Poincaré and Geometrization Conjectures: mathematical connections between String Theory, Ricci Flow and Number Theory.

Poincaré and Geometrization Conjectures: mathematical connections between String Theory, Ricci Flow and Number Theory

Michele Nardelli1,2

1Dipartimento di Scienze della Terra – Università degli Studi di Napoli “Federico II” Largo S. Marcellino, 10, 80138 Napoli (Italy)

2Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”

Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, via Cintia (Fuorigrotta), 80126 Napoli, Italy

Riassunto

La "congettura di Poincaré" afferma che: “Ogni 3-varietà semplicemente connessa chiusa (ossia compatta e senza bordi) ed orientabile è omeomorfa a una sfera tridimensionale”. (In topologia, un omeomorfismo è una corrispondenza biunivoca fra due spazi topologici che ne preserva la topologia. In altre parole, è una funzione tra due spazi topologici con la proprietà di essere continua, invertibile e di avere l’inversa continua. Due spazi omeomorfi godono delle stesse proprietà topologiche (separabilità, connessione, compattezza…). Informalmente, due spazi sono omeomorfi se possono essere deformati l’uno nell’altro senza “strappi”, “sovrapposizioni” o “incollature”). Detto con termini diversi, la congettura dice che la sfera è l'unica varietà tridimensionale "senza buchi", cioè dove qualsiasi cammino chiuso può essere contratto fino a diventare un punto. Il padre dell'enunciato è il matematico francese Henri Poincaré. La Congettura che prende il suo nome nacque nel 1904, mentre lo studioso stava lavorando ai fondamenti di quella che poi sarà chiamata topologia algebrica. L'enunciato di Poincaré tenta di dimostrare che la sfera è il più semplice campo in cui un qualsiasi cammino chiuso possa essere contratto fino a diventare un punto (ogni varietà chiusa n dimensionale omotopicamente equivalente alla n-sfera è omeomorfa alla n-sfera. In topologia, due funzioni continue da uno spazio topologico ad un altro sono dette “omotope”, se una delle due può essere “deformata con continuità” nell’altra e tale trasformazione è detta “omotopia” fra le due funzioni).

Dato che la sfera è la più semplice delle superfici bidimensionali, Poincaré, nel 1904, ipotizzò che così fosse anche in dimensione superiore, congetturando che la tri-sfera è l'unica superficie tridimensionale chiusa (e orientabile, per essere precisi) sulla quale tutte le curve chiuse sono deformabili l'una nell'altra. Una formulazione della congettura di Poincaré a n dimensioni è la seguente: Ogni varietà chiusa n dimensionale omotopicamente equivalente alla n-sfera è omeomorfa alla n-sfera. Questa definizione è equivalente alla congettura di Poincaré nel caso n=3. Le difficoltà maggiori sorgono per le dimensioni n = 3 e n = 4. I matematici cinesi Zhu Xiping e Cao Huaidong, seguendo la strada indicata dal matematico russo G. Perelman, hanno trovato la soluzione di questo grande enigma delle scienze esatte. La congettura di Poincaré avrebbe ripercussioni sulle possibili topologie della teoria delle stringhe e delle varie altre teorie della gravitazione quantistica. Alcune teorie della fisica moderna si formalizzano, infatti, mediante strutture geometriche aventi un numero di dimensioni maggiore di tre: la relatività einsteiniana prevede uno spazio-tempo quadridimensionale, mentre la più recente teoria delle stringhe ipotizza l'esistenza di dieci dimensioni fisiche, sei delle quali «compattificate» in iperspazi minuscoli con una geometria e una topologia molto intricate.

Nella presente tesi vengono evidenziate le connessioni matematiche ottenute tra la Congettura di Poincarè, la Congettura di Geometrizzazione di Thurston, la Teoria di Stringa ed alcuni settori della Teoria dei Numeri.

Vengono prima descritti alcuni fondamentali risultati matematici inerenti la Congettura di Poincarè, ottenuti dal matematico russo G. Perelman e sviluppati dai matematici cinesi Huai-Dong Cao e Xi-Ping Zhu, le cui dimostrazioni possono essere considerate come le conseguenze finali della teoria del flusso di Ricci di Hamilton-Perelman. A questi concetti seguiranno quelli riguardanti la Teoria di Stringa, precisamente, la 3D stringy-gravity per comprendere la congettura di Thurston, i buchi neri tri-dimensionali in teoria di stringa, ed infine l’azione effettiva di una D2-brana frazionaria e quella al contorno di una D3-brana frazionaria

Concluderemo, inoltre, evidenziando le correlazioni ottenute tra alcune equazioni inerenti le Congetture di Poincarè e di Thurston e (i) i settori prima menzionati inerenti la Teoria di Stringa, (ii) alcune formule che riguardano la Teoria dei Numeri, precisamente,

π

e la Costante di Legendre “c”, ed, infine, (iii) il modello di Palumbo applicato alla Teoria di Stringa.

1. Poincarè and Geometrization conjectures. [1]

A Riemannian metric g_ij is called Einstein if R_ij =λg_ij for some constant λ. A smooth manifold M with an Einstein metric is called an Einstein manifold. If the initial metric is Ricci flat, so that

0 =

ij

R , than any Ricci flat metric is a stationary solution of the Ricci flow. This happens, for example, on a flat torus or on any K3-surface with a Calabi-Yau metric.

The equation R_ij+∇_i∇_jf =0, or Ric+∇2f =0, (1.1) is the steady gradient Ricci soliton equation. The equation for a homothetic Ricci soliton (expanding Ricci soliton) is

2 + ∇ + ∇ −2 ij =0 k i jk k j ik ij g V g V g R

λ

, (1.2)

or for a homothetic gradient Ricci soliton,

Rij +∇i∇jf −λgij =0, (1.3)

where λ is the homothetic constant. For λ>0 the soliton is shrinking, for λ<0 it is expanding. The case λ=0 is a steady Ricci soliton, the case V = 0 (or f being a constant function) is an Einstein metric. The Ricci flow on a Kahler manifold is called the Kahler-Ricci flow. A Ricci soliton to the Kahler-Ricci flow is called a Kahler-Ricci soliton.

PROPOSITION A.

On a compact n-dimensional manifold M, a gradient steady or expanding Ricci soliton is necessarily an Einstein metric.

