a non-empty open subset of a cone over a Riemannian manifold. Then ʚ͏Q ͛ʚͨʛʛ is flat for every ͨǨʞRQ͎ʟ. Also extending flows is one of applications of the maximum principle that is important.

Proposition 4.4(see [32]and[20-29] ) let ʚ͇Q ͛ʚͨʛʛ RƙͨƗ͎Ɨ7 , be a Ricci flow with M a compact manifold. Then either the flow extends to an interval ʞRQ ͎ĺʛ for some ͎ĺ Ƙ ͎ or ͌͡ is unbounded on ƐʞRQ ͎ʛ . Now as a last proposition we give another application of maximum principle.

Chapter 4 [MAXIMUM PRINCIPLE]

Hamilton –Ivey pinching

Note that we cannot completely establish a unilateral lower bound on the individual curvatures Q Q , but one can at least show that if one of these curvatures is large and negative , then one of the others must be extremely large and positive ,and so in regions of high curvature, the positive curvature components dominate. This is important phenomenon for Ricci flow that Ivey and Hamilton formalized it.

Proposition4.5 (see[20-29]) Let ʚ͇Q ͛ʚͨʛʛ be a Ricci flow on a compact 3-dimensional manifold on some time interval ʞRQ ͎ʟ . Suppose that the least Eigen value ʚͨQ ͬʛ of the Riemann curvature tensor is bounded below by -1 at times ͨƔR and all ͬǨ͇.then , at all space time points ʚͨQ ͬʛ ǨʞRQ͎ʟƐ͇, we have the scalar curvature bounded ƎX ͌ƚ Vͨ ƍ S And furthermore whenever one has negative curvature in the sense that ʚͨQ ͬʛ ƗR , then one also has the pinching bound ͌ƚT ʚͣ͛͠ ƍ*-%ʚSƍͨʛ ƎUʛ

LI-YAU-HAMILTON ESTIMATES V{tÑàxÜH

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

Chapter 5

Li-Yau-Hamilton Estimates

The classical Harnack inequality from parabolic PDE theory states that for RƗͨͥ Ɨ Ϧ ͨͦ ƙ͎ a non-negative smooth solution ͩǨ̽ ʚ͇ƐʞRQ͎ʟʛ of the linear heat equation "/ͩƔ;ͩ on a closed, connected manifold M satisfies

13. ͩʚT Q ͨͥʛƙ͗',$ ͩʚTQͨͦʛ (5.1)

Where c depends on ͨͥQͨͦ and the geometry of M.

In 1986, peter Li and Shing Tung Yau found a completely new Harnack type result, namely a point wise gradient estimate that can be integrated along a path to find a classical Harnack inequality of the form (5.1) they proved that on a manifold with ͌͗͝ ƚ R and convex boundary ,The differential Harnack expression " ͩ ǧͩ ͦ ͢ ͂ʚͩQ ͨʛ Ȭ / Ǝ ƍ ͩ ͩͦ Tͨ

Is non-negative for any positive solution u of the linear heat equation.

Also Richard Hamilton proved a matrix version of the Li-Yau inequality under slightly different assumptions. Hamilton then found a nonlinear analog for the Ricci flow case. In this section we will introduce Hamilton’s Harnack inequality for the scalar curvature under the Ricci flow.

Theorem5.1 (classical Harnack inequality) let ʚ͇Q ͛ʛ be a compact ͢Ǝdimensional Riemannian manifold with non-negative Ricci curvature .Let ͚ be a positive solution of the heat equation

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

"͚ Ɣ Ǝ;͚Q͚ͣͦR Ɨ ͨ Ɨ ͎ "ͨ

Then for any two points ʚͥQͨͥʛQ ʚͦQͨͦʛ Ǩ͇ƐʚRQ ͎ʛ with ͨͥ Ɨͨͦ , we have

) g ) ͚ͨͥͦʚͥQͨͥʛ ƙ͙ͨ ͚ͨͦͦʚͦQͨͦʛ

ʚ] Q] ʛv Where Ɣ u v and ͘ʚTQTʛ denotes the distance in ʚ͇Q ͛ʛ . /vͯ/u

Proof introduce Ɣ͚ͣ͛͠ , then "͆ ƔƎ+͆ƍ ͆͘ ͦ "ͨ

ͥ %& In fact , using the expression +͚ Ɣ Ǝ "$Ƴǭ͛͛ "&͚Ʒ for the Laplacian, we have :" S S " ͚ %& %& & +͆ Ɣ Ǝ "%Ƴǭ͛͛ "&͆Ʒ Ɣ Ǝ "% ƴǭ͛͛ Ƹ ǭ͛ ǭ͛ ͚ S S S Ǝ" ͚ %& %& % ƔƎ "%Ƴǭ͛͛ "&͚Ʒ Ǝ Ƴǭ͛͛ "&͚Ʒ ʦ ʧ ǭ͛ ͚ ǭ͛ ͚ͦ +͚ " ͚ " ͚ +͚ +͚ Ɣ ƍ͛%& & % Ɣ ƍ͛%&" ͆" ͆Ɣ ƍ ͆͘ ͦ ͚ ͚ ͚ ͚ & % ͚

Derivating respect of t in the definition of ͆, we get

i Ɣ ͥ i! ƔƎά! ʠ̼͙͕͙͖͙͕͙͕͗ͩͧͭͨͥͩͨͣ͜͢͝ i! ƔƎϚ͚ʡ i/ ! i/ ! i/

Ɣ Ǝ+͆ ƍ ͆͘ ͦ

SƔ i Ǝ ͆͘ ͦ ƔƎ+͆ Next define i/ .

So we have "͆ Ɣ͋ƍ ͆͘ ͦ "ͨ

Now we recall Boehner’s formula

Boehner’s formula: for any smooth function on ʚ͇Q ͛ʛ, we have S Ǝ + ͕͚͛ͦ͘ ͦ Ɣ ǧ͚ͦ ͦ Ǝ ˛͕͚͛ͦ͘Q ͕͛ͦ͘ʚ;͚ʛ˜ ƍ͌͗͝ʚ͕͚͛ͦ͘Q ͕͚͛ͦ͘ʛT T

And using Bochner’s formula, let us compute

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

"͋ " "͆ Ɣ ʚƎ;͆ʛ ƔƎ;ƴ ƸƔƎ;͋Ǝ;ʚ ͆͘ ͦʛ "ͨ "ͨ "ͨ

ƔƎ;͋ƎT˛͕͛ͦ͘ʚ;͆ʛQ͕͛ͦ͆͘˜ ƍT ǧͦ͆ ͦ ƍT͌͗͝ʚ͕͛ͦ͆͘Q ͕͛ͦ͆͘ʛ

ƔƎ;͋ƍT˛͕͛ͦ͋͘Q ͕͛ͦ͆͘˜ ƍT ǧͦ͆ ͦ ƍT͌͗͝ʚ͕͛ͦ͆͘Q ͕͛ͦ͆͘ʛ

͗͝ʚ͕͛ͦ͆͘Q ͕͛ͦ͆͘ʛ ƚR i By hypothesis, we have . in order to find a lower bound for i/ we are going to use the well known inequality between the square of the norm and the trace of a symmetric tensor : S S S ǧͦ͆ ͦ ƚ ʚͨͦǧͦ͆ʛͦ Ɣ ʚƎ;͆ʛͦ Ɣ ͋ͦ ͢ ͢ ͢

By this remarks together with ǧͦ͆ ƚR we get

i ƚƎ;͋ƍT˛͕͛ͦ͆͘Q ͕͛ͦ͋͘˜ ƍ ͦ ͋ͦ i/ ) (5.2) by applying the scalar maximum principle , scalar maximum principle: let M be a compact manifold, and let ͛/ a 1- parametric family of smooth metrics on M depending smoothly on t. let ͒/ be a family of smooth vector fields on M depending smoothly on t. Let us consider the partial differential inequations

i! ħ ƚƎ; ͚ ƍ ˛͕͛ͦ͘ ͚ Q͒ ˜ ƍ&͚ͣ i/ "ħ / / / ͨ / (1)

And the associated ordinary differential equation

i#ʚ/ʛ Ɣ &ͣͫͨ͜͜͜͝ʚRʛ Ɣ͢͡͝ʜ͚ ʚͬʛSͬ Ǩ͇ʝ i/ ͤ (2)

Then ͚3ʚͨʛ ƚ ͜ʚͨʛ for every t in an interval ʞRQ ͎ʟ where there is a solution of (1) and(2).

So by the scalar maximum principle, we obtain the following inequality from ()

͋ʚͬQ ͨʛ ƚ ͋($)ʚͨʛ ƚ ʚͨʛ (5.3)

Where ʚͨʛ is the solution of the ODE

ie Ɣ ͦ ͦͫͨ͜͝ʚRʛ Ɣ͋ ʚRʛ i/ ) ($) (5.4)

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

Ȅ ;͆ Ɣ R ͚ ͋ƔƎ;͆ Since M is compact, , then, if is not constant at the start, must be negative at some point at the start , then ͋ʚRʛ Ɣ͋($)ʚRʛ ƗR . From (5.3) and (5.4) we obtain

ʚͨʛ ƚƎ ) ƚƎ) ͋ƚƎ) ͦ/ͯ ġ ͦ/ and ͦ/ ŝʚtʛ

By of Hopf –Rinow theorem, because ͇ is compact manifold, it is also complete. So it follows that there exists a minimal geodesic jointing any pair of points in the manifold .Hence we can take a minimal geodesic parameterized by time ͨ joining ͥ and ͦ, that S ʞͨ Qͨ ʟŴ͇ ʚͨ ʛ Ɣ ʚͨ ʛ Ɣ ͘ʚ Q ʛ Ɣ͆Ƴ Ʒ is ͥ ͦ ,such that ͥ ͥ , ͦ ͦ and ͥ ͦ ʞ/uQ/vʟ .

Denoting $ ȸ $ʚͨʛ the component functions of the geodesic, we compute (using the chain rule)

͆͘ "͆ "͆ ͘ $ʚͨʛ "͆ ʚ ʚͨʛQͨʛ Ɣ ʚ ʚͨʛQͨʛ ƍ Ɣ ʚ ʚͨʛQͨʛ ƍ͆͘Ƴ ĺʚͨʛƷ ͨ͘ "ͨ "ͬ$ ͨ͘ "ͨ "͆ ͢ Ɣ ʚ ʚͨʛQͨʛ ƍ ˛͕͛ͦ͆͘Q ĺʚͨʛ˜ ƚƎ ƍ ͆͘ ͦ ƍ ˛͕͛ͦ͆͘Q ĺʚͨʛ˜ "ͨ Tͨ ͢ ƚƎ ƍ ͆͘ ͦ Ǝ ˛͕͛ͦ͆͘Q ĺʚͨʛ˜ Tͨ

ƚƎ) ƍ ͆͘ ͦ Ǝ ͕͛ͦ͆͘ ĺʚͨʛ ͦ/ (By Cauchy –Schwarz)

ͦ ͦ ͦ ƔƎ) ƍ ʠ ͆͘ Ǝ ͥ ʺ Rʺʡ Ǝ ͥ ʺ Rʺ ƚƎ) Ǝ ͥ ʺ Rʺ ͦ/ ͦ / ͨ / ͦ/ ͨ /

Integrating along the geodesic and using the fundamental theorem of calculus.

ͦ /v ͆͘ /v ͢ S /v ͘ ͆ʚ ʚͨ ʛQͨ ʛ Ǝ͆ʚ ʚͨ ʛQͨ ʛ Ɣǹ ʚ ʚͨʛQͨʛͨ͘ ƚ Ǝ ǹ ͨ͘ Ǝ ǹ ɴ ɴ ͨ͘ ͦ ͦ ͥ ͥ ͨ͘ Tͨ V ͨ͘ /u /u /u

ͦ ͢ ͨ S /v ͘ ƔƎ ͢͠ ƴ ͦƸƎ ǹ ɴ ɴ ͨ͘ T ͨ V ͨ͘ ͥ /u

So we have

/v ͘ ͘ ͘ ͘ʚ Q ʛ ͘ʚ Q ʛ Ɣ͆Ƴ ƷƔǹ ɴ ɴ ͨ͘ Ɣ ɴ ɴ ʚͨ Ǝͨ ʛ ʂ ɴ ɴ Ɣ ͥ ͦ ͥ ͦ ʞ/uQ/vʟ ͨ͘ ͨ͘ ͦ ͥ ͨ͘ ͨ Ǝͨ /u ͦ ͥ

(Because all Riemannian geodesics are constant speed curves)

In conclusion, we get

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

ͦ /v ͘ /v ͘ʚ Q ʛͦ ͘ʚ Q ʛͦ /v ͘ʚ Q ʛͦ ǹ ɴ ɴ ͨ͘ Ɣǹ ͥ ͦ ͨ͘ Ɣ ͥ ͦ ǹͨ͘Ɣ ͥ ͦ Ɣ ͨ͘ ʚͨ Ǝͨ ʛͦ ʚͨ Ǝͨ ʛͦ ͨ Ǝͨ /u /u ͦ ͥ ͦ ͥ /u ͦ ͥ

So we reach

Ǝ͢ ͨͦ ͆ʚ ʚͨͦʛQͨͦʛ Ǝ͆ʚ ʚͨͥʛQͨͥʛ ƚ ͢͠ ƴ ƸƎ T ͨͥ V

!ʚRʚ/ ʛQ/ ʛ and the definition of L given at the beginning of the proof yields have ͢͠ v v ƚ !ʚRʚ/uʛQ/uʛ ͯ) / g ͢͠ ʠ vʡ Ǝ , and, taking exponentials, ͦ /u ͨ

ͯ) ͚ʚͦQͨͦʛ ͨͦ ͦ ͯg ƚƴ Ƹ ͙ ͨ ͚ʚͥQͨͥʛ ͨͥ

So proof is complete.

We have seen that in the Ricci flow the curvature tensor satisfies a nonlinear heat equation, and no-negatively of the curvature operator is preserved by the Ricci flow. Roughly speaking, the Li-Yau-Hamilton estimate say the non-negativity of a certain combination of the derivatives of the curvature up to second order is also preserved by the Ricci flow.

