Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Since the 1980s, a lot of information has also

been gathered concerning matrix elements of

suitable sine-Gordon field quantities between

special quantum states (form factors). Unfortu-

nately, the correlation functions involve infinite

sums of form factors that are quite difficult to

control analytically. Hence, it is not known whether

the correlation functions associated with the form

factors give rise to a Wightman field theory with

the usual axiomatic properties.

The Relation to Relativistic

Calogero–Moser Systems

The behavior of the special classical solutions

discussed earlier is very similar to that of classical

point particles. Furthermore, the picture of classical

solitons, antisolitons, and their bound states scatter-

ing independently in pairs is essentially preserved on

the quantum level, just as one would expect for the

quantization of an integrable particle system.

Next, we note that from the quantum viewpoint

there is no physical distinction between wave

functions and point particles, whereas a classical

wave is a physical entity that is clearly very different

from a point particle. Even so, it is a na tural

question whether there exist classical Hamiltonian

systems of N point particles on the line whose

physical characteristics (charges, bound states, scat-

tering, etc.) are the same as those of the particle-like

sine-Gordon solutions. If so, a second question is

equally obvious: does the quantum version of the

N-particle systems still have the same features as

that of the quantum sine-Gordon excitations?

As we now sketch, the first question has been

answered in the affirmative, whereas the second one

has not been completely answered yet. However, all

of the information on the pertinent quantum

N-particle systems collected thus far points to an

affirmative answer. The systems at issue are relati-

vistic versions of the well-known nonrelativistic

Calogero–Moser N-particle systems.

To begin with the classical two-particle system, its

Hamiltonian is given by

H ¼ðcosh p

þ cosh p

Þcothðð x

 x

Þ=2Þ½47

on the phase space

 ¼fðx; pÞ2R

< x

g;!¼ dx ^ dp ½48

Taking x

þ i yields the particle–antiparticle

Hamiltonian

H ¼ðcosh p

þ cosh p

Þjtanhððx

 x

Þ=2Þj ½ 49

on the phase space

 ¼fðx; pÞ2R

g;!¼ dx ^ dp ½50

The two-antiparticle Hamiltonian is again given by

[47] and [48]. The interaction potential in [47] is

repulsive, whereas it is attractive in [49]. Hence, any

initial point in  gives rise to a scattering state,

whereas points in

 yield scat tering states if and

only if the reduced Hamiltonian

¼ cosh pjtanhðx=2Þj; p ¼ðp

 p

Þ=2

x ¼ x

 x

½51

satisfies

> 1. More specifically, in both cases the

distance jx

(t)  x

(t) j increases linearly as t !1,

the scattering (position shift) being encoded by the

same function [32] as for the sine-Gordon solitons.

The phase-space points on the separatrix {

= 1}

have the same temporal asymptotics as the multipole

solution [24], whereas the bound-state oscillations

for

< 1 match those of the breathers [20].

More generally, the Hamiltonian for N

particles

and N



antiparticles is given by the function

j¼1

coshðp

k¼1

k6¼j

jcothððx

 x

Þ=2Þj





l¼1

jtanhððx

 x



Þ=2Þj þ



l¼1

coshðp







m¼1

m6¼l

jcothððx



 x



Þ=2Þj



j¼1

jtanhððx



 x

Þ=2Þj ½52

on the phase space



ðx

; p

Þ2R

; ðx



; p



2 R



< < x

; x



< < x



½53



¼ dx

^ dp

þ dx



^ dp



½54

This defining Hamiltonian can be supplemented by

þ N



 1) independent Hamiltonians that pair-

wise commute. The action-angle map of this integr-

able system can be used to relate the scattering and

bound-state behavior to that of the sine-Gordon

solutions from an earlier section, yielding an exact

correspondence. Indeed, the variables we used to

describe the particle-like sine-Gordon solutions

amount to the action-angle variables associated to

[52]. Moreover, the matrix L [26] with t = x = 0

582 Sine-Gordon Equation

equals the Lax matrix for the N-particle system, which

is the manifestation of a remarkable self-duality

property of the equal-charge case. There is an equally

close relation between the general particle-like solu-

tions and the general systems encoded in [52].