Let gij be a complete steady gradient Ricci soliton on a manifold M so that

R_ij +∇_i∇_jf =0. Taking the trace, we get R+ f∆ =0. (1.4) Now, for the equation ∇_iR_jk−∇_jR_ik +R_ijkl∇_lf =0,

∇_jR_ij = ∇_iR 2 1 , (1.5) we get ∇_iR−2R_ij∇_jf =0. Then ∇i

(

∇f 2+R

)

=2∇jf

(

∇i∇jf +Rij

)

=0. Therefore R+∇f 2 =C, (1.6)

for some constant C. Taking the difference of (1.4) and (1.6), we get ∆f −∇f 2 =−C. (1.7)

Then, by integrating (1.7) we obtain

∫

∇ =

M f dV 0

2

. Therefore f is a constant and g_ij is Ricci flat.

We assume M is a compact n-dimensional manifold and consider the following functional

(

)

=

∫

(

+∇

)

− M f ij f R f e dV g F , 2 (1.8)

of Perelman defined on the space of Riemannian metrics, and smooth functions on M. Here R is the scalar curvature of gij.

LEMMA 1. (Perelman).

If δg_ij =v_ij and δf =h are variations of g and f respectively, then the first variation of F is _ij given by

(

)

∫

(

)

(

)

_ −      + ∇ − ∆       − + ∇ ∇ + − = M f j i ij ij ij h f f R e dV v f R v h v F 2 2 2 ,

δ

where ij ij v g v= . PROPOSITION 1. (Perelman).

Let g_ij(t) and f(t) evolve according to the coupled flow ij R_ij t g 2 − = ∂ ∂ , f f R t f − ∇ + ∆ − = ∂ ∂ 2 . Then

(

( ) ( )

)

=

∫

+∇∇ − M f j i ij ij t f t R f e dV g F dt d 2 2 ,

and

∫

−

M f

dV

e is constant. In particular F

(

g_ij

( ) ( )

t , f t

)

is non-decreasing in time and the monotonicity is strict unless we are on a steady gradient soliton.

PROPOSITION 2.

On a compact manifold, a steady or expanding breather is necessarily an Einstein metric. Now, we introduce the following important functional, also due to Perelman,

(

)

=

∫

[

(

+∇

)

+ −

]

(

)

− − M f n ij f R f f n e dV g W 2 2 4 , , τ τ πτ (1.9)

where gij is a Riemannian metric, f is a smooth function on M, and

τ

is a positive scale parameter.

The functional W is invariant under simultaneous scaling of

τ

and g_ij, and invariant under diffeomorphism. Namely, for any positive number “a” and any diffeomorphism ϕ

W

(

a

ϕ

∗g_ij,

ϕ

∗f,a

τ

)

=W

(

g_ij,f,

τ

)

. (1.10)

Similar to Lemma 1, we have the following first variation formula for W. LEMMA 2. (Perelman). If v_ij =δg_ij, h=δf , and η=δτ, then

(

)

∫



(

)

+      − ∇ ∇ + − = − − M f n ij j i ij ij ij h v R f g e dV v W 4 2 2 1 , , πτ τ τ η δ

(

)

[

]

(

)

(

)

∫

− −

∫

− −       − ∇ + + − − + ∇ − ∆ +       − − + M M f n f n dV e n f R dV e n f f f R n h v 2 2 2 2 4 2 4 1 2 2 2 τη τ πτ η τ πτ . Here ij ij v g v= as before.

The following result is analogous to Proposition 1.

PROPOSITION 3.

If gij

( )

t , f

( )

t and

τ

( )

t evolve according to the system

ij R_ij t g 2 − = ∂ ∂ , τ 2 2 n R f f t f + − ∇ + ∆ − = ∂ ∂ , =−1 ∂ ∂ t τ ,

(

( ) ( ) ( )

)

=

∫

+∇∇ −

(

)

− − M f n ij j i ij ij t f t t R f g e dV g W dt d 2 2 4 2 1 2 , ,

πτ

τ

τ

τ

and

∫

(

)

− − M f n dV e 2

4

πτ

is constant. In particular W

(

gij

( ) ( ) ( )

t ,f t ,

τ

t

)

is non-decreasing in time and the monotonicity is strict unless we are on a shrinking gradient soliton.

Now we set

(

)

(

)

( )

(

)

_        = ∈ = ∞

∫

− M f n ij ij W g f f C M e dV g 1 4 1 , , , inf , _/2

πτ

τ

τ

µ

(1.11) and

( )

(

)

( )

(

)

_        = > ∈ = ∞

∫

− 1 4 1 , 0 , , , inf W g f f C M _/2 e dV g v ij _n f

πτ

τ

τ

. (1.12)

Note that if we let f/2

e

u= − , then the functional W can be expressed as

(

)

=

∫

[

(

+ ∇

)

− −

]

(

)

− M n ij f Ru u u u nu dV g W 2 2 2 2 2 2 4 log 4 , ,

τ

τ

πτ

and the constraint

∫

(

)

− − =

M f n dV e 1 4

πτ

2 becomes

∫

(

)

− = M n dV u2 4

πτ

2 1. Thus

µ

(

g_ij,

τ

)

corresponds to the best constant of a logarithmic Sobolev inequality. Since the non-quadratic term is sub-critical, it is rather straightforward to show that

[

(

)

]

(

)

(

)

      = − − + ∇

∫

∫

− − M n n M 4 u Ru u logu nu 4 dV u 4 dV 1 inf 2 2 2 2 2 2 2 2

πτ

πτ

τ

is achieved by some nonnegative function u∈H1

( )

M which satisfies the Euler-Lagrange equation

τ

(

−4∆u+Ru

)

−2ulogu−nu=

µ

(

g_ij,

τ

)

u.

Then, for the (1.11), we have

(

)

(

)

( )

(

)

e dV u M C f f g W nu u u Ru u M f n ij         = ∈ = − − + ∆ − ∞

∫

− 1 4 1 , , , inf log 2 4 _/2

πτ

τ

τ

. (1.13)

Now we relating the quantity H (or v) and the W-functional of Perelman defined in (1.9). Observe that v happens to be the integrand of the W-functional,

(

( )

)

=

∫

M

ij t f vdV

g

W , ,

τ

.