We start Li-Yau’s gradient estimate for the linear heat equation

Theorem5.2: let ʚ͇Q ͛ʛ be a compact Riemannian manifold. Assume that on the ball̼ʚRQT͌ʛ,͌͗͝ʚ͇ʛ ƚƎ͟. Then for any ƘS , we have that

v v v v v ^3. ʠ ǧ0 Ǝ 0ħʡ ƙ P ʠ P ƍ :͌͟ʡ ƍ )P & ƍ )P ʚͤQͦʛ 0v 0 v Pvͯͥ ͦʚPͯͥʛ ͦ/ (5.5)

Remark (see[22])if ʚ͇Q ͛ʛ has non-negative Ricci, letting ͌Ŵƍ7 in (5.5) gives the clean estimate (a Hamilton Jacobi inequality):

v ǧ0 Ǝ 0ħ ƙ ) 0v 0 ͦ/ (5.6)

This estimate is sharp in the sense that the equality satisfied for some ʚͬͤQͨͤʛ implies that ʚ͇Q ͛ʛ is isometric to ͌).it can also be easily checked that if ͩ is the fundamental v ) ͥ 3 solution on͌ given by the formula ġ ͙ͬͤ ʠƎ ʡthen the equality holds in ʚͨ_/ʛv ͨ/ ġ 3 v ͩʚͬQ ͨʛ Ɣ ʚVͨʛ v ͙ͬͤ ʠƎ ʡ ͩ (5.6).Because if we set ͨ/ by differentiating the function ,we get

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

3 ǧ ͩƔƎͩ ĝ ǧ ͩƍͩͪ ƔR Θ ͦ/ or Θ % , (5.7)

3 ǧ 0 ͪ Ɣ ĝ ƔƎ ¨ where % ͦ/ 0

Differentiating (5.7) we get

ͩƍͩͪ ƍ 0 ƔR $ % $ % ͦ/ $% (5.8)

To make the expression in (5.8) symmetric in i,j , we multiply ͪ$ to (5.7) and add to (5.8) and obtain

ͩƍͩͪ ƍͩͪ ƍͩͪͪ ƍ 0 ƔR $ % $ % % $ $ % ͦ/ $% (5.9)

i0 Ɣ;ͩ Taking the trace in (5.9) and using the equationi/ , we arrive at "ͩ ͢ ƍTǧͩTͪƍͩ ͪ ͦ ƍ ͩƔR "ͨ Tͨ

ͪƔƎǧ0 Now if we take the optimal vector field 0 ,we recover the equality "ͩ ǧͩ ͦ ͢ Ǝ ƍ ͩƔR "ͨ ͩ Tͨ

Hamilton obtained the following Li-Yau estimate for scalar curvature ͌ʚͬQ ͨʛ.

Recall that under the Ricci flow on a Riemann surface the scalar curvature satisfies the following heat type equation " ƴ Ǝ;Ƹ͌Ɣ͌ͦ "ͨ

By the maximum principle , the positive of the curvature is preserved by the Ricci flow , ͋Ɣ i ͣ͛͌͠ Ǝ ǧͣ͛͌͠ ͦ Hamilton considered the quantity i/ and computed " S S S S S ƴ͋ ƍ Ƹƚ;ƴ͋ƍ Ƹ ƍ Tǧͣ͛͌͠T ǧ ƴ͋ ƍ Ƹƍƴ͋Ǝ Ƹƴ͋ƍ Ƹ "ͨ ͨ ͨ ͨ ͨ ͨ

From the maximum principle, it follows.

Theorem5.3 (Hamilton[33]) let ͛$%ʚͬQ ͨʛbe a complete solution to the Ricci flow with bounded curvature on a surface ͇ . Assume the scalar curvature of the initial metric is positive . Then

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

"͌ ǧ͌ ͦ ͌ Ǝ ƍ ƚR "ͨ ͌ ͨ

In n-dimensions, we have the following generalization of Hamilton’s differential Harnack estimate for surfaces

Theorem5.4 (Hamilton 1993-matrix Harnack for the Ricci flow [33]) if Ƴ͇)Q͛ʚͨʛƷ is a solution to the Ricci flow with non-negative curvature operator, so ͌$%&'͏$%͏&' ƚR for all 2-forms, and either Ƴ͇)Q͛ʚͨʛƷ is compact or complete non-compact with bounded curvature, Then for any 1-form ͑Ǩ̽Ϧʚ<ͥcʛ and 2-form ͏Ǩ̽Ϧʚ+ͦcʛ we have

͇$%͑$͑% ƍT͊+$%͏+$͑% ƍ͌+$%,͏+$͏,% ƚR

Here the 3-tensor ͊ is defined by

͊&$% Ȭ&͌$% Ǝ$͌&%

And the symmetric two tensor M is defined by S S ͇ Ȭ;͌ Ǝ ǧ ͌ƍT͌ ͌ Ǝ͌ ͌ ƍ ͌ $% $% T $ % &$%' &' $+ +% Tͨ $%

We call this Hamilton’s matrix Harnack estimate for Ricci flow.

Consequently For any one-form͐$ , we have "͌ ͌ ƍ ƍT͌T ͐ ƍT͌ ͐ ͐ ƚR "ͨ ͨ $ $ $% $ %

͏ Ɣ ͥ Ƴ͐ ͑ Ǝ͐͑ Ʒ ͑ Because by taking $% ͦ $ % % $ and tracing over $ we immediately get it (where ͐$ Ɣ$͚ for some function.)

Remark in particular by taking ͐ ȸ R we see that the function ͨ͌ʚͬQ ͨʛ is point wise non-decreasing in time

Theorem5.5 (Li-Yau Hamilton for the conjugate heat equation[35]): assume that Ƴ͇Q ͛ʚͨʛƷ is a solution to Ricci flow on͇ƐʞRQ͎ʟ. Note we consider the conjugate heat equation

ʠ i Ǝ;ƍ͌ʡ ͩʚͬQ ͨʛ ƔR ic (7.10)

Ɣ͎Ǝͨ ʠ i Ǝ;ʡ ͪƔR Here . This equation is the adjoint of the heat equation i/ , then we have

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

͚ T͚ ƍ ǧ͚ ͦ Ǝ͌ƍ ƙR c

A fundamental application of the linear trace Harnack estimate is the following classification of eternal solutions. That is the important application of Hamilton’s inequality.

Theorem5.6 if Ƴ͇ͦQ͛ʚͨʛƷ,ͨǨʚƎ7Q 7ʛ, is a complete solution to the Ricci flow with positive curvature and such that ^3.vƐʚͯϦQϦʛ ͌ is attained at some point in space and time , Then Ƴ͇ͦQ͛ʚͨʛƷ is a gradient Ricci Soliton . By the classification theorem ,it must be the cigar soliton

Proof one can compute that

ͦ i ͋Ɣ;͋ƍT˛ǧͣ͛͌͠Q ǧ͋˜ ƍTʺǧǧͣ͛͌͠ ƍ ͥ ͌͛ʺ i/ ͦ (7.11)

ͦ i ͋ƚ;͋ƍ˛ǧͣ͛͌͠Q ǧ͋˜ ƍ͋ͦ ɳ͕ ɳ ƚ Hence i/ ,where we applied the inequality $% ͥ ʚ͕ͨͦʛͦ ͕Ɣǧǧͣ͛͌͠ƍͥ ͌͛ ͢ƔT ) to ͦ with . Now we used our assumption that the solution exists on all of ʚƎ7Q ƍ7ʛ . in particular ,for any Ǩ͌, the solution exists on the interval ʚ Q ƍ7ʛ. Since the solution PDE ͥ͘ Ɣ ͥͦ ͨ͘

*'+ ͥʚͨʛ Ɣ7 ʚͨʛ ƔƎ ͥ With /ŴP is /ͯP , the maximum principle says S d Ɣǧͣ͛͌͠ƍ͌ƚƎ ͨƎ

For all Ƙ . Hence , on all of ͇ͦ Ɛ ʚƎ7Q ƍ7ʛ, by taking ŴƎ7, we get ǧͣ͛͌͠ ƍ ͌ ƚ R

ͦ By our hypothesis ,there’s a point ʚͬͤQͨͤʛ Ǩ͇ Ɛ ʚƎ7Q ƍ7ʛ, such that ͌ʚͬͤQͨͤʛ Ɣ ^3.vƐʚͯϦQͮϦʛ ͌.

At ʚͬͤQͨͤʛ we have

i ƔR ǧ͌ ƔR i/ and

And hence ʚͬͤQͨͤʛ ƔR . Since ͋ƚR ,applying the strong maximum principle to(7.11) we see that

Chapter 5 [LI-YAU-HAMILTON ESTIMATES]

ǧͣ͛͌͠ ƍ ͌ Ɣ ͋ ȸ R

Plugging this back into (7.11), we get S ǧǧͣ͛͌͠ ƍ ͌͛ ȸ R T

On ͇ͦ Ɛ ʚƎ7Q ƍ7ʛ . This says that ͛ʚͨʛ is a gradient Ricci soliton following along ǧͣ͛͌͠ : "͛ Ɣ Ǝ͌͛ Ɣ Tǧǧͣ͛͌͠ "ͨ

As a last theorem we extend it for each ͢ƚS.

Theorem5.7(see[33],[22] )

If Ƴ͇)Q͛ʚͨʛƷ,ͨǨʚƎ7Q 7ʛ, is a complete solution to the Ricci flow with non-negative curvature operator ,positive Ricci curvature and such that ^3.vƐʚͯϦQhʛ ͌ is attained at some point in space and time , Then Ƴ͇)Q͛ʚͨʛƷ is a steady gradient Ricci Soliton.

TWO FUNCTIONAL F AND W OF PERELMAN V{tÑàxÜI

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Chapter6

Two functionals ŀ·Äºő of Perelman

In this chapter, we introduce two functionals of Perelman Đ and ġ, and discuss their relations with the Ricci flow. The functional Đ can be found in the literature on the String Theory, where it describes the flow energy effective action. It was not known whether the Ricci flow is a gradient flow until Perlman showed that the Ricci flow is, in a certain sense, the gradient flow of the functionalĐ. In the theory of dynamical systems we can take the two functionals Đ and ġ as a Lyapunov type.

Theŀ Ǝfunctional

Let ėdenote the space of smooth Riemannian metrics ͛ on ͇. We think of ė formally as an infinite-dimensional manifold. Also ͇ is closed manifold and the tangent space

͎"͇ consists of the symmetric covariant 2-tensorsͪ$% on ͇. ͚ is a function ͚ ȏ ͇ Ŵ ͌ .Letͪ͘ denote the Riemannian volume density associated to a metric ͛. The functional Đȏė Ɛ̽ʚ͇ʛ Ŵ ͌ is given by

Đʚ͛Q ͚ʛ Ɣǹ ʚ͌ƍ ͚ ͦʛ͙ͯ!͐͘

Given ͪ$% Ǩ͎"͇ and ͜Ɣ ͚.The evaluation of the differential͘Đ on ʚ$%Q͜ʛ is written $% ͕ͧ Đʚ$%Q͜ʛT putͪ Ɣ ͛ ͪ$%. Theorem6.1(Perelman) we have ĐƳ Q͜ƷƔȄ ͙ͯ! ʢƎͪ Ƴ͌ ƍ ͚Ʒ ƍ ʠ1 Ǝ͜ʡ ʚT;͚ Ǝ ͚ ͦ ƍ͌ʛʣ ͐͘ $% $% $% $ % ͦ (6.1)

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Proof first we prove ͌ Ɣ ƎϚͪƍ$%ͪ$% Ǝ͌$%ͪ$% .In any normal coordinates at a fixed point, we have " " ͌# Ɣ Ƴ ϙ#ƷƎ Ƴ ϙ#Ʒ $%' "ͬ$ %' "ͬ% $' " S Ɣ Ƭ ͛#(Ƴ ͪ ƍͪ Ǝ ͪ Ʒư "ͬ$ T % '( ' %( ( %' " S Ǝ Ƭ ͛#(ʚ ͪ ƍͪ Ǝ ͪ ʛư "ͬ% T $ '( ' $( ( $'

" S " S ͌ Ɣ Ƭ ͛$(Ƴ ͪ ƍͪ Ǝ ͪ Ʒư Ǝ Ƭ ͛$(ʚ ͪ ƍͪ Ǝ ͪ ʛư %' "ͬ$ T % '( ' %( ( %' "ͬ% T $ '( ' $( ( $' Ɣ ͥ i ƫ ͪ$ ƍͪ$ Ǝ$ͪ ƯƎͥ i ʞ ͪʟ ͦ i3Ĝ % ' ' % %' ͦ i3ĝ ' Therefore $' %' ͌ Ɣ Ƴ͛ ͌%'ƷƔƎͪ%'͌%' ƍ͛ ͌%' S " S " ƔƎͪ ͌ ƍ ƫ'ͪ$ ƍͪ$' Ǝ$ͪƯ Ǝ ƫ%ͪƯ %' %' T "ͬ$ ' ' T "ͬ%

ƔƎͪ%'͌%' ƍ$'ͪ$' Ǝ;ͪ ͦ $% As ͚ Ɣ͛ $͚%͚

ͦ $% We have ͚ ƔƎͪ $͚%͚ƍT˛͚Q ͜˜, as ͐͘ Ɣ ǭ͙ͨ͘ʚ͛ʛͬͥ͘ Uͬ͘) , we have ʚ͐͘ʛ Ɣ 1 ͐͘ ͦ ,so ʚ͙ͯ!͐͘ʛ Ɣ ʠ1 Ǝ͜ʡ ͙ͯ!͐͘ ͦ (6.2) Putting this together gives Đ Ɣ Ȅ ͙ͯ! ʢƎ;ͪ ƍ ͪ Ǝ͌ ͪ Ǝͪ ͚ ͚ƍT˛͚Q ͜˜ ƍ ʚ͌ƍ ͚ ͦʛ ʠ1 Ǝ $ % $% $% $% $% $ % ͦ

͜Ʒʣ͐͘ (6.3)

The goal now is to rewrite the right-hand side of (6.3) so that ͪ$% and ͜ appear algebraically, i.e. without derivatives .as ;͙ͯ! Ɣ ʚ ͚ ͦ Ǝ;͚ʛ͙ͯ! We have

ǹ͙ͯ!ʞƎ;ͪʟ͐͘ Ɣ Ǝ ǹʚ;͙ͯ!ʛͪ͐͘ Ɣ ǹ ͙ͯ!ʚ;͚ Ǝ ͚ ͦʛͪ͐͘

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

ͯ! ͯ! ǹ͙ $%ͪ$%͐͘ Ɣ ǹƳ$%͙ Ʒ ͪ$%͐͘

ͯ! ͯ! ƔƎǹ $Ƴ͙ %͚Ʒͪ$%͐͘ Ɣ ǹ ͙ Ƴ$͚%͚Ǝ$%͚Ʒͪ$%͐͘ Finally

Tǹ ͙ͯ!˛͚Q ͜˜ ͐͘ Ɣ ƎT ǹ ˛͙ͯ!Q͜˜ ͐͘

ƔTǹ ʚ;͙ͯ!ʛ͐͘͜ Ɣ T ǹ ͙ͯ!ʚ ͚ ͦ Ǝ;͚ʛ͐͘͜ Then ͪ ͯ! ͦ Đ Ɣ ǹ ͙ ʢʠ Ǝ͜ʡ ʚT;͚ Ǝ T ͚ ʛ Ǝͪ$%Ƴ͌$% ƍ$%͚Ʒ T ͪ ƍ ʠ Ǝ͜ʡ ʚ͌ƍ ͚ ͦʛʣ ͐͘ T ͪ ͯ! ͦ Ɣǹ ͙ ʢƎͪ$%Ƴ͌$% ƍ$%͚Ʒ ƍ ʠ Ǝ͜ʡ ʚT;͚ Ǝ ͚ ƍ͌ʛʣ ͐͘ T So proof is complete. Remark 6.1(see [38-39]) notice that 1) the functional Đ is invariant under diffeomorphism i.e.Đʚ͇Q &ǳ͛Q ͚ͣ&ʛ ƔĐʚ͇Q ͛Q &ʛ for any diffeomorphisms & 2) Also for any ͗ƘR and b,Đʚ͇Q ͗ͦ͛Q ͚ ƍ ͖ʛ Ɣ͗)͙ͯͦͯĐʚ͇Q ͛Q ͚ʛ. Example (fundamental important example) let ʚ͇Q ͛ʛ be Euclidean space ͇Ɣ͌)and let ͬ ͦ ͢ ͯ) ͯ 3 v ͚ʚͨQ ͬʛ Ɣ ƍ ͣ͛͠V͎ Ɣ Ǝͣ͛͠ ʨʚV͎ʛ ͦ ͙ ͨ ʩ V͎ T

ͯ! Where Ɣͨͤ ƎͨQ notice that ͙ ͬ͘ is the Gaussian measure, which solves the

backward heat equation .If ͨƗͨͤ .This choice of ͛ and ͚ satisfy the equations "͛ ƔƎT͌͗ʚ͛ʛ "ͨ "͚ ƔƎ;͚Ǝ͌ƍ ͚ ͦ "ͨ We can check 80

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

"͚ " ͬ ͦ ͢ ͬ ͦ ͢ Ɣ ʨ ƍ ͣ͛͠V͎ʩ Ɣ Ǝ "ͨ "ͨ V͎ T V͎ͦ T͎

And ͬ ͚ Ɣ T͎ v ͚ ͦ Ɣ 3 ;͚ Ɣ ) So ͨv and ͦ.