As a matter of fact, the connection can be further

strengthened by introducing spacetime trajectories

for the solitons, antisolitons, and breathers, which

are defined in terms of the evolution of an initial

point in 



under the time translation generator

[52] and the space translation generator, obtained

from [52] by the replacement cosh ! sinh . These

point particle and antiparticle trajectories make it

possible to follow the motion of the solitons,

antisolitons, and breathers during the temporal

interval in which the nonlinear interaction takes

place, wher eas for large times the trajectories are

located at the (then) clearly discernible positions of

the individual solitons, antisolitons , and breathers.

Before sketching the soliton-particle correspon-

dence at the quantum level, we add a remark on the

finite-gap solutions of the classical sine-Gordon

equation, already mentioned in the paragraph

containing [11]. These solutions may be viewed as

generalizations of the particle-like solutions dis-

cussed earlier, and they can also be obtained via

relativistic N-particle Calogero–Moser systems. The

pertinent systems are generalizations of the hyper-

bolic systems just described to the elliptic level.

Turning now to the quantum level, we begin by

mentioning that the Poisson-commuting Hamilto-

nians admit a quantization in terms of commuting

analytic difference operators. This involves a special

ordering choice of the p-dependent and x-dependent

factors in the classical Hamiltoni ans, which is

required to preserve commutativity. The resulting

quantum two-body problem can be explicitly solved

in terms of a generalization of the Gauss hypergeo-

metric function. For the case of equal charges, the

scattering is encoded in the sine-Gordon amplitudes



() (cf. [45] and [46]). For the unequal-charge

case, one should distinguish an even and odd

channel. The scattering on these channels is encoded

in the sine-Gordon amplitudes t

þ

()  r

þ

().

Moreover, the bound-state spectrum agrees with

the DHN formula [44] and the bound-state wave

functions are given by hyperbolic functions.

As a consequence of these results, the physics

encoded in the two-body subspace of the sine-

Gordon quantum field theory is indistinguishable

from that of the corresponding two-body relativistic

Calogero–Moser systems. To extend this equivalence

to the arbitrary-N case, one needs first of all

sufficiently explicit solutions to the N-body

Schro¨ dinger equation. To date, this has only been

achieved for the case of N equal charges and the

special couplings for which the reflection amplitude

þ

vanishes. The asymptotics of the pertinent

solutions is factorized in terms of u



(), in agree-

ment with the sine-Gordon picture.

Further Reading

Ablowitz MJ, Kaup DJ, Newell AC, and Segur H (1974) The

inverse scattering transform – Fourier analysis for nonlinear

problems. Studies in Applied Mathematics 53: 249–315.

Coleman S (1977) Classical lumps and their quantum descen-

dants. In: Zichichi A (ed.) New Phenomena in Subnuclear

Physics, Proceedings Erice 1975, pp. 297–421. New York:

Plenum.

Faddeev LD and Takhtajan LA (1987) Hamiltonian Methods in

the Theory of Solitons. Berlin: Springer.

Flaschka HF and Newell AC (1975) Integrable systems of

nonlinear evolution equations. In: Moser J (ed.) Dynamical

Systems, Theory and Applications, Lecture Notes in Physics,

vol. 38, pp. 355–440. Berlin: Springer.

Karowski M (1979) Exact S-matrices and form factors in 1þ1

dimensional field theoretic models with soliton behaviour.

Physics Reports 49: 229–237.

Ruijsenaars SNM (2001) Sine-Gordon solitons vs. Calogero–

Moser particles. In: Pakuliak S and von Gehlen G (eds.)

Proceedings of the Kiev NATO Advanced Study Institute

Integrable Structures of Exactly Solvable Two-Dimensional

Models of Quantum Field Theory, NATO Science Series, vol. 35,

pp. 273–292. Dordrecht: Kluwer.

Scott AC, Chu FYF, and McLaughlin DW (1973) The soliton: a

new concept in applied science. Proceedings of the Institute of

Electrical and Electronics Engineers 61: 1443–1483.