Hence, when M is compact,

∫

 =−

∫

+∇ − ≤      + ∂ ∂ = M v Rv dV M Ric f g udV W d d 0 2 1 2 2 2

τ

τ

τ

τ

,

or equivalently,

(

( ) ( ) ( )

)

(

)

∫

− − ∇ ∇ + = M f n ij j i ij ij t f t t R f g e dV g W dt d 2 / 2 4 1 2 1 2 , ,

πτ

τ

τ

τ

,

which is the same as stated in Proposition 3.

We now use the Ricci-flatness of the metric g~ to interpret the Bishop-Gromov relative volume comparison theorem which will motivate another monotonicity formula for the Ricci flow.

Consider a metric ball in

(

M ~~,g

)

centred at some point

(

p,s,0

)

∈M~. Note that the metric of the sphere SN at τ =0 degenerates and it shrinks to a point. Then the shortest geodesic

γ

( )

τ

between

(

p, s,0

)

and an arbitrary point

(

q,s,

τ

)

∈M~ is always orthogonal to the SN fibre. The length of

( )

τ

γ

can be computed as

∫

( )

_{( )}

∫

(

( )

)

_       + + + = +       + − τ τ τ

τ

τ

τ

γ

τ

τ

τ

γ

τ

0 0 2 3 2 2 2 1 2 2 R d N N R d O N N ij ij g g & & _.

Thus a shortest geodesic should minimize

( )

γ

=

∫

τ

τ

(

+

γ

( )

τ

)

τ

0 2 d R L ij g & _.

Let L(q,

τ

) denote the corresponding minimum. We claim that a metric sphere S_M~

(

2N

τ

)

in M~ of radius 2N

τ

centred at

(

p, s,0

)

is O(N−1)-close to the hypersurface

{

τ

=

τ

}

. Indeed, if

( )

(

x,s',

τ

x

)

lies on the metric sphere S_M~

(

2N

τ

)

, then the distance between

(

x,s',

τ

( )

x

)

and

(

p, s,0

)

is

( )

(

( )

)

_       + + = −2 3 , 2 1 2 2 L x x O N N x N N

τ

τ

τ

which can be written as

( )

(

,

( )

)

(

2

)

( )

1 2 1 − − = + − = − L x x O N O N N x

τ

τ

τ

.

This shows that the metric sphere S_M~

(

2N

τ

)

is O

( )

N−1 −close to the hypersurface

{

τ

=

τ

}

. Thus, we have

(

(

)

)

(

)

∫

(

)

(

)

      + − ≈ − M M N N N M L x O N dV N N N S Vol ~ 2 , 2 2 1 2 2

τ

ω

τ

τ

(

)

∫

(

)

( )

      + − ≈ − M M N N N dV N o x L N N 2 , 1 2 1 2

ω

τ

τ

,

where

ω

_N is the volume of the standard N-dimensional sphere. Now the volume of Euclidean sphere of radius 2N

τ

in Rn+ N+1 is

(

(

)

)

(

)

_{n N} n N R N N S Vol n N + + = + + 1 2

τ

2

τ

2

ω

.

Thus we have

(

(

)

)

(

)

(

)

( )

(

)

M M n n R M dV x L N const N S Vol N S Vol N n       − ⋅ ⋅ ≈ −

∫

− + +

τ

τ

τ

τ

τ

, 2 1 exp 2 2 2 2 ~ 1 .

Since the Ricci curvature of M~ is zero (modulo N−1), the Bishop-Gromov volume comparison theorem suggests that the integral

( )

∫

(

)

(

)

      − = − ∆ M M n dV x L V

τ

τ

τ

π

τ

, 2 1 exp 4 ~ 2 ,

which we will call Perelman’s reduced volume, should be non-increasing in

τ

.

Now the Li-Yau-Perelman distance l=l

(

q,

τ

)

is defined by l

(

q,

τ

)

=L

(

q,

τ

)

/2

τ

. We thus have the following

LEMMA 3.

For the Li-Yau-Perelman distance l

(

q,

τ

)

defined above, we have

K R l l 2 / 3 2 1

τ

τ

τ

=− + + ∂ ∂ , (1.14) l2 R l 1_3/2 K

τ

τ

− + − = ∇ , (1.15) l R n _3/2 K 2 1 2

τ

−

τ

+ − ≤ ∆ , (1.16)

in the sense of distributions. Moreover, the equality in (1.16) holds if and only if we are on a gradient shrinking soliton.

COROLLARY 3.1

Let gij

( )

τ

,

τ

≥0, be a family of metrics evolving by the Ricci flow gij =2Rij

∂ ∂

τ

on a compact

n-dimensional manifold M. Fix a point p in M and let l

(

q,

τ

)

be the Li-Yau-Perelman distance from

(

p,0

)

. Then for all

τ

,

{

(

)

}

2 ,

minl q

τ

q∈M ≤n.

As consequence of Lemma 3, we obtain

0 2 2 ≥ + − ∇ + ∆ − ∂ ∂

τ

τ

n R l l l , or equivalently

(

4

)

2exp

( )

≤0     −       + ∆ − ∂ ∂ − l R n

τ

π

τ

.

If M is compact, we define Perelman’s reduced volume by

V

( )

(

)

[

l

(

q

)

]

dV

( )

q M n τ

τ

πτ

τ

=

∫

4 − exp− , ~ 2 ,

where dV_τ denotes the volume element with respect to the metric gij

( )

τ

. Note that Perelman’s reduced volume resembles the expression in Huisken’s monotonicity formula for the mean curvature flow. It follows that

(

)

(

(

)

)

( )

_∫

(

)

(

(

)

)

(

)

(

(

)

)

( )

∫

_ ≤      − +       − ∂ ∂ = − − − − M n n M n q dV q l R q l q dV q l d d τ τ

π

τ

τ

π

τ

τ

τ

τ

τ

π

τ

4 2exp , 4 2exp , 4 2exp ,

∫

(

)

(

(

)

)



( )

=      − ∆ ≤ − M n q dV q l , 0 exp 4

π

τ

2

τ

_τ .

This says that if M is compact, then Perelman’s reduced volume V~

( )

τ

in nonincreasing in

τ

; moreover, the monotonicity is strict unless we are on a gradient shrinking soliton.

THEOREM 1 (Perelman’s Jacobian comparison theorem).