ī v ġ Ȅ ͙ xč ͐͘ Ɣ ʚV͎ʛv Also because we know ġ . By differentiating with respect to gives v 3 v ī ġ ) Ȅ ͙ xč ͐͘ Ɣ ʚV͎ʛv ġ ͨv ͦ. So ͢ ǹ ͚ ͙ͦͯ!ͪ͘ Ɣ ġ T͎ ) ) Then Đʚͨʛ Ɣ Ɣ .In particular, this is non-decreasing as a function of ͨǨ ͦ ͦʚ/tͯ/ʛ

ʞRQ ͨͤʛ.

Theorem6.2 let ͛$% and ͚ʚͨʛ evolve according to the coupled flow "͛ $% ƔƎT͌ "ͨ $% "͚ ƔƎ;͚Ǝ͌ƍ ͚ ͦ "ͨ Ȅ ͙ͯ!͐͘ Then is constant. Proof by the chain rule we have " " ͐͘ Ɣ ǯ͙ͨ͛͘ ͬͥ͘ Ȁͬͦ͘ ȀUȀͬ͘) "ͨ "ͨ $% S $% ͥ ͦ ) Ɣ ͛ ƳƎT͌$%Ʒ͙ͨ͛ͬ͘͘ Ȁͬ͘ ȀUȀͬ͘ ƔƎ͌͐͘ Tǭ͙ͨ͛͘$% ͙̻ͨ͘ Ɣ ʚ̻ͯͥʛ$% ʠ Ĝĝʡ ͙̻ͨ͘ Because we have / / Hence ͘ "͚ ʚ͙ͯ!͐͘ʛ Ɣ͙ͯ! ƴƎ Ǝ͌Ƹ͐͘Ɣʚ;͚ Ǝ ͚ ͦʛ͙ͯ!͐͘ Ɣ Ǝ;ʚ͙ͯ!ʛ͐͘ ͨ͘ "ͨ Because ;ʚ͙ͯ!ʛ Ɣ ʚ ͚ ͦ Ǝ;͚ʛ͙ͯ!. So it then follows that

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

͘ ǹ͙ͯ!͐͘ Ɣ Ǝ ǹ ;ʚ͙ͯ!ʛ͐͘ ͨ͘ Because ͇ is closed manifold according to divergence theorem

ǹ;ʚ͙ͯ!ʛ͐͘ Ɣ R So this finishes the proof of the theorem. ʠ1 Ǝ͜ʡ ʚT;͚ Ǝ ͚ ͦ ƍ͌ʛ we would like to get rid of the ͦ term in (6.1) . we can do this 1 Ǝ͜ƔR by restricting our variations so that ͦ .From (6.2),this amounts to assuming that assuming͙ͯ!͐͘ is fixed. We now fix a smooth measure͘͡on ͇ and relate f to ͛ by requiring that͙ͯ!͐͘ Ɣ ͘͡ . Equivalently, we define a section ͧȏė Ŵė Ɛ ̽ʚ͇ʛ

ͧʚ͛ʛ Ɣƴ͛Q͢͠ʠ ʡƸ Đ( Ɣ Đͣͧ ė by ( . Then the composition is a function on and its deferential is given by

͘Đ(Ƴͪ ƷƔȄ ͙ͯ!ƫƎͪ Ƴ͌ ƍ ͚ƷƯ͐͘ $% $% $% $ % (6.4) Defining a formal Riemannian metric on ė by S $% ˛ͪ$%Qͪ$%˜" Ɣ ǹͪͪ$%͘͡ T The gradient flow of Đ(on ė is given by i ͛ ƔƎTƳ͌ ƍ ͚Ʒ i/ $% $% $ % (6.5) The induced flow equation for ͚ is "͚ S " Ɣ ͛$% ͛ ƔƎ;͚Ǝ͌ "ͨ T "ͨ $% As with any gradient flow, the function Đ( is non-decreasing along the flow line with its derivative being given by the length squared of the gradient, i.e. " ( ͦ Đ ƔTǹ ɳ͌$% ƍ$%͚ɳ ͘͡ "ͨ

We can check that if ͛$%ʚͨʛ and ͚ʚͨʛ evolve according to the coupled flow

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

i"Ĝĝ ƔƎT͌$% ʰ i/ (6.7) i! ƔƎ;͚ƍ ͚ ͦ Ǝ͌ i/ ͦ ĐƳ͛ ʚͨʛQ ͚ʚͨʛƷ Ɣ T Ȅ ɳ͌ ƍ ͚ɳ ͙ͯ! ͐͘ Then / $% $% $ % .Now we prove that to obtaining a solution of (6.5)and (6.6) is to show that it is some how equivalent to the decoupled system of (6.7). ǳ Proposition6.1(see [39]) defining ͛Ȥʚͨʛ Ɣ ʚͨʛ / ʚ͛ʚͨʛʛ,we have i"Ȥ i" Ɣʚͨʛ ǳʚ͛ʛ ƍʚͨʛ ǳ ʠ ʡ ƍʚͨʛ ǳʚĖ ͛ʛ i/ / / i/ / (6.9) Also for some function͚S ͇ Ŵ ͌, we then have

ĖʚO!ʛ͛ƔT͙͂ͧͧʚ͚ʛ Theorem6.3 the solutions of (6.5) and (6.6) may be generated by pulling back solutions of (6.7) by an appropriate time-dependent diffeomorphism.

Proof By previous proposition we know Ė͛ƔƎT͙͂ͧͧʚ͚ʛ where ͒ʚͨʛ ƔƎ͚.We fix ʚͨʛ ȸ S. Now we define ͛Ȥʚͨʛ by (6.8), it will evolve, by (6.9),according to

"͛Ȥ Ɣ ǳʚͨʛƫƎT͌͗͝ʚ͛ʛ ƎT͙͂ͧͧ ʚ͚ʛƯ "ͨ / "

Where ͙͂ͧͧ"ʚ͚ʛ is the Hessian of ͚ with respect to the metric ͛ .Keeping in mind that ɸ /S ʚ͇Q ͛Ȥʛ Ŵ ʚ͇Q ͛ʛ is an isometry, we may then write ͚ Ȭ͚ͣ / to give "͛Ȥ ƔƎTƳ͌͗͝ʚ͛Ȥʛ ƍ͙͂ͧͧ ʚ͚ɸʛƷ "ͨ "Ȥ The evolution of ͚ɸ is then found by the chain rule "͚ɸ "͚ ʚͬQ ͨʛ Ɣ ʚ ʚͬʛQͨʛ ƍ͒ʚ͚ʛʚ ʚͬʛQͨʛ "ͨ "ͨ / / ͦ ͦ ƔƫƳƎ;"͚ƍ ͚ Ǝ͌"ƷƎ ͚ Ưʚ /ʚͬʛQͨʛ ɸ ƔƫƎ;"Ȥ͚ Ǝ͌"ȤƯʚͬQ ͨʛ ͦ (Here we have used of this fact thatĖO!͚Ɣ ͚ ) Thus we have a solution to "͚ɸ ƔƎ; ͚ɸ Ǝ͌ "ͨ "Ȥ "Ȥ We would like to use this to develop a controlled quantity for Ricci flow, but we need to eliminate͚. This can be accomplished by taking an infimum, defining

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

ʚ͇Q ͛ʛ Ɣ͚͢͝Đʚ͇Q ͛Q ͚ʛ !S ę Ͱͥ ȄĆ

Lemma6.1(see[39]) let ͛$%ʚͨʛ͕͚͘͢ʚͨʛ evolve according to the coupled flow

"͛$% ƔƎT͌$% Ʀ "ͨ "͚ ƔƎ;͚ƍ ͚ ͦ Ǝ͌ "ͨ

Then ĐƳ͛$%ʚͨʛQ͚ʚͨʛƷ is non-decreasing in time and monotonicity is strict unless we are on a steady gradient solution. Proof according to previous computations we can show that " ͦ ͯ! ĐƳ͛$%ʚͨʛQ ͚ʚͨʛƷ Ɣ T ǹ ɳ͌$% ƍ$%͚ɳ ͙ ͐͘ "ͨ So proof is complete. So by previous lemma we obtain

ʠ͛$%ʚͨʛʡ ƙĐƳ͛$%ʚͨʛQ͚ʚͨʛƷƙĐƳ͛$%ʚͨͤʛQ͚ʚͨͤʛƷ Ɣ ʠ͛$%ʚͨͤʛʡ (6.10)

ͨƗͨ Ȅ ͙ͯ!͐͘ Ɣ S For ͤ and .

Definition6.1 a steady breather is a Ricci flow solution on an interval ʞͨͥQͨͦʟthat ǳ satisfies the equation ͛ʚͨͦʛ Ɣ& ͛ʚͨͥʛ for some &Ǩ͚͚̾͝ʚ͇ʛ. Now we show that a steady breather on a compact manifold is a gradient steady soliton.

Theorem6.4 (Perelman[39]) a steady breather is a gradient steady soliton. ǳ Proof .We have Ƴ͛ʚͨͦʛƷƔʚ& ͛ʚͨͥʛʛ ƔƳ͛ʚͨͥʛƷ.Because is invariant under diffeomorphism. So by (6.10),Đʚ͛ʚͨʛQ ͚ʚͨʛʛ must be constant in ͨ.From " ͦ ͯ! ĐƳ͛$%ʚͨʛQ ͚ʚͨʛƷ Ɣ T ǹ ɳ͌$% ƍ$%͚ɳ ͙ ͐͘ "ͨ

We conclude, ͌$% ƍ$%͚ƔR .Then ͌ƍϚ͚ƔR and so the system

"͛$% ƔƎT͌$% Ʀ "ͨ "͚ ƔƎ;͚ƍ ͚ ͦ Ǝ͌ "ͨ Change to

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

"͛ $% ƔƎT͌ "ͨ $% "͚ Ɣ ͚ ͦ "ͨ This is a gradient expanding soliton. Lemma 6.2(see [38-39])we have " T ƚ ͦʚͨʛ "ͨ ͢

Definition6.2 an expanding breather is a Ricci flow solution on an interval ʞͨͥQͨͦʟthat ǳ satisfies the equation ͛ʚͨͦʛ Ɣ͗& ͛ʚͨͥʛ for some ͗ƘSand &Ǩ͚͚̾͝ʚ͇ʛ. Proposition6.2 an expanding breather is a gradient expanding solution. Lemma6.3 (see [43])ʚ͇Q ͛ʛ is finite Lemma 6.4(see [43])ʚ͇Q ͛ʛ is the least number for which one has the inequality

ͦ ͦ ͦ ǹV ͩ " ƍ͌ ͩ ͐͘ ƚ ʚ͇Q ͛ʛ ǹ ͩ ͐͘

ͥ ͦ ͦ For all ͩ in the sobolev space ͂ ʚ͇ʛ.(note Ĺ͚Ĺ u Ɣ Ȅ Ƴ ͚ " ƍ͚ Ʒ͐͘for ̽ͥfunctions)

The ő Ǝ Functional

we know that the metric͛$%ʚͨʛ evolving by Ricci flow is called a breather, if for someͨͥ Ɨͨͦand ƘR the metrics ͛$%ʚͨͥʛand ͛$%ʚͨͦʛ differ only by 0 diffeomorphism; the cases Ɣ SQ Ɨ SQ Ƙ S correspond to steady, shrinking and expanding breathers, respectively. In order to handle the shrinking case when ʚ͇Q ͛ʛ Ƙ R we need to replace our functional Đ by its generalization, which contains explicit insertions of the scale parameter, to be denoted by . Thus consider the functional.

ͯ) ͦ ͯ! ġƳ͛$%Q͚QƷƔǹ ʞʚ ͚ ƍ͌ʛ ƍ͚Ǝ͢ʟʚVʛ ͦ ͙ ͐͘ Restricted to ͚ satisfying

ͯ) ǹ ʚVʛ ͦ ͙ͯ!͐͘ Ɣ S

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

ƘR. Where ͛$%is a Riemannian metric, ͚ is a smooth function on ͇ and is a positive scale parameter. Also for any positive number ͕ and any diffeomorphism ǳ ǳ ġƳ͕ ͛$%Q ͚Q ͕Ʒ Ɣ ġƳ͛$%Q͚QƷ.

Now we start with first variation for ġ.