Smirnov FA (1992) Form Factors in Completely Integrable

Models of Quantum Field Theory. Advanced Series in

Mathematical Physics, vol. 14. Singapore: World Scientific.

Thacker HB (1981) Exact integrability in quantum field theory

and statistical systems. Reviews of Modern Physics 53:

253–285.

Zamolodchikov AB and Zamolodchikov AlB (1979) Factorized

S-matrices in two dimensions as the exact solutions of certain

relativistic quantum field theory models. Annals of Physics

(NY) 120: 253–291.

Sine-Gordon Equation 583

Singularities of the Ricci Flow

M Anderson, State University of New York at

Stony Brook, Stony Brook, NY, USA

Introduction

Fix a closed n-dimensional manifold M, and let M

be the space of Riemannian metrics on M. As in the

reasoning leading to the Einstein equations in

general relativity, there is basically a unique simple

and natural vector field on the space M. Namely, the

tangent space T

M consists of symmetric bilinear

forms; besides multiples of the metric itself, the Ricci

curvature Ric

of g is the only symmetric form that

depends on at most the second derivatives of the

metric, and is invariant under coordinate chan ges,

that is, a (0, 2)-tensor formed from the metric. Thus,

consider

¼ Ric

þ g

where ,  are scalars. Setting  = 2, the corre-

sponding equation for the flow of X is

gðtÞ¼2Ric

gðtÞ

þ gðtÞ½1

The Ricci flow, introduced by Hamilton (1982),is

obtained by setting  = 0:

gðtÞ¼2Ric

gðtÞ

½2

Rescaling the metric and time variable t transforms

[2] into [1], with  = (t). For example, rescaling the

Ricci flow [2] so that the volume of (M, g(t)) is

preserved leads to the flow equation [1] with

 = 2

R, twice the mean value of the scalar

curvature R.

The Ricci flow [2] bears some relation with the

metric part of the -function or renormalization

group (RG) flow equation

gðtÞ¼ðgð tÞÞ

for the two-dimensional sigma model of maps



! M.The-function is a vector field on M,

invariant under diffeomorphisms, which has an

expansion of the form

ðgÞ¼Ric

þ "Riem

þ

where Riem

is quadratic in the Riemann curvature

tensor. The Ricci flow corresponds to the one-loop

term or semiclassical limit in the RG flow

(cf. D’Hoker (1999) and Friedan (1985)).

Recently, G Perelman (2002, 2003a, b) has deve-

loped new insights into the geometry of the Ricci flow

which has led to a solution of long-standing mathe-

matical conjectures on the structure of 3-manifolds,

namely the Thurston geometrization conjecture

(Thurston 1982), and hence the Poincare´ conjecture.

Basic Properties of the Ricci Flow

In charts where the coordinate functions are locally

defined harmonic functions in the metric g(t), [2]

takes the form

¼ g

þ Q

ðg;@gÞ

where  is the Laplace operator on functions with

respect to the metric g = g(t)andQ is a low er-order

term quadratic in g and its first-order partial

derivatives. This is a nonlinear heat-type equation

for g

and leads to the existence and uniqueness of

solutions to the Ricci flow on some time interval

starting at any smooth initial metric. This is the

reason for the minus sign in [2]; a plus sign gives a

backwards heat-type equation, which has no solu-

tions in general.

The flow [2] gives a natural method to try to

construct canonical metrics on the manifold M.

Stationary points of the flow [2] are Ricci-flat

metrics, while stationary points of the flow [1] are

(Riemannian) Einstein metrics, where Ric

= (R=n)g,

with R the scalar curvature of g. One of Hamilton’s

motivations for studying the Ricci flow were results

on an analogous question for nonlinear sigma

models. Consider maps f between Riemannian

manifolds M, N with Lagrangian given by the

Dirichlet energy. Eells–Sampson studied the heat

equation for this action and proved that when the

target N has nonpositive curvature, the flow exists

for all time and converges to a stationary point of

the action, that is, a harmonic map f

: M ! N. The

idea is to see if an analogous program can be

developed on the space of metrics M .