Let gij

( )

τ

be a family of complete solutions to the Ricci flow gij =2Rij

∂ ∂

τ

on a manifold M with

bounded curvature. Let

γ

:

[ ]

0,

τ

→M be a shortest L-geodesic starting from a fixed point p. Then

Perelman’s reduced volume element

(

πτ

)

(

l

( )

τ

) ( )

J

τ

n

−

−

exp

4 2 is nonincreasing in

τ

along

γ

.

THEOREM 2 (Monotonicity of Perelman’s reduced volume).

Let g be a family of complete metrics evolving by the Ricci flow _ij g_ij =2R_ij

∂ ∂

τ

on a manifold M

with bounded curvature. Fix a point p in M and let l

(

q,

τ

)

be the reduced distance from

(

p,0

)

. Then (i) Perelman’s reduced volume

( )

=

∫

(

)

−

(

−

(

)

)

( )

M n q dV q l

V~

τ

4

πτ

2exp ,

τ

_τ is finite and nonincreasing

in

τ

; (ii) the monotonicity is strict unless we are on a gradient shrinking soliton. Now, we have lim 2

( )

1 0 = − → +

τ

τ

τ J n , (1.17) and l

( )

0 = v2. (1.18) Combining (1.17) and (1.18) with Theorem 1, we get

V

( )

=

∫

(

)

−

(

−l

(

q

)

)

dV

( )

q ≤

∫

(

)

−

(

−l

( )

) ( )

J ₌ dv= M TM n n p 0 2 2exp , 4 exp 4 ~ τ τ

πτ

τ

τ

τ

πτ

τ

=

( )

−

∫

_n

( )

− <+∞ R n dv v2 2 exp 4

π

. (1.18a)

This proves that Perelman’s reduced volume is always finite and hence well defined. THEOREM 3 (No local collapsing theorem I).

Given any metric g on an n-dimensional compact manifold M. Let ij gij

( )

t be the solution to the

Ricci flow on

[

0,T

)

, with T <+∞, starting at g . Then there exist positive constants _ij κ and ρ₀

such that for any t₀∈

[

0,T

)

and any point x₀∈M, the solution g_ij

( )

t is κ-noncollapsed at (x₀,t₀)

We have that for k large enough,

(

2

)

2 2 ~ n k k k k r V ε ≤ ε . (Step 1) We estimate the integral of V~k

(

ε

krk2

)

as follows,

(

)

_∫

(

)

(

(

)

)

( )

_∫

(

)

(

(

)

)

( )

            _≤ − + − = − = − ₋ − ₋ M L k k t r n k k r t k k n k k k k k k r k k v k k k k k k q dV r q l r q dV r q l r r V 2 2 / 1 4 1 2 2 exp 2 2 2 2 2 2 2 , exp 4 , exp 4 ~ ε ε ε ε

πε

ε

ε

πε

ε

∫

(

)

(

(

)

)

( )

            _≤ − − − − + 2 2 / 1 4 1 2 exp \ 2 2 2 , exp 4 k r k k v k k k L M k k t r n k kr l q r dV q ε ε ε

ε

πε

. (1.19)

The second term on the right hand side of (3.19) can be estimated as follows

(

)

(

(

)

)

( )

(

)

(

( )

) ( )

(

)

∫

∫

      _≤ − −       > = − − − − ≤ − 2 2 / 1 4 1 1/2 2 exp \ 4 1 0 2 2 2 2 exp 4 , exp 4 k k k v k k k k r L M _v n r t k k n k kr l q r dV q l I dv ε _ε τ ε ε

τ

τ

πτ

ε

πε

( )

∫

( )

      > − − ≤ − = 2 / 1 4 1 2 2 2 exp 4 k v n k n dv v ε ε π , (1.20)

for k sufficiently large. Combining (1.19)-(1.20), we have the relation of Step 1. We next have that (Step 2)

( ) (

=

)

−

∫

(

−

(

)

)

( )

> M k n k k k t t l q t dV q C V~ 4π 2 exp , ₀ ' (1.20a)

for all k, where C' is some positive constant independent of k. It suffices to show the Li-Yau-Perelman distance l

( )

⋅,tk is uniformly bounded from above on M. Combining Step 1 with Step 2,

and using the monotonicity of Vk

( )

τ

~ , we get '_< ~

( )

_≤ ~

(

2

)

_≤2 2 _→0 n k k k k k k t V r V C ε ε as k→∞. DEFINITION 1

A solution gij

( )

x, to the Ricci flow on the manifold M, where either M is compact or at each time t t

the metric g_ij

( )

⋅,t is complete and has bounded curvature, is called a singularity model if it is not flat and of one of the following three types:

Type I: The solution exists for t∈

(

−∞,Ω

)

for some constant Ω with 0<Ω<+∞ and

(

t

)

Rm ≤Ω/ Ω− everywhere with equality somewhere at t = 0;

Type II: The solution exists for t∈

(

−∞,+∞

)

and Rm ≤1 everywhere with equality somewhere at t = 0;

Type III: The solution exists for t∈

(

−A,+∞

)

for some constant A with 0< A<+∞ and

(

A t

)

A

Rm ≤ / + everywhere with equality somewhere at t = 0.

We state a result of Sesum on compact Type I singularity models. Recall that Perelman’s functional W is given by

(

)

=

∫

(

)

−

[

(

∇ +

)

+ −

]

− M g f n dV e n f R f f g W , ,τ 4πτ 2τ 2

with the function f satisfying the constraint

∫

(

)

− − =

M g f n dV e 1 4πτ 2 .

Furthermore, we have also that

(

( )

)

(

( )

)

(

(

)

)

_{( )}       = − − =

∫

− − M gt f n dV e t T t T f t g W t g inf , , 4π 2 1 µ .

As shown by Natasa Sesum, Type I assumption guarantees the boundedness of

µ

(

g

( )

t

)

, while the compactness assumption of the rescaling limit guarantees the existence of the limit for the minimizing functions f

( )

⋅,t . Therefore we have

THEOREM 4 (Sesum).

Let

(

M,gij

( )

t

)

be a Type I singularity model obtained as a rescaling limit of a Type I maximal solution. Suppose M is compact: then

(

M,g_ij

( )

t

)

must be a gradient shrinking Ricci soliton.

THEOREM 5 (Long-time existence theorem proposed by Perelman).