Theorem6.5 (Perelman)assume that ͛$% Ɣͪ$% and ͚ Ɣ ͜. Put Ɣ then we have

ġƳͪ Q͜QƷƔȄ ʢʚ͌ƍ ͚ ͦʛ Ǝͪ Ƴ͌ ƍ ͚Ʒ ƍ ͜ ƍ ʞʚT;͚ Ǝ ͚ ͦ ƍ $% $% $% $ % ġ 1 )a ͯ! ͌ʛ ƍ͚Ǝ͢ʟ ʠ Ǝ͜Ǝ ʡʣ ʚVʛ v ͙ ͐͘ ͦ ͦc i ͐͘ Ɣ 1 ͐͘ Proof Byi/ ͦ . We see ͯ) ͪ ͢ ͯ) ʠʚVʛ ͦ ͙ͯ!͐͘ʡ Ɣ ʠ Ǝ͜Ǝ ʡ ʚVʛ ͦ ͙ͯ!͐͘ T T But we know ͪ ͯ! ͦ ĐƳ$%Q͜ƷƔǹ ͙ ʢƎͪ$%Ƴ͌$% ƍ$%͚Ʒ ƍ ʠ Ǝ͜ʡ ʚT;͚ Ǝ ͚ ƍ͌ʛʣ ͐͘ T We obtain ͪ ͦ ͦ ġ Ɣ ǹ ʢʚ͌ƍ ͚ ʛ ƍʠ Ǝ͜ʡ ʚT;͚ Ǝ T ͚ ʛ Ǝͪ$%Ƴ͌$% ƍ$%͚Ʒ ƍ ͜ T ͪ ͢ ͯ) ƍ ʞʚ͌ƍ ͚ ͦʛ ƍ͚Ǝ͢ʟ ʠ Ǝ͜Ǝ ʡʣ ʚVʛ ͦ ͙ͯ!͐͘ T T But we know ;͙ͯ! Ɣ ʚ ͚ ͦ Ǝ;͚ʛ͙ͯ! Therefore

ġƳͪ$%Q͜QƷ

ͦ Ɣǹ ʢʚ͌ƍ ͚ ʛ Ǝͪ$%Ƴ͌$% ƍ$%͚Ʒ ƍ ͜ ͪ ͢ ͯ) ƍ ʞʚT;͚ Ǝ ͚ ͦ ƍ͌ʛ ƍ͚Ǝ͢ʟ ʠ Ǝ͜Ǝ ʡʣ ʚVʛ ͦ ͙ͯ!͐͘ T T So proof is complete. Definition6.3 the arguments ͛Q ͚ and are called compatible if ͙ͯ! ǹ ) ͐͘ Ɣ S ʚVʛͦ

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Lemma6.5(see[39]) under the transformation ʚ͛Q ͚Q ʛ Ŵ ʚͯͥ͛Q ͚Q Sʛcompatibility is preserved, as is the functional : ġʚ͛Q ͚Q ʛ Ɣġʚͯͥ͛Q ͚Q Sʛ Note that on ͌), the Gaussian measure ͘ is defined in terms of the lebesgue measure ͬ͘ by ͯ) ͯ 3 v ͘ Ɣ ʚTʛ ͦ ͙ ͦ ͬ͘ The normalization being chosen so that

ǹ͘ƔS ġ Lemma 6.6(see [42])(L.Gross) If ͪS ͌) Ŵ͌ is, say, smooth and satisfies ͪ, ͪ Ǩ ͆ͦʚ͘ʛthen ͥ ͦ ǹͪͦ͢͠ ͪ ͘ ƙ ǹ ͪ ͦ͘ ƍ ʦǹͪͦ͘ʧ ͢͠ ʦǹͪͦ͘ʧ so if we choose ͪ so that Ȅ ͪͦ͘ Ɣ S,then inequality becomes

ǹͪͦ͢͠ ͪ ͘ ƙ ǹ ͪ ͦ͘

That called log-sobolev inequality. Theorem6.6(see[39]) (Perelman) Let ͛ denote the flat metric on ͌).if ͚ and are compatible with ͛, then ġʚ͛Q ͚Q ʛ ƚR Proof let ͚S ͌) Ŵ͌ be compatible with ͛ and , which in this situation means that ͙ͯ! ǹ ) ͬ͘ Ɣ S ġ ʚVʛͦ

v ͥ ę ī ͯę Ɣ Ȅ ͬ͘ Ɣ S ͪ Ɣ ͙ x v If we set we obtain ġ ġ .If we define ,we have ͦ ʚͦ_ʛv

3 v ͯ) ͯ 3 v ͯ) ͯ! ͪͦ͘ Ɣ ͙ ͦ ʚTʛ ͦ ͙ ͦ ͬ͘ Ɣ ʚTʛ ͦ ͙ͯ!ͬ͘ Ȅ ͪͦͬ͘ Ɣ S Ȅ ͪͦ͢͠ ͪ ͘ ƙ So ġ . Therefore, by the log-sobolev inequality we obtain,

Ȅ ͪ ͦ͘. By computing the left-hand side and right hand side of this inequality we get

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

3 v ͬ ͦ ͚ ͯ) ͯ 3 v ǹͪͦ͢͠ ͪ ͘ Ɣ ǹ ͙ ͦ ͯ! ʦ Ǝ ʧ ʚTʛ ͦ ͙ ͦ ͬ͘ V T

ͬ ͦ ͚ ͙ͯ! Ɣǹ ʦ Ǝ ʧ ͬ͘ V T ) ʚTʛͦ

v 3 ǧ! ī ͯę ǧͪ Ɣ ʠ Ǝ ʡ ͙ x v Also ͦ ͦ . Which gives us ͬ ͦ ͬT ǧ͚ ǧ͚ ͦ 3 v ǧͪ ͦ Ɣ ʦ Ǝ ƍ ʧ ͙ ͦ ͯ! V T V therefore ͬ ͦ ͬT ͚ ͚ ͦ ͙ͯ! ͪ ͦ͘ Ɣ ʦ Ǝ ƍ ʧ ͬ͘ V T V ) ʚTʛͦ the ͙͕ͨ͛ͦͨͣ͢͢͝͝ Ǝ ͖ͭ Ǝ ͕ͤͦͨͧ formula gives us ͬT ͚ ͙ͯ! S ͬ͘ Ǝǹ ͬ͘ Ɣ ǹͬTǧʚ͙ͯ!ʛ T ) T ) ʚTʛͦ ʚTʛͦ But we can compute that T ͬ Ɣ ͢ so ͬT ͚ ͙ͯ! Ǝ͢ ͙ͯ! Ǝǹ ͬ͘ Ɣ ǹ ͬ͘ T ) T ) ʚTʛͦ ʚTʛͦ So ͬ ͦ ͢ ͚ ͦ ͙ͯ! ǹ ǧͪ ͦ͘ Ɣ ǹ ʦ Ǝ ƍ ʧ ͬ͘ V T V ) ʚTʛͦ And the log-sobolev inequality gives us ͬ ͦ ͚ ͙ͯ! ͬ ͦ ͢ ͚ ͦ ͙ͯ! ǹ ʦ Ǝ ʧ ͬ͘ ƙ ǹ ʦ Ǝ ƍ ʧ ͬ͘ V T ) V T V ) ʚTʛͦ ʚTʛͦ So S S ͙ͯ! ġƴ͛Q͚Q ƸƔǹ Ƭ ͚ ͦ ƍ͚Ǝ͢ư ͬ͘ ƚ R T T ) ʚTʛͦ

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

ġʚ͛Q ͚Q ʛ Ɣġʠ ͥ ͛Q ͚Q ͥʡ ʚ͌)Q͛ʛ by the scale invariance ͦc ͦ and because is preserved under the homothetic scaling ,we conclude ġʚ͛Q ͚Q ʛ ƚR and proof will be complete. Remark 6.2(see[38]) to easily we can check that for any͚ ,compatible with ͛ v ġʚ͛Q ͚Q ʛ ƔR ͚ʚͬʛ ȸ 3 , if and only if ͨc . We have a analogous theorem forġ,like Đ.We see that ġ,is increasing under the Ricci flow when ͚ and are made to evolve appropriately.

Theorem6.7 (Perelman) if ͛$%ʚͨʛ,͚ʚͨʛand ʚͨʛevolve according to the system

"͛$% ˫ ƔƎT͌$% ˮ "ͨ "͚ ͢ ƔƎ;͚ƍ ǧ͚ ͦ Ǝ͌ƍ ˬ "ͨ T ˮ " Ɣ ƎS ˭ "ͨ Then we have the identity ͘ S ͦ ͯ) ͯ! ġƳ͛$%ʚͨʛQ͚ʚͨʛQ ʚͨʛƷ Ɣ ǹ T ɴ͌$% ƍ$%͚Ǝ ͛$%ɴ ʚVʛ ͦ ͙ ͐͘ ͨ͘ T ġ ͯ! Ȅ ʚVʛ v ͙ ͐͘ ġƳ͛ ʚͨʛQ͚ʚͨʛQʚͨʛƷ and is constant .In Particular .in particular $% is non-decreasing in time and monotonicity is strict unless we are on a shrinking gradient soliton. Now we give an example of a gradient shrinking soliton . Example consider ͌) with the flat metric, constant in time ͨǨʚƎ7Q Rʛ and let ƔƎͨ v ͚ʚQ ͬʛ Ɣ 3 and ͨc.

v īv 3 ͯ! ͯ ͚ʚQ ͬʛ Ɣ ͙ Ɣ͙ xś proof because ͨc so we get . To easily we can checkʚ͛ʚͨʛQ͚ʚͨʛQʚͨʛʛ satisfies the following system

"͛$% ˫ ƔƎT͌$% ˮ "ͨ "͚ ͢ ƔƎ;͚ƍ ǧ͚ ͦ Ǝ͌ƍ ˬ "ͨ T ˮ " Ɣ ƎS ˭ "ͨ

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

ġ ͯ! and Ȅ ʚVʛ v ͙ ͐͘ Ɣ S

v v v ʚ ǧ͚ ͦ ƍ͌ʛ ƍ͚Ǝ͢Ɣ 3 ƍ 3 Ǝ͢Ɣ 3 Ǝ͢ now ͨcv ͨc ͦc so it follows from of this fact that we have

ͯ 3 v ) ǹ͙ͨc ͐͘ Ɣ ʚVʛͦ ġ and ͯ 3 v ͬ ͦ ) ͢ ǹ͙ͨc ͐͘ Ɣ ʚVʛͦ Vͦ T

Therefore ġʚͨʛ ƔR for all ͨ.So proof is complete. ġ ͯ! Remark6.3 now if ͛ is the Euclidean metric and we let ͩƔʚVʛ v ͙ , we see that Ǝ͢ ͣ͛ͩ͠ Ɣ ͣ͛͠ʚVʛ Ǝ͚ T ǧͩ ͦ Ɣ ʚVʛͯ) ǧ͚ ͙ͦͯͦ! v ǧ͚ ͦ Ɣ ǧ0 So 0v Therefore ǧͩ ͦ ͢ ġʚ͇Q ͛Q ͚Q ʛ Ɣǹ ʨ Ǝͩͣ͛ͩ͠ʩ ͬ͘ Ǝ ͣ͛͠ʚVʛ Ǝ͢ ͩͦ T but we know that ġƚR so it implies a log-sobolev inequality v Ȅ ǧ0 ͬ͘ ƚ Ȅ ͩͣ͛ͩͬ͘͠ ƍ ) ʚVʛ ƍ͢ 0v ͦ If we set &ͦ Ɣͩ S ͢ V ǹ ǧ& ͦͬ͘ ƚ ǹ&ͦͣ͛͠ &ͦͬ͘ ƍ ͣ͛͠ʚVʛ ƍ͢ T

For the general case, we have v ġʚ͇Q ͛Q ͚Q ʛ Ɣ Ȅ ʢ ʠ͌ͩ ƍ ǧ0 ʡ Ǝͩͣ͛ͩ͠ʣ ͐͘ Ǝ ) ͣ͛͠ʚVʛ Ǝ͢ 0v ͦ (6.11) one can show that ġʚ͇Q ͛Q ͚Q ʛ ƚƎ͗ʚ͇Q ͛Q ʛ so according to (6.11)we get Ȅ ͌&ͦ͐͘ ƍ Ȅ V ǧ& ͦ͐͘ ƚ Ǝ͗ ƍ Ȅ &ͦͣ͛͠&ͦ͐͘ ƍ ) ͣ͛͠ʚVʛ ƍ͢ ͦ (6.12) Definition6.4 we set

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

S Ϧ ͯ! Ƴ͛$%QƷƔ͚͢͝ƥġƳ͛$%Q͚QƷS͚ Ǩ̽ ʚ͇ʛQ ) ǹ͙ ͐͘ Ɣ SƩ ʚVʛͦ Which according to (6.12) is the best possible constantƎ͗. Remark 6.4(see [38]) is finite.

Definition6.5 a shrinking breather is a Ricci flow solution on ʞͨͥQͨͦʟthat satisfies ǳ ͛ʚͨͦʛ Ɣ͗& ͛ʚͨͥʛ for some ͗ƗS and& Ǩ ͚͚̾͝ʚ͇ʛ. Definition6.6 a shrinking soliton lives on a time interval ʚƎ7Q Rʛ.a gradient shrinking soliton satisfies the equations "͛ ͛ $% ƔƎT͌ ƔT ͚ƍ $% "ͨ $% $ % ͨ "͚ Ɣ ǧ͚ ͦ "ͨ

Remark 6.5(see [39])one can show that for any time ͨƙͨͤ we have

ʚ͇Q ͛ʚͨʛQʚͨʛʛ ƙġʚ͇Q ͛ʚͨʛQ͚ʚͨʛQʚͨʛʛ ƙʚ͇Q ͛ʚͨͤʛQʚͨͤʛʛ (6.13) Theorem6.8 A shrinking breather is a gradient shrinking soliton. ͨ Ɣ /vͯ/u Ɣͨ Ǝͨ Ɣͨ Ǝͨ Ɣ͗ Proof put ͤ ͥͯ .Then if ͥ ͤ ͥand ͦ ͤ ͦ , we get ͦ ͥ.Therefore because the functional ġ is invariant under simultaneous scaling of and ͛$% are invariant under diffeomorphism, so ͦ ǳ ǳ ʚ͛ʚͨͦʛQͦʛ Ɣƴ & ͛ʚͨͥʛQͦƸƔʚ& ͛ʚͨͥʛQͥʛ Ɣʚ͛ʚͨͥʛQͥʛ ͥ so by (6.13) and this fact that ͘ S ͦ ͯ) ͯ! ġƳ͛$%ʚͨʛQ͚ʚͨʛQ ʚͨʛƷ Ɣ ǹ T ɴ͌$% ƍ$%͚Ǝ ͛$%ɴ ʚVʛ ͦ ͙ ͐͘ ͨ͘ T it follows that the solution is a gradient shrinking soliton .

Remark6.6(see [43]):Ƴ͛$%ʚͨʛQƎͨƷ is non-decreasing along the Ricci flow . Proposition6.3(see[43]) ʚ͛Q ʛ is negative for small ƘR and tends to zero as ŴR.

RecentdevelopmentsonPerelman’sfunctionalŀ·Äºő

Definition6.7 in [36] Jun-Fang Li introduced the following ĐƎfunctional,

ͦ ͯ! Đ&ʚ͛Q ͚ʛ Ɣǹ ʚ͌͟ ƍ ǧ͚ ʛ͙ ͘

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Where ͟ƚS.when͟ƔS, this is the ĐƎfunctional. The following theorem is analogous result, like,Đ. Theorem6.9 (see[36])suppose the Ricci flow of ͛ʚͨʛ exists for ʞRQ ͎ʛ, then all the functional Đ&ʚ͛Q ͚ʛ will be monotone under the following coupled system, i.e.