There are a number of well-known obstructions to

the existence of Einstein metri cs on manifolds, in

particular, in dimensions 3 and 4. Thus, the Ricci

flow will not exist for all time on a general

manifold. Hence, it must develop singularities. A

fundamental issue is to try to relate the structure of

the singularities of the flow with the topology of the

underlying manifold M.

A few simple qualitative features of the Ricci flow

[2] are as follows: if Ric(x, t) > 0, then the flow

contracts the metric g(t) near x, to the future, while

584 Singularities of the Ricci Flow

if Ri c(x, t) < 0, then the flow expands g(t) near x.At

a general point, there will be directions of positive

and negative Ricci curvature, along which the metric

locally contracts or expands. The flow preserves

product structures of metrics, and preserves the

isometry group of the initial metric.

The form of [2] shows that the Ricci flow

continues as long as Ricci curvature remains

bounded. On a bounded time interval where Ric

g(t)

is bounded, the metrics g(t) are quasi-isometric, that

is, they have bounded distortion compar ed with the

initial metric g(0). Thus, one needs to consider

evolution equations for the curvature, induced by

the flow for the metric. The simplest of these is the

evolution equation for the scalar curvature R:

R ¼ R þ 2jRicj

½3

Evaluating [3] at a point realizing the minimum R

min

of R on M shows that R

min

is monotone nondecreas-

ing along the flow. In particular, the Ricci flow

preserves positive scalar curvature. Moreover, if

min

(0) > 0, then

t 

min

ð0Þ

½4

Thus, the Ricci flow exists only up to a maximal

time T  n=2R

min

(0) when R

min

(0) > 0. In contrast,

in regions where the Ricci curvature stays negative

definite, the flow exists for infinite time.

The evolution of the Ricci curvature has the same

general form as [3]:

¼ R

½5

The expression for

Q is much more complicated

than the Ricci curvature term in [3] but involves

only quadratic expressions in the curvature.

However,

Q involves the full Riemann curvature

tensor Riem of g , and not just the Ricci curvature (as

[3] involves Ricci and not just scalar curvature). An

important feature of dimension 3 is that the full

Riemann curvature Riem is determined algebraically

by the Ricci curvature. So the Ricci flow has a much

better chance of ‘‘working’’ in dimension 3. For

example, an analysis of

Q shows that the Ricci flow

preserves positive Ricci curvature in dimension 3; if

Ric

g(0)

> 0, then Ric

g(t)

> 0, for t > 0. This is not the

case in higher dimensio ns. On the other hand, in any

dimension >2, the Ricci flow does not preserve

negative Ricci curvature, or even a general lower

bound Ric , for >0. For the remainder of the

article, we usually assume then that dim M = 3.

The first basic result on the Ricci flow is the

following, due to Hamilton (1982).

Space-form theorem.Ifg(0) is a metric of positive

Ricci curvature on a 3-manifold M, then the volume

normalized Ricci flow exists for all time, and

converges to the round metric on S

=, where  is

a finite subgroup of SO(4), acting freely on S

Thus, the Ricci flow ‘‘geometrizes’’ 3-manifolds of

positive Ricci curvature. There are two further

important structural results on the Ricci flow.

Curvature pinching estimate (Hamilton 1982,

Ivey 1993). For g(t) a solution to the Ricci flow on

a closed 3-manifold M, there is a nonincreasing

function  :(1, 1) ! R, tending to 0 at 1, and a

constant C, depending only on g(0), such that,

Riemðx; tÞC  ðRðx; t ÞÞ  jRðx; tÞj ½6

This estimate does not imply a lower bound on

Riem(x, t) uniform in time. However, when com-

bined with the fact that the scalar curvature R(x, t)

is uniformly bounded below (cf. [3]), it implies that

jRiemj(x, t)  1 only where R(x,t) 1. To control

the size of jRiemj, it thus suffices to obtain just an upper

bound on R. This is remarkable, since the scalar

curvature is a much weaker invariant of the metric

than the full curvature. Moreover, at points where the

curvature is sufficiently large, [6]

shows that

Riem(x, t)=R(x,t)  ,for small. Thus, if one scales

the metric to make R(x,t)= 1, then Riem(x,t) .In

such a scale, the metric then has almost non-negative

curvature near (x, t).