For any fixed constant ε >0, there exist nonincreasing (continuous) positive functions δ~

( )

t and

( )

t

r~ , defined on

[

0,+∞

)

, such that for an arbitrarily given (continuous) positive function

δ

( )

t with

( )

t δ

( )

t

δ ≤ ~ on

[

0,+∞

)

, and arbitrarily given a compact orientable normalized three-manifold as initial data, the Ricci flow with surgery has a solution with the following properties: either

(i) it is defined on a finite interval

[

0,T

)

and obtained by evolving the Ricci flow and by performing a finite number of cutoff surgeries, with each δ -cutoff at a time t∈

(

0,T

)

having

δ

=

δ

( )

t , so that the solution becomes extinct at the finite time T, and the initial manifold is diffeomorphic to a connected sum of a finite copies of S2×S1 and S3/Γ

(the metric quotients of round three-sphere); or

(ii) it is defined on

[

0,+∞

)

and obtained by evolving the Ricci flow and by performing at most countably many cutoff surgeries, with each δ -cutoff at a time t∈ ,

[

0+∞

)

having

( )

t

δ

δ

= , so that the pinching assumption and the canonical neighbourhood assumption (with accuracy

ε

) with r=r~

( )

t are satisfied, and there exist at most a finite number of surgeries on every finite time interval.

The famous Poincarè conjecture states that every compact three-manifold with trivial fundamental group is diffeomorphic to 3

S . Let M be a compact three-manifold with trivial fundamental group. In particular, the three-manifold M is orientable. Arbitrarily given a Riemannian metric on M, by scaling we may assume the metric is normalized. With this normalized metric as initial data, we consider the solution to the Ricci flow with surgery. If one can show the solution becomes extinct in finite time, it will follow from Theorem 5 (i) that the three-manifold M is diffeomorphic to the three-sphere S3. Such finite extinction time result was first proposed by Perelman, and, recently, Minicozzi has published a proof to it. The combination of Theorem 5 (i) and Colding-Minicozzi’s finite extinction result, gives a complete proof of the Poincarè conjecture.

THEOREM 6 (Perelman).

For any ε >0 and 1≤ A<+∞, one can find

κ

=

κ

(

A,

ε

)

>0, K₁

(

A,

ε

)

<+∞, K₂ =K₂

(

A,

ε

)

<+∞

and r =r

(

A,

ε

)

>0 such that for any t₀<+∞ there exists δA =δA

( )

t0 >0 (depending also on

ε

),

nonincreasing in t , with the following property. Suppose we have a solution, constructed by 0

Theorem 5 with the nonincreasing (continuous) positive functions δ~

( )

t and r~

( )

t , to the Ricci flow with δ-cutoff surgeries on time interval

[

0,T

]

and with a compact orientable normalized three-manifold as initial data, where each δ -cutoff at a time t satisfies δ =δ

( )

t ≤δ~

( )

t on

[

0,T

]

and

( )

t δA δ δ = ≤ on _      0 0 , 2 t t

; assume that the solution is defined on the whole parabolic

neighbourhood

(

)

{

( )

(

)

[

0

]

}

2 0 0 0 0 2 0 0 0 0,t ,r , r x,t x B x ,r ,t t r ,t x P − =∆ ∈ _t ∈ − , 0 2 0 2r <t , and satisfies 02 − ≤ r Rm on P

(

x0,t0,r0,−r02

)

, and

(

(

)

)

3 0 1 0 0, 0 0 B x r A r Volt t − ≥ . Then

(i) the solution is κ-noncollapsed on all scales less than r in the ball ₀

(

₀, ₀

)

0 x Ar

B_t ;

(ii) every point

(

₀, ₀

)

0 x Ar

B

x∈ _t with R

(

x,t₀

)

≥K₁r₀−2 has a canonical neighbourhood B, with

(

,σ

)

(

,2σ

)

0 0 x B B x Bt ⊂ ⊂ t for some

( )

(

0

)

2 1 1 ,

0<σ <C ε R− x t , which is either a strong

ε

-neck or an

ε

-cap; (iii) if r0 ≤r t0 then 2 0 2 − ≤K r R in Bt₀

(

x0,Ar0

)

.

Here C1

( )

ε

is the positive constant in the canonical neighbourhood assumption. Now, we have that every shortest L-geodesic from

(

x,t₀

)

to the ball 2

(

₀, ₀

)

0 0

r x

B_{t −r} is necessarily admissible. By combining with the assumption that

(

(

₀, ₀

)

)

1 ₀3

0

0 B x r A r

Vol_{t t} ≥ − , we conclude that Perelman’s reduced volume of the ball 2

(

0, 0

)

0 0

r x

B_t−r satisfies the estimate

(

(

)

)

(

)

(

(

)

)

( )

( )

( )

∫

− ≥ − = − ₋ − 0 0 2 0 0 2 0 0 2 0 0 2 0 , 2 0 2 3 2 0 0 0, 4 exp , ~ r x B t r r t r r t A c q dV r q l r r x B V π (1.21)

for some positive constant c

( )

A depending only on A. The union of all shortest L-geodesics from

(

x,t0

)

to the ball 2

(

0, 0

)

0 0 x r B_{t −r} , defined by CB_{t r}2

(

x₀,r₀

) ( )( )

{

y,t y,t 0 0− =

lies in a shortest L-geodesic from

(

x,t₀

)

to a point in 2

(

₀, ₀

)}

0 0

r x

B_{t −r} , forms a cone-like subset in space-time with vertex

(

x,t₀

)

. Denote by B

( )

t the intersection of the cone-like subset 2

(

0, 0

)

0 0 x r

CB_t−r with the time-slice at t. Perelman’s reduced volume of the subset B

( )

t is given by

(

( )

)

(

(

)

)

(

(

)

)

( )

( )

∫

− − − = − −t _Bt t t B t t t l q t t dV q V 2 ₀ 3 0 exp , 4 ~ 0

π

.