"͛$% ƔƎT͌$% Ʀ "ͨ "͚ ƔƎ;͚ƍ ͚ ͦ Ǝ͌ "ͨ ͘ ͦ ͯ! ͦ ͯ! Đ&Ƴ͛$%Q ͚Ʒ Ɣ Tʚ͟ Ǝ Sʛ ǹ ͌͗ ͙ ͘ ƍ T ǹ ɳ͌$% ƍ$%͚ɳ ͙ ͘ ƚR ͨ͘ define &ʚ͛ʛ Ɣ͚͢͝Đ&ʚ͛Q ͚ʛ, where infimum is taken over all smooth ͚, satisfying Ȅ ͙ͯ!͘ Ɣ S ʚ͛ʛ Ɣʚ͛ʛ .and we assume ͥ . Theorem6.10(see [36]and[46]) ʚ͛ʛ is the lowest eigenvalue of the parameter ƎV; ƍ ͌ and the non-decreasing of the Đ functional implies the non-decreasing of ʚ͛ʛ. as an application ,Perelman was able to show that there is no non-trivial steady or expanding Ricci breathers on closed manifolds. Theorem6.11(see[46]) on a compact Riemannian manifold ʚ͇Q ͛ʚͨʛʛQ where ͛ʚͨʛ satisfies the Ricci flow equation for ͨǨʞRQ͎ʛQ the lowest eigenvalue & of the operatorƎVϚ ƍ͌͟ is non-decreasing under the Ricci flow. The monotonicity is strict unless the metric is Ricci-flat. X. D. Cao considered the eigenvalues of the operator

͌ ƎϚ ƍ T On manifolds with nonnegative cuvvature operator. He showed that the eigenvalues of these manifolds are non-decreasing along the Ricci flow.

Corollary On a compact Riemannian manifold, the lowest eigenvalues of the operator ƎϚ ƍ ͦare non-decreasing under the Ricci flow . ͟ƔT ͥ ƎϚ ƍ proof let , then ͨ ͦ is the lowest eigenvalue of ͦ and the result will follows.

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Theorem6.12 (see [46])Let͛ʚͨʛQ ͨ Ǩ ʞRQ ͎ʛ, be a solution to the Ricci flow on a closed Riemannian manifold ͇).Assume that there is a ̽ͥ-family of smooth functions ͚ʚͨʛ Ƙ R, which satisfy

S ʚͨʛ͚ʚͨʛ ƔƎ; ͚ʚͨʛ ƍ ͌ ͚ʚͨʛ "ʚ/ʛ T "ʚ/ʛ

ͦ ǹ͚ʚͨʛ͘"ʚ/ʛ ƔS

Where ʚͨʛ is a function of ͨ only .Then ͘ T ʚͨʛ ƔVǹ ͌ ǧ$͚ǧ%͚͘ƍTǹ ͌͗ ͚ͦͦ ͘ ͨ͘ $%

ͦ ͯe ͦ ͯe Ɣ ǹ ɳ͌$% ƍ$%ɳ ͙ ͘ ƍ ǹ ͌͗ ͙ ͘ ƚ R

Entropyfunctionalfordiffusionoperator

Let ʚ͇Q ͛ʛ be a compact Riemannian manifold, &Ǩ̽ͦʚ͇ʛ.Let ͆ Ɣ ; Ǝ ǧ&T ǧQ͘ Ɣ ͙ͯm͐͘ Let ͙ͯ! ͩƔ ( ʚVͨʛ ͦ be a positive solution of

ʚ"/ Ǝ͆ʛͩƔR Inspired by the work of Perelman and Ni, we have the following results.

Theorem 6.13(X.-D. Li 2006) let

͡ ͡ ͂(ʚͩQ ͨʛ Ɣ ǹ ͩͣ͛ͩ͘͠ Ǝ ʠ ͣ͛͠ʚVͨʛ ƍ ʡ T T ͙ͯ! ͦ ġʚͩQ ͨʛ Ɣǹ ʚͨ ǧ͚ ƍ͚Ǝ͡ʛ ( ͘ ʚVͨʛ ͦ

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Then

͘ ͡ ͂(ʚͩQ ͨʛ ƔƎǹ ʠ͆ͣ͛ͩ͠ ƍ ʡ ͩ͘ ͨ͘ Tͨ

͘ ġʚͩQ ͨʛ Ɣ Ƴͨ͂ ʚͩQ ͨʛƷ ͨ͘ ( Theorem6.14 (X.-D. Li) let u be a positive solution of the heat equation " ƴ Ǝ͆ƸͩƔR "ͨ Suppose that

&̂& ͌͗͝ ʚ͆ʛSƔ͌͗͝ƍͦ&Ǝ ƚR (Q) ͡Ǝ͢ Then ͘ġʚͩQ ͨʛ ͛ ͦ ͦ ƔƎTǹ ƴʺ ͚Ǝ ʺ ͩ͘ ƍ ͌͗͝(Q)ʚ͆ʛʚǧ͚Q ǧ͚ʛͩƸ ͘ ͨ͘ T T ͡Ǝ͢ ͦ Ǝ ǹʠǧ&T ǧ͚ ƍ ʡ ͩ͘ ͡Ǝ͢ T

Corollary (X.-D.Li 2006) supposes that ͌͗͝(Q)ʚ͆ʛ ƚR then Ŵʚʛis decreasing along the heat diffusionʚ"c Ǝ͆ʛͩƔR.

Perelmanfunctionalŀ·ÄºőforextendRicciflowsystem We consider a system of evolution equations such that the stationary points satisfy ͌͗ʚ͛ʛ ƔTͩ̂ͩ͘͘ ;"ͩƔR to this end Bernhard list in [37]extended the Ricci flow to the system i"ʚ/ʛ ƔƎT͌͗Ƴ͛ʚͨʛƷƍVͩ̂ͩ͘͘ i/ (6.14) i0ʚ/ʛ Ɣ;"ʚ/ʛͩʚͨʛ i/ For a Riemannian metric͛ʚͨʛ, a function ͩʚͨʛQ and given initial data ͛ʚRʛ and ͩʚRʛT this is a quasilinear, weakly parabolic, coupled system of second order. Here ͩ͘ ɬ ͩ͘

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

$ % is the tensor "$ͩ"%ͩͬ͘ ɬ ͬ͘ and the laplacian of a function ͩ with respect to ͛ is " $% & given byϚ ͩ Ɣ͛ ʚ"$"%ͩƎϙ$% "&ͩʛ. Definition6.8(see[37]) let Ǩ͌ be a positive real number. Then the entropyġ of a configuration ʚ͛QͩQ͚QʛǨėʚ͇ʛ Ɛ̽ Ϧʚ͇ʛ Ɛ̽ Ϧʚ͇ʛ Ɛ͌ ͮ is defined to be

ͯ) ġʚ͛Q ͩQ ͚Q ʛ Ȭǹ ʞʚ͍ƍ ͚͘ ͦʛ ƍ͚Ǝ͢ʟ ʚVʛ ͦ ͙ͯ!͐͘ i" ͍Ȭ͌ƎT ͩ͘ ͦ Ĝĝ Ɣ Where (So the evolution of the metric can then be written as i/

ƎT͍$%) now we prove thatġ is scaling invariant . Lemma6.7 (see[37])let ƘR be a constant and be a diffeomorphism of ͇.then the entropy ġis invariant under simultaneous scaling of ͛ and by in the sense that ġʚ ͛QͩQ͚Q ʛ Ɣġʚ͛Q ͩQ ͚Q ʛ and invariant under diffeomorphisms ġʚǳ͛Q ǳͩQ ǳ͚Q ʛ Ɣġʚ͛Q ͩQ ͚Q ʛ Proof the invariance under diffeomorphisms is clear .also we have ġʚ ͛QͩQ͚Q ʛ

$% $% Ɣǹ ƫ Ƴ͌ʚ ͛ʛ ƎTʚ ͛ʛ "$ͩ"%ͩƍʚ ͛ʛ "$͚"%͚Ʒ ƍ ͚ ͯ) Ǝ͢Ư ʚV ʛ ͦ ͙ͯ!ǭ͙ͨ͘ʚ ͛ʛͬ͘

Ɣǹ ʞ ʚ ͯͥ͌ƎT ͯͥ ͩ͘ ͦ ƍ ͯͥ ͚͘ ͦʛ ƍ͚ ͯ) ͯ) ) Ǝ͢ʟ ͦ ʚVʛ ͦ ͙ͯ! ͦ͐͘ Ɣ ġʚ͛Q ͩQ ͚Q ʛ

Theorem6.15 (see[37])let ͇ be a closed Riemannian manifold and assume that ͛Q ͩQ ͚ and satisfy on ʞRQ ͎ʛ Ɛ͇ the evolution equations

"/͛ƔƎT͍4

"/ͩƔ;ͩ " ͚ƔƎ;͚ƍ ǧ͚ ͦ Ǝ͍ƍ ) / ͦc

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

"/ƔƎS

(͍ is the trace of symmetric tensor field͍4 ) then the following monotonicity formula holds: S ͦ ͦ ͦ "/ġʚͨʛ Ɣǹ ʦT ɴ͍4 ƍ ͚Ǝ ͛ɴ ƍV ǧͩ Ǝ ͩ͘ʚǧ͚ʛ ʧ ͘͡ ƚ R T Remark6.7(see[37]) the entropy ġ is non-decreasing and equality holds if and only if the solution is a homothetic shrinking gradient soliton. In this case ʚ͛Q ͩQ ͚Q ʛʚͨʛsatisfies at everyͨǨʞRQ͎ʛ. ͍ ƍ͚ͦƎ ͥ ͛ƔR ǧͩ Ǝ ͩ͘ʚǧ͚ʛ ƔR 4 ͦc and Theorem6.16(see[37]) if we varyġalong the variation given by the following evolution equations ͦ "/͛ƔƎT͍4 ƎTǧ ͚

"/ͩƔ;ͩƎ˛ͩ͘Q ͚͘˜ " ͚ƔƎ;͚Ǝ͍ƍ) / ͦc

"/ƔƎS then we have

S ͦ ͦ ͦ "/ġʚ͛Q ͩQ ͚Q ʛʚͨʛ Ɣ ǹ ʦT ɴ͍4 ƍ ͚Ǝ ͛ɴ ƍV ;ͩ Ǝ ˛ͩ͘Q ͚͘˜ ʧ ͘͡ ƚ R T Definition 6.9(see[37])let ʚ͛Q ͩQ ʛ Ǩ ėʚ͇ʛ Ɛ̽ Ϧʚ͇ʛ Ɛ͌ ͮ be a configuration .then we define

ͯ) Ȭʚ͛Q ͩQ ʛ Ɣ',$ ʪġʚ͛Q ͩQ ͚Q ʛ ǹ ʚVʛ ͦ ͙ͯ!͐͘ Ɣ Sʫ öʚ ʛ !Ǩ We investigate the remaining case of shrinking breathers.

Theorem6.17. (see[37])suppose ʚ͛Q ͩʛʚͨʛ is a solution to (6.14) on ʞRQ ͎ʛ Ɛ͇ where ͇ is closed. Fix a ǨʞRQ͎ʛ and define ʚͨʛ ȏƔ ƎͨTthen ʚ͛Q ͩQ ʛʚͨʛ is non- ͨ i ʚͨʛ ƔR decreasing in . If i/ the solution is a gradient shrinking soliton. Proposition6.4(see[37]) Let ʚ͛Q ͩʛʚͨʛbe a shrinking breather on a closed manifold ͇. Then it necessarily is a gradient shrinking soliton.

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Perelman’sFunctionalŀ·ÄºőonRicciYangMillsflow

The Yang-Mills heat flow was first used by Atiyah (in [45])and Bott and simon Donaldson. Donaldson used it to give an analytic proof of a Theorem of Narasimhan and Seshadri. also Atiyah and Bott used the Yang-Mills heat flow to study the topology of minimal Yang Mills connections. The Yang-Mills heat flow is a gauge-theoretic heat flow; that is, it is a differential equation for a field on a principal fiber bundle. In the study of geometric evolution equations monotonic quantities have always played an important role. Here we give a Review of the Ricci Yang- Mills flow using energy functional, but as for Ricci flow the resulting equations are not a-priori gradient equations. In other words we would like to follow the ideas of Perelman in order to write the Ricci Yang-Mills flow as a gradient flow. We claim that our coupled system is the gradient flow of some functional Đʚ͛Q͕Q͚ʛ analogous to that of Perelman.

For definition of the Ricci Yang-Mills flow at first, we recall the ͏ʚ͡ʛ Ǝvector bundle.

Let G be a unitary group ͏ʚ͡ʛ ɛ ́͆ʚ͡Q ͌ʛ ɛ ́͆ʚT͡Q ͌ʛ, a ́Ǝvector bundle is a complex vector bundle of rank m together with Hermitian metric, a smooth function which assigns to each ͤ Ǩ ͇ a map

˾Q ˿+ ȏ ̿+ Ɛ̿+ Ŵ̽ Which satisies the axioms

1. ˾ͪQ ͫ˿+is complex linear in ͫ and conjugate linear in ͪ.

2. ˾ͪQ ͫ˿+ Ɣ˾ͫQ ͪ˿+ 3. ˾ͪQ ͫ˿+ ƚR, with equality holding only if ͪ Ɣ R

Definition6.10(see [44]) Let ͊ be a ͏ʚSʛ Ǝbundle over a compact manifold. One can choose a metric on ͊ such that the Ricci flow equations, with the additional hypothesis that size of the fiber remains fixed, yield the Ricci Yang-Mills flow: "͛ ƔƎT͌͗ƍ̀ͦ "ͨ

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

"̻ ƔƎ̾ǳ̀ʚ̻ʛ "ͨ here “A” is a connection on ͊ and ̾ǳis adjoint of the exterior derivative ̾T also ̀ is a two-form on ͇ where ̀ʚ̻ʛ Ɣ ̻̾ that ̻̾ denote the exterior covariant derivative. Recall that if G is a lie sub group of ́͆ʚ͡Q ͌ʛQa ́Ǝvector bundle is a rank m vector bundle whose transition functions take their values in ́.

Definition6.11(see [44]) Let ʚ͇Q ͛ʛ be a Riemannian manifold and let ̿Ŵ͇ denote a principal ͅƎbundle (ͅ is a lie group) over ͇ with connection̻. In this section will always refer to the Levi-Civita connection of ͛. Consider the functional S Đʚ͛Q ̻Q ͚ʛ Ɣǹ ƴ͌Ǝ ̀ ͦ ƍ ǧ͚ ͦƸ͙ͯ!͐͘ V where ͌ is the scalar curvature of the base metric, ͚Ǩ̽Ϧʚ͇ʛ and ͐͘ denotes the volume form of ͛. We use the notation to refer to the first variation at 0 of other quantities with respect to the parameterͨ.