Harnack estimate (Hamilton 1982). Let (N, g (t))

be a solution to the Ricci flow with bounded and

non-negative curvature Riem  0, and suppose g (t)

is a complete Riemannian metric on N. Then for

0 < t

 t

Rðx

; t

Þ

exp 

ðx

; x

2ðt

 t

Rðx

; t

Þ½7

where d

is the distance function on (M, g

). This

allows one to control the geometry of the solution at

different spacetime points, given control at an initial

point.

Singularity Formation

The deeper analysis of the Ricci flow is concerned

with the singularities that arise in finite time.

Equation [3] shows that the Ricci flow will not

exist for arbitrarily long time in general. In the case

of initial metrics with positive Ricci curvature, this is

resolved by rescaling the Ricci flow to constant

volume. However, the general situation is necessarily

much more complicated. For example, any manifold

which is a connected sum of S

= or S

 S

factors

has metrics of positive scalar curvature. For obvious

Singularities of the Ricci Flow 585

topological reasons, the volume normalized Ricci

flow then cannot converge nicely to a round metric;

even the renormalized flow must develop

singularities.

The usual meth od to understand the structure of

singularities, particularly in geometric PDEs, is to

rescale or renormalize the solution on a sequence

converging to the singularity to make the solution

bounded, and try to pass to a limit of the

renormalization. Such a limit solution models the

singularity formation, and one hopes (or expects)

that the singularity models have special features

making them much simpler than an arbitrary

solution of the flow.

A singularity forms for the Ricci flow only where

the curvature becomes unbounded. Suppose then

that 

= jRiemj(x

, t

) !1, on a sequence of points

2 M, and times t

! T < 1. Consider the

rescaled or blow-up metrics and times



Þ¼





gðtÞ;



¼ 

ðt  t

Þ½8

where 

are diffeomorphisms giving local dilations

of the manifold near x

by the factor 

The flow



is also a solution of the Ricci flow,

and has bounded curvature at (x

, 0). For suitable

choices of x

and t

, the curvature will be bounded

near x

, and for nearby times to the past,



 0; for

example, one might choose points (x

, t

) where the

curvature is maximal on (M, g(t)), t  t

The rescaling [8] expands all distances by the

factor 

, and time by the factor 

. Thus, in

effect one is studying very small regions, of

spatial size on the order of r

= 

1

about (x

, t

and ‘‘using a microscope’’ to examine t he small-

scalefeaturesinthisregiononascaleofsize

about 1.

A limit solution of the Ricci flow, defined at least

locally in space and time, will exist provided that the

local volumes of the rescalings are bounded below

(Gromov compactness). In terms of the original

unscaled flow, this requires that the metric g(t)

should not be locally collapsed on the scale of its

curvature, that is,

vol B

ðr

; t

Þr

½9

for some fixed but arbitrary >0. A maximal

connected limit (N,



t ), x) containing the base point

x = lim x

, is then called a ‘‘singularity model.’’

Observe that the topology of the limit N may well

be distinct from the original manifold M, most of

which may have been blown off to infinity in the

rescaling.

To see the potential usefulness of this proces s,

suppose one does have local noncollapse on the scale

of the curvature, and that base points of maximal

curvature in space and time t  t

have been chosen.

At least in a subsequence, one then obtains a limit

solution to the Ricci flow (N,



t), x), based at x,

defined at least for times (1, 0], with



t )a

complete Riemannian metric on N. Such solutions

are called ancient solutions of the Ricci flow. The

estimate [6] shows that the limit has non-negative

curvature in dimension 3, and so [7] holds on N.