Since the cone-like subset 2

(

₀, ₀

)

0

0 x r

CB_t−r lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison Theorem 1 and the estimate (1.21) to conclude that

V~_t−t

(

B

( )

t

)

≥V~_r

(

B_{t −r}2

(

x₀,r₀

)

)

≥c

( )

A 0 0 2 0 0 (1.22)

for all t∈

[

t₀−r₀2,t₀

]

. Consider B~

(

t₀−

ξρ

2

)

, the subset at the time-slice

{

t= t₀−ξρ2

}

where every point can be connected to

(

x,t₀

)

by an admissible shortest L-geodesic. Perelman’s reduced volume of B~

(

t₀−

ξρ

2

)

is given by

(

(

−

)

)

=

∫

₍

₎

(

)

(

−

(

)

)

( )

= − − − 2 0 2 0 2 ~ 2 2 3 2 2 0 4 exp , ~ ~ ξρ ξρ ξρ B t ξρ _Bt πξρ l qξρ dVt q V

(

)

(

(

)

)

( )

(

)

( )

∫

            − ≤ ∩ − − − + − = ₂ 2 1 4 1 2 0 2 0 exp ~ 2 2 3 2 , exp 4 ξρ ξρ ξρ ξ ξρ πξρ v L t B l q dVt q

(

)

(

(

)

)

( )

(

)

( )

∫

            − ≤ − − − − + ₂ 2 1 4 1 2 0 2 0 exp \ ~ 2 2 3 2 , exp 4 ξρ ξρ ξρ ξ ξρ πξρ v L t B l q dVt q . (1.23)

Note that the whole region P

(

x,t₀,ρ,−ρ2

)

is unaffected by surgery because ~

( )

₀ 2 1 t r η ρ≥ and

( )

, 0 2 ~ , ~ , , 0 0 0 >            ε

δ L t r t r t is sufficiently small. Then, there is a universal positive constant ξ0 such

that when 0<ξ≤ξ0, there holds

( )

ξρ

(

ρ

)

ξ , exp 0 2 1 2 4 1 B x L t v ⊂         ≤ −

and the first term on right hand side of (1.23) can be estimated by

(

)

(

(

)

)

( )

( )

(

)

( )

∫

            − ≤ ∩ − − − − ≤ − 2 2 1 4 1 2 0 2 0 exp ~ 2 3 2 3 2 2 3 2 4 , exp 4 ξρ ξρ ξ ξρ ξ

ξ

π

ξρ

πξρ

v L t B C t q e dV q l (1.24)

for some universal constant C; while the second term on right hand side of (1.23) can be estimated by

(

)

(

(

)

)

( ) ( )

( )

(

)

( )

∫

∫

            − ≤ − −         > − − − − ≤ − 2 2 1 4 1 2 0 2 1 2 0 exp \ ~ 4 1 2 2 3 2 2 3 2 exp 4 , exp 4 ξρ ξρ ξρ ξ ξ

π

ξρ

πξρ

v L t B l q dVt q v v dv. (1.25)

Since B

(

t₀ −

ξρ

2

)

⊂B~

(

t₀−

ξρ

2

)

, the combination of (1.22)-(1.25) bounds

ξ

from below by a positive constant depending only on A. This proves the statement (i).

2. Ricci flow on compact four-manifolds with positive isotropic curvature. [2]

Let M4 be a compact four-manifold with no essential incompressible space-form and with a metric

ij

THEOREM 2.1

There exists a positive constant

κ

₀ with the following property. Suppose we have a four-dimensional (compact or noncompact) ancient

κ

-solution with restricted isotropic curvature pinching for some κ>0. Then either the solution is

κ

₀-noncollapsed for all scales, or it is a metric quotient of the round cylinder R×S3.

Let g_ij

( )

x,t , x∈M4 and t∈

(

−∞,0

]

, be an ancient

κ

-solution with restricted isotropic curvature pinching for some κ >0. For arbitrary point

(

p,t₀

)

∈ M4×

(

−∞,0

]

, we define that

τ

=t₀−t, for

0 t t< ,

(

)

=

{

∫

τ

(

(

γ

( )

−

)

+

γ

( )

_{( −)}

)

γ

[ ]

τ

→

γ

( )

=

γ

( )

τ

=

}

τ

τ

0 4 2 0 : 0, 0 , , inf 2 1 , 0 q p with M ds s s t s R s q l _{g t s} ij & _, and

( )

=

∫

₄

(

)

−

(

−

(

)

)

₋

( )

0 , exp 4 ~ 2 M l q dVt q V

τ

πτ

τ

_τ , where _{g (t s)} ij − ⋅ ⋅ 0

is the norm with respect to the metric g_ij

(

t₀−s

)

and _−τ 0

t

dV is the volume element with respect to the metric g_ij

(

t₀−

τ

)

. Here, l is called the reduced distance and V~

( )

τ

is called the reduced volume. For any A≥1, one can find B=B

( )

A <+∞ such that for every τ >1 there holds l

(

q,

τ

)

<B and

τ

R

(

q,t₀−

τ

)

≤B (2.1) whenever

τ

≤

τ

≤A

τ

2 1 and d _τ q q

τ

A

τ

t ≤           − , ₂ 2 2 0 .

Considering the reduced volume V~

( )

τ

of the ancient

κ

-solution, we have from Perelman’s Jacobian comparison theorem that

( )

=

∫

(

)

− − ₋ ≤

∫

( )

− − = 4 4 2 0 4 1 4 ~ 2 2 M TM X t l p dX e dV e V

τ

πτ

τ

π

. Now, we denote by

(

( )

)

4 1 1 , 0 0 B p Vol_{t t} =

ε

. For any v∈T_pM4, we have that one can find a L-geodesic

( )

τ

γ

, starting at p, with +

( )

=v

→

τ

γ

τ

τ 0 &

lim , which satisfies the following L-geodesic equation

( )

2

(

,

)

0 2 1 = ⋅ + ∇ −

τ

τ

γ

γ

τ

τ

& R Ric &

d d

. (2.2)

By integrating the L-geodesic equation we see that as

τ

≤ with the property that

ε

( )

( )

,1 0 p

B_t

∈

σ

γ

for

σ

∈

(

0,

τ

]

, there holds

τ

γ

&

( )

τ

−v ≤C

ε

(

v +1

)

_(2.3) for some universal positive constant C.