Lemma6.8(see [44]) Let ͛$% Ɣͪ$%, ̻$ Ɣ $, ͚ Ɣ ͜,then ĐʚͪQ Q ͜ʛ S ͯ! ǳ $ Ɣǹ ͙ ƬƎͪ$% ƴ͌͗$% Ǝ $% ƍ$%͚Ƹ Ǝ %Ƴ ̀% Ǝ ͚̀$%Ʒ T ͪ S ƍ ʠ Ǝ͜ʡ ƴT;͚ Ǝ ǧ͚ ͦ ƍ { Ǝ | ͦƸư ͐͘ T V & where $% Ɣ ̀$ ̀&% Theorem6.18 (see [44])Given ʚ͛ʚͨʛQ̻ʚͨʛQ͚ʚͨʛʛ a solution to following system ͛ ƔƎT͌͗ ƍ ƎT ͚ / $% $% $% $ % ̻ ƔƎ͘ǳ̀ ƍ$͚̀ / $ $% (6.15) ͚ƔƎ;͚Ǝ͌ƍͥ ̀ ͦ / ͦ Then the functional Đ is monotonically increasing in ͨ. In particular

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

͘ S ͦ ǳ $ ͦ ͯ! ĐƔǹ ʦT ɴ͌͗$% Ǝ $% ƍ$%͚ɴ ƍ ɳ͘ ̀ Ǝ ͚̀$%ɳ ʧ ͙ ͐͘ ƚ R ͨ͘ T Definition6.12: (we define a metric on our configuration space to be

ͯ! ˛ʚ͛ͥQ̻ͥʛQ ʚ͛ͦQ̻ͦʛ˜ Ɣǹ ƳTʚ̻ͥQ̻ͦʛ ƍTʚ͛ͥQ͛ͦʛƷ͙ ͐͘ then the gradient flow of Đ becomes (6.15)

ĐƳ͛ʚͨʛQ̻ʚͨʛQ͚ʚͨʛƷƚ Remark6.8(see[44]) under equations (6.15) we know / RT ͌ Ǝ ͥ ̀&̀ ƍǧǧ ͚ƔR ͘ǳ̀ Ǝ͚̀ ƔR Equality is attained precisely when $% ͦ $ &% $ % and % $ $% i.e. When ʚ͛Q ̻ʛ is a steady gradient Ricci Yang-Mills soliton. Remark6.9 The solutions to equations (6.15) are equivalent to the system ͛ ƔƎT͌͗ ƍ ̀ ̀ / $% $% $& %& ̻ ƔƎ͘ǳ̀ / $ $ (6.16) ͚ƔƎ;͚ƍ ǧ͚ ͦ Ǝ͌ƍͥ ̀ ͦ / ͦ corollary under equations (6,16) ͘ ĐƳ͛ʚͨʛQ̻ʚͨʛQ͚ʚͨʛƷƚR ͨ͘

Proposition6.5 (see[44])there exists a unique minimizer ͚ of Đʚ͛Q͕Q͚ʛ subject to the constraint

ǹ͙ͯ!͐͘ Ɣ S

Definition 6.13according to this proposition we can then define

ʚ͛Q ̻ʛ Ɣ ͚͢͝ ʪĐʚ͛Q ̻Q ͚ʛS͚ Ǩ̽ϦQǹ ͙ͯ!͐͘ Ɣ Sʫ

Proposition6.6(see[44]) if ʚ͛ʚT ʛQ ̻ʚT ʛʛ is a solution to the Ricci Yang - Mills flow, then ʚ͛ʚͨʛQ ̻ʚͨʛʛ is non - decreasing in time.

Remark6.10(see[44]) the minimum value of ʚ͛Q ̻ʛis equal toͥʚ͛Q ̻ʛ, ʚ͛Q ̻ʛ ƎV; ƍ ͌ Ǝ ͥ ̀ ͦ where ͥ is the smallest eigenvalue of the elliptic operator ͨ .

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

Then the minimizer,͚ͤ, of Đ satisfies the Euler - Lagrange equation S ʚ͛Q ̻ʛ ƔT;͚ Ǝ ǧ͚ ͦ ƍ͌Ǝ ̀ ͦ ͤ ͤ V Definition6.14 A solution ʚ͛ʚͨʛQ ̻ʚͨʛʛ to the Ricci Yang -Mills flow is called a breather if there exist timesͨͥ Ɨͨͦ, a constant , and a diffeomorphismS ͇ Ŵ ͇such that ǳ ǳ ͛ʚͨͦʛ Ɣ & ͛ʚͨͥʛ,̻ʚͨͦʛ Ɣ & ̻ʚͨͥʛ ƘSQ ƗS͕͘͢ ƔS correspond to ʚ͛ʚͨʛQ ̻ʚͨʛʛ being a expanding, shrinking or steady breather respectively. Theorem6.19(see[44]) let ʚ͇)Q ͛ʚͨʛQ ̻ʚͨʛʛbe a solution to the Ricci Yang-Mills flow on a closed manifold. if there existͨͥ Ɨͨͦ with ʚ͛ʚͨͥʛQ ̻ʚͨͥʛʛ Ɣ ʚ͛ʚͨͦʛQ ̻ʚͨͦʛʛthen ʚ͛ʚͨʛQ̻ʚͨʛʛ is a steady gradient Ricci Yang-Mills soliton, which must have ̀ ͦ ƔR and be scalar flat. Definition 6.15(see[44])we define S ͯ) ġʚ͛Q ̻Q ͚Q ʛ Ɣǹ ƴƴ ǧ͚ ͦ ƍ͌ƍ ̀ ͦƸƍ͚Ǝ͢ƸʚVʛ ͦ ͙ͯ!͐͘ V where Ɣ͎Ǝͨ and ʚ͛ʚͨʛQ ̻ʚͨʛʛis a solution to Ricci Yang - Mills flow which exists on a maximal time interval of the form ʞRQ ͎ʟ where ͎Ɨ7T

Theorem6.20(see[44]) letͪ$% Ɣ ͛$%, ̻$ Ɣ $, ͚ Ɣ ͜ and Ɣ .Then ġʚͪQ Q ͜Q ʛ

ͯ) S Ɣǹ ʚVʛ ͦ ͙ͯ!͐͘ Ƭƴ ǧ͚ ͦ ƍ͌ƍ ̀ ͦƸ V S Ǝͪ ƴ͌͗ ƍ ƍǧǧ ͚Ƹ Ǝ Ƴ͘ǳ̀ Ǝ$͚̀ Ʒƍ͜ $% $% T $% Η Θ % % $% S ͪ ͢ ƍ ƬƴT;͚Ǝ ǧ͚ ͦ ƍ͌Ǝ ̀ ͦƸƍ͚Ǝ͢ưʠ Ǝ͜Ǝ ʡư V T T Remark 6.11(see[44]) Consider the following system of equations ͛ ƔƎTʠ͌͗ Ǝ ͥ ƍǧǧ ͚ʡ / $% $% ͦ $% Η Θ ̻ ƔƎ͘ǳ̀ ƍ$͚̀ / $ $ $% ͚ƔƎ;͚Ǝ͌ƍͥ ̀ ͦ ƍ ) / ͦ ͦc ƔƎS / for a solution to this system we have

100

Chapter 6 [TWO FUNCTIONALS F,W OF PERELMAN]

͘ġ S ͦ S ǳ $ ͦ ͦ Ɣǹ ʨT ɴ͌͗$% Ǝ ͛$% ƍǧΗǧΘ͚ɴ ƍɳ͘ ̀% ƍ ͚̀$%ɳ ƍ ̀ ͨ͘ T V S ͯ) Ǝ ͦư ʚVʛ ͦ ͙ͯ! ͐͘ T

Definition6.15(see[44]) let ʚ͇Q ͛ʚͨʛQ ̻ʚͨʛʛbe a solution to RYM-flow which exists on a maximal time interval ͎ƗWT ʚ͇Q ͛ʚͨʛQ ̻ʚͨʛʛis a low - energy solution if ͦ *'+ʚ͎Ǝͨʛ ̀ t ƔR /Ŵ ʚħʛ So according to this Definition we get the following corollary. Corollary(see[44]) let ʚ͇Q ͛ʚͨʛQ ̻ʚͨʛʛ be a low energy solution to RYM flow on

ʞRQ ͎ʛthen there exists ͨͤ Ɨ͎ such that for all ͨͤ ƙͨƗ͎, we have ͘ġ ƚR ͨ͘ Definition 6.16Given ʚ͇Q͛Q̻ʛ at ͨǨ͌ let

ͯ) ʚ͛Q ̻Q ʛ Ɣ',$ ʪġʚ͛Q ̻Q ͚Q ʛ ǹ ʚVʛ ͦ ͙ͯ!͐͘ Ɣ Sʫ ! Corollary (see[44])let ʚ͇Q ͛ʚͨʛQ ̻ʚͨʛʛbe a low - energy solution to RYM flow ʞRQ ͎ʛT ͨ Ɨ͎ ͨ ƙͨƗ͎ [ ƚR on Then there exists ͤ such that for all ͤ we have / .

101

PERELMAN REDUCED VOLUME AND REDUCED LENGTH V{tÑàxÜJ

Chapter 7 [Perelman Reduced volume and reduced length]

Chapter 7

Perelman Reduced volume and reduced length

In this chapter we discuss Perelman’s notions of the ĖƎlength in the context of Ricci flows. This is a functional defined on paths in space –time parametrized by backward time, denoted .The main purpose of this chapter is to use the Li-Yau-Perelman distance to define the perelman’s reduced volume, which was introduced by Perelman, and prove the monotonicity property of the reduced volume under the Ricci flow. The reduced distance, i.e. the ͠Ǝfunction. The ͠Ǝfunction is defined in terms of a natural curve energy along the Ricci flow ,which is analogous to the classical curve energy employed in the study of geodesics ,but involves the evolving metric .as well as the scalar curvature as a potential term. There are two applications of this theory of Perelman .We use the theory of ĖƎgeodesics and the associated notion of reduced volume to establish non-collapsing results .The second application will be to Ǝnon- collapsed solutions of bounded non-negative curvature .A reader who wants to focus on the Poincare conjecture or the Geometrization conjecture could in principle start with Chapter 7.

Perelman reduced distance

102 Chapter 7 [Perelman Reduced volume and reduced length] Definition7.1: suppose that either ͇ is compact or ͛$%ʚʛare complete and have

uniformly bounded curvature .The ĖƎlength of a smooth space curve S ʞͥQͦʟŴ͇ is defined by

c Ėʚ ʛ Ɣ Ȅ v :Ƴ͌Ƴ ʚʛƷƍ ʖʚʛ ͦʛƷ͘ (7.1) cu

Ɣʚͨʛ c ƔƎS ͌ʚ ʚʛʛ Where satisfying / and the scalar curvature and the norm

ʖʚʛ are evaluated using the metric at time ͨƔͨͤ Ǝ.

We consider an 1-parameter family of curves .SʞͥQͦʟŴ͇ parameterized by

ͧǨʚƎ Q ʛ .Equivalently, We have a map ȭʚͧQ ʛ with ͧǨʚƎ Q ʛ and ǨʞͥQͦʟ ͒ƔiRȭ ͓ƔiRȭ ʞ͒Q ͓ʟ ƔR ͓Ǝ ͒Ǝʞ͒Q ͓ʟ Ɣ .Putting ic and i. ,We have .(Because

R).This implies that ͓Ɣ͒ .

Writing as shorthand for ʺ ,and restricting to the curve ʚʛ Ɣ ȭʚRQ ʛ .we have . .Ͱͤ

ʚ ͓ʛʚʛ Ɣ ͓ʚʛ and ʚ ͒ʛʚʛ Ɣ ʚ͓ʛʚʛ .Set ͧƔ: one sees immediately with respect to the variable ͧ .

The ĖƎ functional is

ͦ :c ͥ R Ėʚ ʛ Ɣ Ȅ v ʦ ʺ ʺ ƍTͧͦ͌Ƴ ʚͧʛƷʧ ͧ͘ (7.2) :cu ͦ .

Remark7.2(Ė on Riemannian Products) suppose that we are given a Riemannian

)u )v product solution ʠ͈ͥ Ɛ͈ͦ Qͥ͜ʚʛ ƍͦ͜ʚʛʡto the backward Ricci flow and a ̽ͥ Ǝ Ɣʚ Q ʛSʞ Q ʟŴ͈ Ɛ͈ Ė ʚ ʛ ƔĖ ʚ ʛ ƍĖ ʚ ʛ Path ͥ ͦ ͥ ͦ so #uͮ#v #u #v .

Example suppose that Ricci flow is a constant family of Euclidean metrics on ͌) Ɛ ʞRQ ͎ʟ. Then we have ͌Ƴ ʚʛƷƔR .So according to (7.2) we get

S c ͘ ͦ Ėʚ ʛ Ɣ ǹ : ɴ ɴ ͘ T ͤ ͘

R ͥ R Ɣ : Ɣ ͘ Ɣ Tͧͧ͘ If we change , c ͦ:c . ,

103 Chapter 7 [Perelman Reduced volume and reduced length] v v S . S ͘ ͦ S . ͘ ͦ Ėʚ ʛ Ɣ ǹͧ ɴ ɴ Tͧͧ͘ Ɣ ǹ ɴ ɴ ͘ ͧ ͦ T ͤ Vͧ ͧ͘ T ͤ ͧ͘

So we arrive to (7.2) formula.

First variation formulae for ņ Ǝgeodesics

At first recall the classical variation formulae that give the first derivative of the metric

function ͘ʚͬͤQͬʛon a Riemannian manifoldʚ͇Q ͛ʛ.