Thus, the limit is indeed quite special. The topology

of complete manifolds N of non-negative curvature

is completely understood in dimension 3. If N is

noncompact, then N is diffeomorphic to R

, S

 R,

or a quotient of these spaces. If N is compact, then

a slightly stronger form of the space-form theo-

rem implies N is diffeomorphic to S

=, S

 S

,or



The study of the formation of singulariti es in

the Ricci flow was initiated by Hamilton (1995).

Recently, Perelman has obtained an essentially

complete understanding of the singularity behavior

of the Ricci flow, at least in dimension 3.

Perelman’s Work

Noncollapse

Consider the Einstein–Hilbert action

RðgÞ¼

RðgÞdV

½10

as a functional on M. Critical points of R are Ricci-

flat metrics. It is natural and tempting to try to

relate the Ricci flow with the gradient flow of R

(with respect to a natural L

metric on the space M).

However, it has long been recognized that this

cannot be done directly. In fact, the gradient flow of

R does not even exist, since it implies a backwards

heat-type equation for the scalar curvature R

(similar to [3] but with a minus sign before ).

Consider however the following functional

extending R:

Fðg; f Þ¼

ðR þjrf j

Þe

f

½11

as a functional on the larger space M  C

(M, R), or

equivalently a family of functionals on M,parame-

trized by C

(M, R). The functional [11] also arises in

string theory as the low-energy effective action; the

scalar field f is called the dilaton. Fix any smooth

measure dm on M and define the Perelman coupling

by requiring that (g, f )satisfy

f

¼ dm ½12

586 Singularities of the Ricci Flow

The resulting functional

ðg; f Þ¼

ðR þjrf j

Þdm ½13

becomes a funct ional on M. (This coupling does not

appear to have been considered in string theory.)

The L

gradient flow of F

is given simply by

¼2ðRic

f Þ½14

where

f is the Hessian of f with respect to

g.The

evolution equation [14] for

g is just the Ricci flow [2]

modified by an infinitesimal diffeomorphism:

f = (d=dt)(



g), where (d=dt) 

rf . Thus, the

gradient flow of F

is the Ricci flow, up to

diffeomorphisms. The evolution equation for the

scalar field f,

¼

f 

R ½15

is a backward heat equation (balancing the forward

evolution of the volume form of

g(t)). Thus, this

flow will not exist for general f, going forward in t.

However, one of the basic points of view is to let the

(pure) Ricci flow [2] flow for a time t

> 0. At t

one may then take an arbitrary f = f (t

) and flow

this f backward in time ( = t

 t) to obtai n an

initial value f (0) for f. The choice of f (t

) deter-

mines, together with the choice of volume form of

g(0)), (or g(t

)), the measure dm and so the choice

of F

. The process of passing from F to F

corresponds to a reduction of the symmetry group of

all diffeomorphisms D of F to the group D

volume-preserving diffeomorphisms; the quotient

space D=D

has been decoupled into a space

(M, R) of parameters.

The functionals F

are not scale invariant. To

achieve scale invariance, Perelman includes an

explicit insertion of the scale parameter, related to

time, by setting

Wðg; f ;Þ¼

ðjrf j

þ RÞþf  n



ð4Þ

n=2

f

dV ½16

with coupling so that dm = (4)

n=2

f

dV is fixed.

The entropy functional W is invariant under

simultaneous rescaling of  and g, and 

= 1.

Again, the gradient flow of W is the Ricci flow

modulo diffeomorphisms and rescalings and the

stationary points of the gradient flow are the

gradient Ricci solitons,

Ric

þ D

f 

2

g ¼ 0

for which the metrics evolve by diffeomorphisms

and rescalings. Gradient solitons arise naturally as

singularity models, due to the rescalings and

diffeomorphisms in the blow-up procedure [8].An

important example is the cigar soliton on R

 R,

(or R

 S

g ¼ð1 þ r

1

Eucl

þ ds

½17

Perelman then uses the scalar field f to probe the

geometry of g(t). For instance, the collapse or

noncollapse of the metric g(t ) near a point x 2 M

can be detected from the size of W(g(t)) by choosing

f

to be an approximation to a delta function

centered at (x, t). The more collapsed g(t) is near x,

the more negative the value of W(g(t)). The collapse

of the metric g(t) on any scale in finite time is then

ruled out by combining this with the fact that the

entropy functional W is increasing along the Ricci

flow.