Now, for v∈TpM4 with 2

1

4 1 −

≤

ε

v and for

τ

≤ with the property that

ε

( )

( )

,1 0 p Bt ∈

σ

γ

for

(

τ

]

σ

∈ 0, , we have

(

( )

)

≤

∫

τ

( )

< −

∫

τ =

σ

σ

ε

σ

σ

γ

τ

γ

0 0 2 1 1 2 1 , 0 d d p d_t & _. This shows

( )

( )

,1 4 1 exp 0 2 1 p B v L ⊂ t       ≤

ε

−

ε

(2.4)

where Lexp

( )( )

⋅

ε

denotes the exponential map of the L distance with parameter

ε

. We decompose the reduced volume V~

( )

ε

as

( )

(

)

( )

( )

(

)

( )

( )

∫

∫

∫

        ≤ − −         ≤ − − − + − − ≤ − = ₄ 2 1 0 2 1 4 0 4 1 exp 2 4 1 exp \ 2 exp 4 exp 4 ~ M l dVt L v M L v l dVt V ε ε ε ε ε ε

πε

πε

ε

(2.5)

The first term on right hand side of (2.5) can be estimated by

(

)

( )

(

)

( )

( ) ( )

∫

∫

        ≤ − − − − − ≤ − ≤ ε ε ε ε

πε

πε

2 1 0 0 0 4 1 exp ,1 2 4 2 exp 4 exp 4 v L l dVt e B_t p l dVt 4

( )

4 2 2

(

( )

,1

)

4

( )

4 2 2 0 0

π

ε

ε

π

ε ε − − − = ≤e Volt Bt p e . (2.6)

where we used (2.4) and the equivalence of the evolving metric over

( )

,1 0 p

B_t . While the second term on the right hand side of (2.5) can be estimated as follows

(

)

( )

(

)

( ) ( )

( )

∫

∫

        ≤         > = − − − − − ≤ − − ε ε 2

πε

ε ε

πτ

τ

τ 1 4 2 1 0 4 1 exp \ 4 1 0 2 2 exp 4 exp 4 v L M l dVt v l J dv (2.7)

by Perelman’s Jacobian comparison theorem, where J

( )

τ

is the Jacobian of the L-exponential map. To evaluate l

( )

⋅,

τ

at τ =0, we use (2.3) again to get

( )

⋅ =

∫

τ

(

+

γ

( )

)

→

τ

τ

0 2 2 2 1 , s R s ds v l & _{, as} + → 0

τ

, thus l

( )

⋅,0 = v2. (2.8)

Hence by combining (2.7)-(2.8) we have

(

)

( )

( )

( )

( )

∫

∫

        ≤         > − − − − − ≤ − − < ε ε 2

πε

ε

π

ε

ε

1 4 2 1 0 4 1 exp \ 4 1 2 2 2 2 exp 4 exp 4 v L M l dVt v v dv . (2.9)

By summing up (2.5), (2.6) and (2.9), we obtain

V~

( )

ε

<2

ε

2. (2.10)

(

)

(

(

)

)

(

(

)

)

( )

(

(

)

)

(

(

)

)

( )

( ) ( )

∫

∫

− ≥ − ≥ − = − − − − M k k t B q k k t k k k k t k k q l q dV q dV q l V τ τ τ τ τ

β

τ

τ

π

τ

τ

π

τ

, 2 2 2 2 0 0 0 4 2 exp ,2 2 , exp 2 4 2 ~

for some universal positive constant

β

. By applying the monotonicity of the reduced volume and (2.10), we deduce that

β

≤V~

(

2

τ

_k

)

≤V~

( )

ε

<2

ε

2. This proves

(

( )

,1

)

₀ 0 0 0 B p ≥

κ

> Vol_{t t}

for some universal positive constant

κ

₀. LEMMA 2.1

Given

100 1

0<

ε

< , 0<δ <ε and 0< T<+∞, there exists a radius 0< h<δσ , depending only on

( )

T r

,

δ

and the pinching assumption, such that if we have a solution to the Ricci flow with surgery, with a compact four-manifold

(

M4,g_ij

( )

x

)

with no essential incompressible space form and with positive isotropic curvature as initial data, defined on

[

0,T

)

, going singular at the time T, satisfies the a priori assumptions and has only a finite number of surgery times on

[

0,T

)

, then for each point x with h

( )

x =R−2

( )

x ≤h

1

in an

ε

-horn of

(

Ω,g_ij

)

with boundary in Ω , the neighbourhood _σ

( )

(

x h x

)

{

y dist

(

y x

)

h

( )

x

}

B ij g T 1 1 ,

,

δ

− = ∈Ω ≤

δ

− is a strong δ -neck (i.e.,

( )

(

( )

)

[

( )

]

{

y,t y∈B_T x,_δ−1h x ,t∈T−h2 x,T

}

is, after scaling with factor h−2

( )

x , δ -close (in C

[ ]

δ−1 topology) to the corresponding subset of the evolving standard round cylinder S3×R

over the time interval

[

−1,0

]

with scalar curvature 1 at the time zero).

PROPOSITION 2.1

Given a compact four-manifold with positive isotropic curvature and no essential incompressible space form and given ε >0, there exist decreasing sequences ε >~ >rj 0, κj >0,

{

, ,

}

~ 0,

minε2 δ₀ δ >δj > j=1,2,..., with the following property. Define a positive function δ

( )

t ~

on

[

0,+∞

)

by δ

( )

t δj

~ ~

= when t∈

[

(

j−1

)

ε2, jε2

)

. Suppose we have a solution to the Ricci flow with surgery, with the given four-manifold as initial datum defined on the time interval

[

0,T

)

and with a finite number of δ -cutoff surgeries such that any δ -cutoff surgery at a time t∈

(

0,T

)

with

δ

=

δ

( )

t

satisfies 0<δ

( )

t ≤δ~

( )

t . Then on each the time interval

[

(

j₋1

)

_ε2,j_ε2

]

I

[

0,T

)

, the solution satisfies the κj-noncollapsing condition on all scales less than

ε

and the canonical neighbourhood

assumption (with accuracy

ε

) with r=r~_j. LEMMA 2.2

For a given compact four-manifold with positive isotropic curvature and no essential incompressible space form and given ε >0, suppose we have constructed the sequences, satisfying the above proposition for 1≤ j≤l. Then there exists κ >0, such that for any r, 0< r<ε , one can

find

δ

~ with 0<

δ

~<min

{

ε

2,

δ

₀,

δ

}

, which depends on r,

ε

and may also depend on the already constructed sequences, with the following property. Suppose we have a solution, with the given four-manifold as initial data, to the Ricci flow with surgery defined on a time interval

[

0,T

]

with

(

)

2

2

1ε ε ≤T < l+

l such that the assumptions and conclusions of Proposition 2.1 hold on

[

0,lε2

)

, the canonical neighbourhood assumption (with accuracy

ε

) with r holds on

[

lε2,T

]

, and each

δ

( )

t -cutoff surgery in the time interval t∈

[

(

l−1

)

ε2,T

]

has 0<δ

( )

t <δ~. Then the solution is κ -noncollapsed on

[

0,T

]

for all scales less than

ε

.