ͥ Let S ʞRQSʟ Ŵ ͇ ranges over all ̽ ƎCurves from ͬͤ to ͬ ,and the Dirichlet energy ̿ʚ ʛof the curve is given by the formula

S ͥ " ͦ ̿ʚ ʛ Ɣ ǹ ͒ "͘ ͙͙ͨͫͦ͒͜ Ɣ T ͤ "ͨ

We know that the distance ͘ʚͬͤQͬʛ on a Riemannian manifold ʚ͇Q ͛ʛ can be defined by the energy-minimisation formula

ͥ ͘ʚͬ Qͬʛͦ Ɣ',$ ̿ʚ ʛ ͦ ͤ R (7.3)

Also if S ʞRQ ͨʟŴ͇, ʚRʛ Ɣͤ, , ƳͨƷƔͥ

ͦ͘ʚͤQ ͥʛ ͚͢͝ƣ̿ʚ ʛ S ʞRQ ͨʟŴ͇Q ʚRʛ ƔͤQQ ƳͨƷƔͥƧƔ ͨ

Lemma 7.1The first variation formula for Dirichlet energy ̿ʚ ʛis

ͥ ̿ʚ ʛ Ɣ ˛͒Q ͓˜ ͥ Ǝ Ȅ ˛͒Q ǧ ˜ͨ͘ . /Ͱͤ ͤ ΍ (7.4)

iR iR ̾ ͒ ̾ ͒Ɣǧê ͒ ͒Ɣ ʖ Ɣ Proof: consider / means / and i. , i/ . ê²

S ͥ ̿ʚ ʛ Ɣ ǹ ˛ ʖQ ʖ˜ͨ͘ T ͤ

104 Chapter 7 [Perelman Reduced volume and reduced length] ͥ i ͥ iR iR Ɣ ʺ Ȅ ˛ Q ˜ "2 ͦ i. .Ͱͤ ͤ i/ i/

ͥ Ɣ ͥ Ȅ ˛̾ ʠiRʡ Q iR˜ ƍ ˛iR Q ̾ ʠiRʡ˜ "2 ͦ ͤ . i/ i/ i/ . i/

ͥ " " Ɣǹ ˛̾. ƴ ƸQ ˜ "2 ͤ "ͨ "ͨ

ͥ Ɣ Ȅ ˛̾ ͒Q ʖ˜"2 ͤ /

With integration by parts we get

ͥ ͥ Ɣ ˛͒Q ʖ˜ ͤ Ǝǹ ˛͒Q ̾/ ʖ˜"2 ͤ

ͥ ͥ Ɣ ˛͒Q͓˜ ͤ Ǝǹ ˛͒Q ǧ΍˜ ͤ

Remark7.3 if we fix the end points of ,then the first term of the right hand side of (7.4) vanishes .If we consider arbitrary variations ͒ of with fixed end points,we thus conclude that in order to be a minimiser for (7.3),that must obey the geodesic flow equation

͓ƔR

Now we develop analogous variational formulae for ĖƎlength (reduced distance)on Ricci flow

Theorem 7.1(Perelman [39]) the first variation formula for ĖƎlength is

cv cv ͥ ʚĖʛ ƔT:˛͒Q ͓˜ɳ ƍ Ȅ : ˛͓Q ǧ͌ Ǝ TǧΌ͒ƎV͌͗͝ʚTQ͒ʛ Ǝ ͒˜ ͘ (7.5) cu cu c

Where ˛TQT˜ denotes the inner product with respect to the metric͛$%ʚʛ.

Proof let ͒ʚʛ Ɣ ʖʚʛ,so according to (7.1) we obtain

cv ʚĖʛ Ɣǹ :ʚǧ΍͌ƍT˛ǧΌ͓Q ͒˜ʛ͘ cu

105 Chapter 7 [Perelman Reduced volume and reduced length] cv Ɣǹ :ʚ˛ǧ͌Q ͓˜ ƍT˛ǧΌ͓Q ͒˜ʛ͘ cu

" ic ƔƎS Ĝĝ ƔT͌ Because i/ so c $% ,using this fact we get

˛Q˜ Ɣ ˛ ͓Q ͒˜ ƍ ˛͓Q ͒˜ ƍT͌͗͝ʚ͓Q͒ʛ c (7.6)

˛Q˜ Because we can break c into two parts :first assumes that the metric is constant and the second deals with the variation with of the metric .The first contribution is the usual formula

͘˛͓Q ͒˜ Ɣ ˛ ͓Q ͒˜ ƍ ˛͓Q ͒˜ ͘

We show last term in equation (7.6) is that come from differentiating the metric with respect to and Ricci flow equation we get

͘˛͓Q ͒˜ ƔT͌͗͝ʚ͓Q͒ʛ ͘

So we arrive to equality of (7.6)

We continue the proof of (7.5)

cv ǹ :ʚ˛͓Q ǧ͌˜ ƍT˛ǧΌ͓Q ͒˜ʛ͘ Ɣ cu

cv ͘˛͓Q ͒˜ Ɣ ǹ : ʦ˛͓Q ǧ͌˜ ƍT ƎT˛͓Q ǧ ͒˜ ƎV͌͗͝ʚ͓Q͒ʛʧ ͘ ͘ Ό cu

cv cv S Ɣ T:˛͒Q ͓˜ɳ ƍǹ : ˛͓Q ǧ͌ Ǝ TǧΌ͒ƎV͌͗͝ʚ͒Q T ʛ Ǝ ͒˜ ͘ cu cu

Lemma7.2 (Perelman) we consider a variation ʚQ ͧʛwith fixed endpoints, so that

͓ʚͥʛ Ɣ͓ʚͦʛ ƔR.so from (7.5) the ĖƎshortest geodesic ʚQ ͧʛ for Ǩ ʞͥQͦʟ satisfies the following ĖƎ geodesic equation

106 Chapter 7 [Perelman Reduced volume and reduced length] ǧ ͒Ǝͥ ǧ͌ ƍ ͥ ͒ƍT͌͗͝ʚ͒Q T ʛ ƔR Ό ͦ ͦc (7.7)

This equation is called Euler-Lagrange Equation.

Therefore we can say for any ͦ Ƙͥ ƘR ,there is always an ĖƎgeodesic ʚʛ for

ǨʞͥQͦʟ .

Written with respect to the variable ͧƔ: the ĖƎgeodesic equation becomes

Ȩ ͦ Ȩ ǧȨ ͒ ƎTͧ ǧ͌ ƍ Vͧ͌͗͝Ƴ͒QTƷƔR

͒Ȩ Ɣ R ƔTͧ͒ Where .

Lemma7.3(Perelman) let S ʞͥQͦʟŴ͇be anĖ Ǝgeodesic. Then *'+cŴͤ :͒ʚʛexists.

Proof: multiplying the ĖƎgeodesic equation (7.7) by: , we get

c ǧ Ƴ:͒Ʒ Ɣ : ǧ͌ Ǝ T:͌͗͝ʚ͒Q T ʛ ʞ Q ʟ Ό ͦ On ͥ ͦ

or equivalently

c Ƴ:͒Ʒ Ɣ : ͌ Ǝ T͌͗͝Ƴ:͒QTƷ ʞ Q ʟ c ͦ On ͥ ͦ

Thus if a continuous curve ,defined on ʞͥQͦʟ ,satisfies the ĖƎgeodesic equation on ͮ every subinterval RƗͥ ƙƙͦ ,then :ͥ͒ʚͥʛhas a limit as ͥ ŴR .

Remark7.4 for a fixed ͤ Ǩ ͇,by taking ͥ ƔR and ʚRʛ Ɣͤ ,the vector

ͪƔ*'+cŴͤ :͒ʚʛis well-defined in ͎͇ .TheĖ Ǝ exponential map Ė͙ͬͤcS͎͇Ŵ͇ sends ͪ to ʚʛ.

Note that for any vector ͪǨ͎͇ ,we can find an ĖƎgeodesic ʚʛwith

*'+cŴͤ~ : ʖʚʛ Ɣͪ.

Now we give an Estimate for speed of ĖƎgeodesics

Theorem(see[38])7.2: letʚ͇)Q͛ʚʛʛ, Ǩ ʞRQ ͎ʟ,be a solution to the backward Ricci flow

with bounded sectional curvature and ͕ͬ͡ʜ ͌͡ Q ͌͗ ʝ ƙ̽ͤ Ɨ on ͇ƐʞRQ͎ʟ.There

107 Chapter 7 [Perelman Reduced volume and reduced length] exists a constant ̽ʚ͢ʛ Ɨ depending only on ͢ such that given Rƙͥ ƙͦ Ɨ͎,if

S ʞͥQͦʟŴ͇is an ĖƎgeodesic with

͘ *'+ ʚʛƔͪǨ͎ ͇ : Rʚcuʛ cŴcu ͘

Then for any ǨʞͥQͦʟ ,

͘ ͦ ̽ʚ͢ʛ͎ ͪt ͦ ͪt ɴ ʚʛɴ ƙ͙ ͪ ƍ ͯͥ ʚ͙ ƎSʛ ͘ "ʚcʛ ͢͡͝ƣ͎ Ǝ ͦQ̽ͤ Ƨ

ͪ ͦ Ɣ ͪ ͦ Where "ʚcuʛ .

Also we give a lemma about existence of ĖƎgeodesics between any two space-time endpoints .

Lemma7.4(see[39]) letʚ͇)Q͛ʚʛʛ, Ǩ ʞRQ ͎ʟ, be a complete solution to the backward

Ricci flow with bounded sectional curvature .Given ͤQ ͥ Ǩ͇ and RƗͥ ƙͦ Ɨ

͎,There exists a smooth path ʚʛS ʞͥQͦʟ Ŵ͇ from ͤ to ͥ such that has the minimal of ĖƎlength among all such paths .Furthermore ,all ĖƎlength minimizing paths are smooth ĖƎgeodesics.

Corollary7.1 (see[43]):if we extend the curve for piecewise smooth curves

(whereR ƙ ͥ Ɨͦ ƙͨͤ )then for first variation formula of breaking points ͤ Ɣͥ Ɨ ͦ ƗˆƗ& Ɣͦ ,we have

cv ͥ cv ƳĖʚ ʛƷƔȄ : ˛ǧ͌ Ǝ TǧΌ͒ƎV͌͗͝ʚ͒ʛ Ǝ ͒Q ͓˜ ͘ ƍ T:˛͒Q ͓˜cɳ ƍ cu c cu ?&ͯͥ Tǭ ˛͒ͯʚ ʛ Ǝ͒ͮʚ ʛQ͓ʚ ʛ˜ $Ͱͦ $ $ $ $ cĜ

Definition7.2: Fixing a point ͤ ,we denote by ͆ʚͥQʛ the Ė Ǝlength of the ĖƎshortest curve ʚʛ ,Rƙƙ ,joining ͤ and ͥ .In other words

͆ʚͥQʛ Ɣ͚͢͝ʜĖʚ ʛ S ʞRQ ʟ Ŵ ͇ͫͨ͜͝ ʚRʛ Ɣ ͤQ ʚʛ Ɣ ͥʝ

Theorem7.3 (Perelman[39]) suppose that

108 Chapter 7 [Perelman Reduced volume and reduced length] "͌ ͌ ͂ʚ͒ʛ ƔƎ Ǝ ƎT˛ǧ͌Q ͒˜ ƍT͌͗͝ʚ͒Q ͒ʛ "

And

c ͧ ͅƔǹ ͦ ͂ʚ͒ʛ͘ ͤ

Then

ǧ͆ ͦ ƔƎV͌ƍ ͦ ͆Ǝ ͨ ͅ a) :c :c i ƔT:͌Ǝ ͥ ͆ƍͥ ͅ b) ic ͦc c

Proof: The first variation formula in (7.5) implies that

ǧ΍͆ʚͥQʛ Ɣ ˛Tǭ͒ʚʛQ͓ʚʛ˜

So ǧ͆ʚͥQʛ ƔT:͒ʚʛ

At first we prove

i ʚ ʚʛQʛ Ɣ ͆ʚ ʚʛQʛʺ Ǝ ˛ǧ͆Q ͒˜ (7.8) ic c cͰc

ʚʛ ƔƎi ƍ͒ʚʛ Since i/ ,So the chain rule implies

͘ ͘ ͆ʚ ʚʛQʛɴ Ɣ ͆ʚ ʚʛQʛ ƍ ˛ǧ͆Q ͒˜ ͘ cͰc ͘

Therefore we get to (7.8)

Also from (7.1)and (7.8)we obtain

"͆ ʚ ʚʛQʛ Ɣ ǭʚ͌ƍ ͒ ͦʛ Ǝ ˛ǧ͆Q ͒˜ "

And because ǧ͆ʚͥQʛ ƔT:͒ʚʛ so

109 Chapter 7 [Perelman Reduced volume and reduced length] "͆ ʚ ʚʛQʛ Ɣ ǭʚ͌ƍ ͒ ͦʛ Ǝ ǭ ͒ ͦ "

ƔT:͌Ǝ:ʚ͌ƍ ͒ ͦʛ (7.9)

Now we compute the ƍ ͒ ͦ .

ʚʛ Ɣ i ƍ͒ʚʛ According to (7.5) and this fact that ic ,we get

͘ "͌ ʚ͌ʚ ʚʛQʛ ƍ ͒ʚʛ ͦʛ Ɣ ƍ ˛ǧ͌Q ͒˜ ƍ ͒ʚʛ ͦ ͘ "

Also we know ͒ʚʛ ͦ Ɣ ˛͒ʚʛQ͒ʚʛ˜

͒ʚʛ ͦ Ɣ ˛͒ʚʛQ͒ʚʛ˜ ƔT˛ǧ ͒Q ͒˜ ƍT͌͗͝ʚ͒Q͒ʛ So from c c Ό

ʚ͌ʚ ʚʛQʛ ƍ ͒ʚʛ ͦʛ Ɣ i ƍ ˛ǧ͌Q ͒˜ ƍT˛ǧ ͒Q ͒˜ ƍT͌͗͝ʚ͒Q͒ʛ We get c i/ Ό

ǧ ͒Ɣͥ ǧ͌ Ǝ ͥ ͒ƎT͌͗͝ʚ͒Q T ʛ Also we have Ό ͦ ͦc so

͘ "͌ S S ʚ͌ʚ ʚʛQʛ ƍ ͒ʚʛ ͦʛ Ɣ ƍ ͌ƍT˛ǧ͌Q ͒˜ ƎT͌͗͝ʚ͒Q ͒ʛ Ǝ ʚ͌ƍ ͒ ͦʛ ͘ "ͨ

S Ɣ Ǝ͂ʚ͒ʛ Ǝ ʚ͌ƍ ͒ ͦʛ

w w ͦ ͥ ͦ ʦvʚ͌ƍ ͒ ʛʧ˄ Ɣ :ʚ͌ƍ ͒ ʛ Ǝv͂ʚ͒ʛ So c ͦ cͰc

S ͘ ͧ Ɣ ͆ʚ ʚʛQʛɴ Ǝͦ͂ʚ͒ʛ T ͘ cͰc

w ͥ vʚ͌ƍ ͒ ͦʛ Ɣ ͆ʚͥQʛ Ǝͅ So ͦ where

c ͧ ͅƔǹ ͦ ͂ʚ͒ʛ͘ ͤ

Therefore from This fact that ǧ͆ ͦ ƔV ͒ ͦ ƔƎV͌ƍVʚ͌ ƍ ͒ ͦʛ and () we conclude ͕and ͖ .So proof is complete.

110 Chapter 7 [Perelman Reduced volume and reduced length] Second variation formulae for ņ Ǝgeodesics

Proposition 7.1if we compute the second variation of energy when is a geodesic, we get

ͦ͘ c ̿ʚ ʛ Ɣ ˛ ͓Q ˜ c ƍǹ ʚ˛ǧ ǧ Q ˜ ƍ ˛ǧ ͒Q ǧ ͒˜ʛͨ͘ ͦ ͤ ΍ ΍ ΍ ΍ ͧ͘ ͤ

Now turn to the second spatial variation of the reduced length.

Theorem 7.4(Perelman[39])for any ĖƎgeodesic ,we have

ͦ ʚĖʛ Ɣ c T ˛ ͓Q ͒˜ c ƍ Ȅ ʞT ͓ ͦ ƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜ ƍ ͌ƍT ͌͗͝ʚ͓Q ͓ʛ Ǝ : ͤ ͤ :

V͌͗͝ʚ͓Q ͒ʛʟ͘ .

ͦ Proof to compute the second variation ʚĖʛ.we start the first variation formula .

We know

c ʚĖʛ Ɣǹ:ʚǧ΍͌ƍT˛ǧΌ͓Q ͒˜ʛ͘ ͤ

c ͦʚĖʛ Ɣ͓ʠȄ ʚǧ ͌ƍT˛ǧ ͓Q ͒˜ʛ͘ʡ So ͤ : ΍ Ό

c ǧ ͌Ɣ͓͌ ͦʚĖʛ Ɣ͓ʠȄ ʚ͓͌ ƍ T˛ǧ ͓Q ͒˜ʛ͘ʡ We know ΍ ,so ͤ : Ό

c ͦ Ɣ ʦǹ :Ƴ͓Ƴ͓ʚ͌ʛƷƍT˛͓Q ͒˜ ƍT ǧΌ͓ Ʒ͘ʧ ͤ

Because ǧΌ͓Ɣǧ΍͒ we get

c Ɣ ʠȄ Ƴ͓Ƴ͓ʚ͌ʛƷƍT˛ ͓Q ͒˜ ƍT ǧ ͓ ͦƷ͘ʡ ͤ : Ό

Also we know ͔Ǝ͔ƎʞQʟ͔Ɣ͌ʚ͒Q ͓ʛ͔ for all vectors ͒Q ͓Q ͔.so

T˛͓Q ͒˜ ƔT˛͓Q ͒˜ ƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜

111 Chapter 7 [Perelman Reduced volume and reduced length] For continuing proof, we prove the following equality

i ˛ ͓Q ͒˜ Ɣ ˛ ͓Q ͒˜ ƍ ˛ ͓Q ͒˜ ƍT͌͗͝ʚ ͓Q ͒ʛ ƍ ˛ i ͓Q ͒˜ ic ic (7.10)

i ˛ ͓Q ͒˜ We can break ic into two parts: the first assumes that the metric is constant and the second deals with the variation with of the metric .The first contribution is the usual formula

" ˛ ͓Q ͒˜ Ɣ ˛ ͓Q ͒˜ ƍ ˛ ͓Q ͒˜ "

This gives the first two terms of the right-hand side of the equation .we show that the last two terms in that equation come from differentiating the metric with respect to .To

do this recall that in local coordinates ,writing the metric as ͛$% ,we have

& $ $ & ' % ˛͓Q ͒˜ Ɣ͛$%Ƴ͓ "&͓ ƍϙ&'͓ ͓ Ʒ͒

There are two contributions coming from differentiating the metric with respect to

.The first is when we differentiate ͛$% .This leads to

& $ $ & ' % T͌͗͝$%Ƴ͓ "&͓ ƍϙ&'͓ ͓ Ʒ͒ ƔT͌͗͝ʚ͓Q ͒ʛ

The other contribution is from differentiating the Christoffel symbols .This yields

"ϙ$ ͛ &' ͓&͓'͒% $% "

ϙ$ Ɣ ͥ ͛.$ʚ" ͛ ƍ"͛ Ǝ"͛ ʛ Differentiating the formula &' ͦ & .' ' .& . &' leads to

"ϙ$ ͛ &' ƔƎT͌͗͝ ϙ$ ƍ͛ ͛.$ʚ" ͌͗͝ ƍ"͌͗͝ Ǝ"͌͗͝ ʛ $% " $% &' $% & .' ' .& . &'

$ ƔƎT͌͗͝$%ϙ&' ƍ"&͌͗͝%' ƍ"'͌͗͝%& Ǝ"%͌͗͝&'

Thus ,we have

112 Chapter 7 [Perelman Reduced volume and reduced length] "ϙ$ ͛ &' ͓&͓'͒% ƔƳƎT͌͗͝ ϙ$ ƍ" ͌͗͝ Ʒ͓&͓'͒% $% " $% &' & %'

Ɣ͌͗͝ʚ͓Q ͓ʛ

So we obtain

͘ " T˛ ͓Q ͒˜ ƔT ˛ǧ ͓Q ͒˜ ƎV͌͗͝ʚ ͓Q ͒ʛ ƎT˛ǧ ͓Q ǧ ͒˜ ƎT˛ ǧ ͓Q ͒˜ ͘ ΍ ΍ Ό " ΍

ƍT˛͓Q ͒˜ ƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜

Also we can compute

˛ i ͓Q ͒˜ ƔTʚ ͌͗͝ʛʚ͓Q ͒ʛ Ǝ ʚ ͌͗͝ʛʚ͓Q ͓ʛ ic (7.11)

Hence from (7.10) and (7.11)we get

i ˛ ͓Q ͒˜ Ɣ ˛ ͓Q ͒˜ ƍ ˛ ͓Q ͒˜ ƍT͌͗͝ʚ ͓Q ͒ʛ ƍTʚ ͌͗͝ʛʚ͓Q ͒ʛ Ǝ ic

ʚ͌͗͝ʛʚ͓Q ͓ʛ (7.12)

Suppose ͓ʚRʛ ƔR and the fact that :͒ʚʛhas a limit as ŴR are used to get the third equality below .So applying (7.12) and integrating by parts, we arrive

c ͦ ͦ ƳĖʚ ʛƷƔʦǹ :Ƴ͓Ƴ͓ʚ͌ʛƷƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜ ƍT ǧ΍͒ Ʒ͘ʧ ͤ c " ƍTǹ : Ƶ ˛͓Q ͒˜ Ǝ ˛͓Q ͒˜ ƎT͌͗͝ʚǧ΍͓Q ͒ʛ ͤ "

ƎTʚ͌͗͝ʛʚ͓Q ͒ʛ ƍ ʚ͌͗͝ʛʚ͓Q ͓ʛʧ ͘

c Ɣ ʠȄ Ƴ͓Ƴ͓ʚ͌ʛƷƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜ ƍT ǧ ͒ ͦƷ͘ʡ ƍ ͤ : ΍ c T Ȅ ƳƎ˛ ͓Q ͒˜ ƎT͌͗͝ʚǧ ͓Q ͒ʛ Ǝ ͤ : ΍ c Tʚ ͌͗͝ʛʚ͓Q ͒ʛ ƍ ʚ ͌͗͝ʛʚ͓Q ͓ʛƷ͘ ƍ T:˛ ͓Q ͒˜ɳ Ǝ ͤ c ͥ c Ȅ ˛͓Q ͒˜͘ Ɣ T:˛͓Q ͒˜ ƍ ʠȄ :Ƴ͓Ƴ͓ʚ͌ʛƷƎ͓T ͌ ƍ T˛͌ʚ͓Q ͒ʛ͓Q ͒˜ ƍ ͤ :c ͤ

113 Chapter 7 [Perelman Reduced volume and reduced length] ͦ T ǧ΍͒ Ʒʡ ͘ ƍ

c ͥ ͥ T Ȅ : ƴƎ ˛ǧ ͓Q ʢǧ ͒ƍT͌͗͝ʚ͒ʛ Ǝ ǧ͌ ƍ ͒ʣ˜ ƎTʚ ͌͗͝ʛʚ͓Q ͒ʛ ƍ ͤ ΍ Ό ͦ ͦc

ʚ͌͗͝ʛʚ͓Q ͓ʛƸ͘

c ͦ ͦ ƔTǭ˛͓Q ͒˜ ƍ ʦǹ :Ƴǧ΍Q΍͌ƍT˛͌ʚ͓Q ͒ʛ͓Q ͒˜ ƍT ǧ΍͒ Ʒ͘ʧ ͤ c ƍǹ :ƳƎVʚǧ΍͌͗͝ʛʚ͓Q ͒ʛ ƍTʚǧΌ͌͗͝ʛʚ͓Q ͓ʛƷ͘ ͤ

ǧ ͒Ǝͥ ǧ͌ ƍ ͥ ͒ƍT͌͗͝ʚ͒Q T ʛ ƔR ͙͂ͧͧʚ͚ʛʚ͒Q ͓ʛ Ɣ Because we know Ό ͦ ͦc and ʚ͚ʛ Ǝ ʚ͚ʛ Ođ ,so proof is complete .

ņ and Riemannian distance:

Theorem7.5 (see[43]) let S ʞRQ ʟŴ͇ , ǨʚRQ͎ʟ,be a ̽ͥ Ǝpath starting at ͤ and ending at q.

i) (bounding Riemannian distance byĖ )for any ǨʞRQʟ we have

ͧ T̽ͤ͢ ͦ͘ ʚͤQ ʚʛʛ ƙT:͙ͦtc ƴĖʚ ʛ ƍ ͦƸ "ʚͤʛ U

Where Ƴ͇)Q͛ʚʛƷQ Ǩ ʞRQ ͎ʟ , denote a complete solution to the backward Ricci flow, and ͤǨ͇ shall be a base point .also we assume the curvature bound

+ 6 ʜ ͌͡ʚͬQ ʛ Q ͌͗ʚͬQ ʛ ʝ ƙ̽ͤ Ɨ7 ʚ3Q/ʛǨƐʞͤQʟ

Remark for i) when M is noncompact .from i) we conclude for any ǨʚRQ͎ʟ .

*'+ ͆ʚͥQʛSƔ *'+Tǭ͆ʚͥQʛ Ɣ7 ,ŴϦ ,ŴϦ

ii) (bounding speed at some time byĖ ) there exists ͨǳ ǨʚRQʛ such that

114 Chapter 7 [Perelman Reduced volume and reduced length] ͦ ͦ ʺiR ʚ ʛʺ Ɣ ʺiQ ʚ ʛʺ ƙ ͥ Ėʚ ʛ ƍ )t ǳ ic ǳ ia ǳ ͦ:c ͧ "ʚcǳʛ "ʚcǳʛ

Where ʚʛ Ȭ ʚʛ , Ɣ T: ,and ǳ ƔT:ǳ . iii) (bounding L by Riemannian distance) for any ͥ Ǩ͇ and ƘR

ͦ ͧ ͘"ʚcʛʚͤQ ͥʛ T̽ͤ͢ ͆ʚͥQʛ ƙ͙ͦtc ƍ ͦ T: U

For proof of this theorem, we start with a remark

Remark7.5: for ͥ Ɨͦ and ͬǨ͇

ͯͦtʚcvͯcuʛ ͦtʚcvͯcuʛ ͙ ͛ʚͦQͬʛ ƙ͛ʚͥQͬʛ ƙ͙ ͛ʚͦQͬʛ

Proof i) suppose ƔT: and ʚʛ Ȭ ʚʛ .At first we compute

ͦ:c " ͦ :c " ͦ c ǹ ɴ ʚʛɴ ͘ ƔĖʚ ʛ Ǝǹ ɴ ʚʛɴ ͘ Ǝǹ ǭ͌ʚ ʚʛQʛ ͘ " " av ͤ ͦ:c "ʦ ͨ ʧ ͤ

ͧ T̽ͤ͢ ƙĖʚ ʛ ƍ ͦ U

ͦtc Because ͌ƚƎ̽ͤ͢.Hence, since ͛ʚRʛ ƙ͙ ͛ʚʛ for ǨʞRQʟ we get

ͦ ͦ:c " ͦ͘ ƳͤQ ʚʛƷƙ͙ͦtc ʬǹ ɴ ʚʛɴ ͘ʭ "ʚͤʛ " av ͤ "ʦ ͨ ʧ

ͦ:c " ͦ T̽͢ ͧ ͦ c ͦ c ͤ ƙ͙ t T: ǹ ɴ ʚʛɴ v ͘ ƙT:͙ t ƴĖʚ ʛ ƍ ͦƸ " a U ͤ "ʦ ͨ ʧ

ii) by taking Ɣ in proof of i) we get

S ͦ:c " ͦ S ̽͢ ǹ ɴ ʚʛɴ ͘ ƙ Ėʚ ʛ ƍ ͤ av T: " "ʦ ʧ T: U ͤ ͨ

Now with using the mean value Theorem for integrals, There exists ͨǳ ǨʚRQʛsuch that

115 Chapter 7 [Perelman Reduced volume and reduced length] " ͦ S ̽͢ ɴ ʚ ʛɴ ƙ Ėʚ ʛ ƍ ͤ " ǳ U "ʚcǳʛ T:

iii) Let S ʞRQT:ʟŴ͇ be a minimal geodesic from ͤ to ͥ with respect to metric ͛ʚʛ.Then because on͛ʚʛ we have

T:͆ʚ ʚʛQʛ ƔT:ĖƳ ƷƔ ͦ͘ ʚͤQ ͥʛ ʞͤQcʟ "ʚcʛ

ͦ:c ͦ ͦ ͘ ͦ ͆ʚͥQʛ ƙĖʚʛ Ɣǹ ʦ ͌ ʦʚʛQ ʧ ƍ ɴ ɴ ʧ ͘ ͤ V V ͘ "ʚcʛ

ͦ:c ͦ ͦ ͧ ͦtc ̽ͤ͢ ͘ T̽ͤ͢ ͙ ͦtc ͦ ͦ ƙǹ ʦ ƍ͙ ɴ ɴ ʧ ͘ ƙ ƍ ͘"ʚcʛʚͤQ ͥʛ ͤ V ͘ "ʚcʛ U T:

So proof is complete.

theĖ-Jacobi equation :consider a family ʚQ ͩʛ of Ė-geodesics parameterized by ͧand

defined on ʞͥQͦʟwith Rƙͥ ƙͦ .Let ͓ʚʛ be a vector field along defined by

" ͓ʚʛ Ɣ ʚQ ͧʛɴ "ͧ .Ͱͤ

Now from second variation Theorem we obtain ͓ʚʛsatisfies the ĖƎJacobi equation

͓ƍ͌ʚ͓Q ͒ʛ͒Ǝͥ ʚ͌ʛ ƍ ͥ ͓ƍTʚ ͌͗͝ʛʚ͒Q T ʛ ƍT͌͗͝ʚ ͓Q T ʛ ƔR ͦ ͦc (7.13)

This is a second –order linear equation for ͓ .supposing that ͥ ƘR ,there is a unique

vector field along solving this equation ,vanishing at ͥ with a given first-order

derivative along at ͥ.similarly ,there is a unique solution to this equation

,vanishing at ͦ and with a given first order derivative at ͦ .

Definition 7.3 a field ͓ʚʛalong an ĖƎgeodesic is called an ĖƎjacobi field if it

satisfies the ĖƎjacobi equation ,equation (7.13) ,and if it vanishes at ͥ.

116 Chapter 7 [Perelman Reduced volume and reduced length] Notation: for every vector field along we denote by ͕̈́͗ʚ͓ʛthe expression on the left-hand side of equation(7.13) .

In one of remarks we considered that the vector *'+cŴͤ :͒ʚʛexists .Now we give a