CLAIM 2.1

For any L<+∞ one can find

δ

~=

δ

~

(

L,r,~rl,

ε

)

>0 with the following property. Suppose that we

have a curve γ , parametrized by t∈

[

T₀,t₀

]

,

(

)

2 _{0 0}

1 T t

l− ε ≤ < , such that γ

( )

t₀ =x₀, T is a surgery ₀ time and γ

( )

T₀ lies in a 4h-collar of the middle three-sphere of a δ -neck with the radius h obtained in Lemma 2.1, where the δ-cutoff surgery was taken. Suppose also each

δ

( )

t -cutoff surgery in the time interval t∈

[

(

l−1

)

ε2,T

]

has 0<δ

( )

t <δ~. The we have an estimate

∫

0−0

(

(

(

−

)

−

)

+

(

−

)

_{( − )}

)

≥ 0 0 2 0 0 0 , T t t g d L t t t R ij τ τ γ τ τ γ τ & _{τ , (2.11)} where τ =t₀−t∈

[

0,t₀ −T₀

]

.

Now choose L = 100 in (2.11), then it follows from Claim 2.1 that there exists

δ

~>0, depending on r and r~l, such that as each δ -cutoff surgery at the time interval t

[

(

l 1

)

,T

]

2

ε −

∈ has

δ

<

δ

~, every barely admissible curve γ with endpoints

(

x₀,t₀

)

and

( )

x,t , where t∈

[

(

l−1

)

ε2,t₀

)

, has

( )

∫

−

(

(

( )

)

( )

_{( )}

)

− ≥ + − = t t R t _{g t} d L ij 0 0 0 2 0 100 , τ γ τ τ τ γ τ γ & _{τ ,}

which implies the reduced distance from

(

x₀,t₀

)

to

( )

x,t satisfies l≥25ε−1. (2.12)

We also observe that, there exists a minimizing curve γ of l_min

(

t₀− l

(

−1

)

ε2

)

, defined on

(

)

[

2

]

0 1

,

0 ε

τ∈ t − l− with γ

( )

0 =x₀, such that

L

( )

γ ≤2⋅

(

2 2ε

)

<10ε . (2.13)

Now we want to get a lower bound for the reduced volume of a ball around x of radius about r~_l at some time-slice slightly before t . Since the solution satisfies the canonical neighbourhood assumption on the time interval

[

(

l−1

)

ε2,lε2

)

, it follows that

for those

( )

      − ∈ −1 −1~2 64 1 , ~ 16 1 , , ,t P x t rl rl

x η η for which the solution is defined. Thus by combining (2.13) and (2.14), the reduced distance from

(

x₀,t₀

)

to each point of the ball 

     − − −r l t x r B l ~ 16 1 , 1 ~ 64 1_η 1 2 η

is uniformly bounded by some universal constant.

We want to get a lower bound estimate for the volume of the ball

(

₀, ₀

)

0 x r

Bt . The reduced distance

from

(

x₀,t₀

)

to each point of the ball       − − −r l t r x B l ~ 16 1 , 1 ~ 64

1_η1 2 η is uniformly bounded by some universal constant. We may assume ε >0 is very small. Then it follows from (2.12) that the points in the ball

      − − −_r l t r x B l ~ 16 1 , 1 ~ 64

1_η1 2 η can be connected to

(

x0,t0

)

by shortest L- geodesics, and all of these L- geodesics are admissible. The union of all shortest L- geodesics from

(

x₀,t₀

)

to the ball

      − − −_r l t r x B l ~ 16 1 , 1 ~ 64 1_η1 2 η denoted by _     − − −_r l t r x CB l ~ 16 1 , 1 ~ 64

1_η 1 2 η , forms a cone-like subset in space-time with the vertex

(

x₀,t₀

)

. Denote B(t) by the intersection of 

     − − −_r l t r x CB l ~ 16 1 , 1 ~ 64

1_η 1 2 η with the time-slice at t. The reduced volume of the subset B(t) is defined by

(

( )

)

(

(

)

)

(

(

)

)

( )

( )

∫

− − − = − −t _Bt t t B t t t l q t t dV q V~ 4 ₀ 2exp , ₀ 0 π .

Since the cone-like subset 

     − − −_r l t x r CB l ~ 16 1 , 1 ~ 64

1_η1 2 η lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison to conclude that

(

( )

)

_l

(

_{l l}

)

r t r t t t t B t V B x r c r V l l ~ , ~ 16 1 , ~ ~ 1 ~ 64 1 ~ 64 1 1 2 1 2 0 0 _{η η} η ≥ κ              ≥ − − + − − − − (2.15) for all ∈_ − − _ 0 2 1 , ~ 64 1 t r t

t η l , where c

(

κl,~rl

)

is some positive constant depending only on κl and r~l.

Denote by

(

(

)

)

4 1 0 0 0 1 0 Vl0 B0 x ,r r− _{t t} =

ξ

. Our purpose is to give a positive lower bound for ξ. We may assume 4 1 < ξ , thus ₀2 ₀ 1~2 64 1

0<ξr <t −t+ η−r_l . Furthermore, we denote by B~

(

t₀ −

ξ

r₀2

)

the subset of the points at the time-slice

{

t=t₀−ξr₀2

}

where every point can be connected to

(

x₀,t₀

)

by an admissible shortest L- geodesic. Clearly B

(

t₀ −

ξ

r₀2

)

⊂B~

(

t₀−

ξ

r₀2

)

. Since r r

η 2 1 0 ≥ and

(

ε

)

δ

δ

~= ~ r,~r_l, sufficiently small, the region P

(

x₀,t₀,r₀,−r₀2

)

is unaffected by surgery. Then by the exactly same argument as deriving (2.4), we see that there exists a universal positive constant ξ₀ such that as 0<ξ ≤ξ₀, there holds

( )

₀2

(

_{0 0}

)

4 1 , exp 0 2 1 r B x r L _t v ⊂         ≤ − ξ ξ . (2.16)