Much more detailed information can be obtained

by studying the path integral associated to the

evolution equation [15] for f, given by

LðÞ¼



ﬃﬃﬃ



½j

ðÞj

þ RððÞÞd

where R and j

()j are computed with respect to the

evolving metrics g(). In particular, the study of the

geodesics and the associated variational theory of

the lengt h functional L are important in under-

standing the geometry of the Ricci flow near the

singularities.

Singularity Models

A major accom plishment of Perelman is essentially a

classification of all compl ete singularity models

(N, g(t)) that arise in finite time. In the simple case

where N is compact, then as noted above, N is

diffeomorphic to S

=, S

 S

,orS



In the much more important case where N is

complete and noncompact, Perelman proves that the

geometry of N near infinity is that of a union of

"-necks. Thus, at time 0, and at points x with

r(x) = dist(x, x

) 1, for a fixed base point x

region of radius "

1

about x, in the scale where

R(x) = 1, is "-close to such a region in the standard

round product metric on S

 R; " may be made

arbitrarily small by choosing r(x) sufficiently large.

For example, this shows that the cigar soliton [17]

cannot arise as a singularity model. Moreover, this

structure also holds on a time interval on the order

of "

1

to the past, so that on such regions the

solution is close to the (backwards) evolving Ricci

flow on S

 R.

Singularities of the Ricci Flow 587

Perelman shows that this structural result for the

singularity models themselves also holds for the

solution g( t) very near any singulari ty time T. Thus,

at any base point (x, t) where the curvature is

sufficiently large, the rescaling as in [8] of the

spacetime by the curvature is smoothly close, on

large compact domains, to corresponding large

domains in a complete singul arity model. The

‘‘ideal’’ complete singularity models do actually

describe the geometry and topology near any

singularity. Consequently, one has a detailed under-

standing of the small-scale geometry and topology in

a neighborhood of every point where the curvature

is large on (M, g(t)), for t near T.

The main consequence of this analysis is the

existence of canonical, almost round 2-spheres S

any region of (M, g(t)) where the curvature is

sufficiently large; the radius of the S

’s is on the

order of the curvature rad ius. One then disconnects

the manifold M into pieces, by cutting M along a

judicious choice of such 2-spheres, and gluing in

round 3-balls in a natural way. This surgery process

allows one to excise out the regions of (M, g(t))

where the Ricci flow is almost singular, and thus

leads to a naturally defined Ricci flow with surgery,

valid for all times t 2 [0, 1).

The surgery process disconnects the original

connected 3-manifold M into a collection of disjoint

(connected) 3-manifolds M

, with the Ricci flow

running on each. However, topologically, there is a

canonical relation between M and the components

; M is the connected sum of {M

}. An analysis of

the long-time behavior of the volume-normalized

Ricci flow confirms the expectation that the flow

approaches a fixed point, that is, an Einstein metric,

or collapses along 3-manifolds admitting an S

fibration. This then leads to the proof of Thurston’s

geometrization conjecture for 3-manifolds and

consequently the proof of the Poincare´ conjecture.

It gives a full classification of all closed 3-manifolds,

much like the classification of surfaces given by the

classical uniformization theorem.

See also: Einstein Manifolds; Evolution Equations: Linear

and Nonlinear; Minimal Submanifolds; Renormalization:

General Theory; Topological Sigma Models.

Further Reading

Anderson M (1997) Scalar curvature and geometrization con-

jectures for 3-manifolds. In: Grove K and Petersen P (eds.)

Comparison Geometry, MSRI Publications, vol. 30, pp. 49–82.

Cambridge: Cambridge University Press.

Cao HT, Chow B, Cheng SC, and Yau ST (eds.) (2003) Collected

Papers on Ricci Flow. Boston: International Press.

D’Hoker E (1999) String theory. In: Deligne P et al. (eds.)

Quantum Fields and Strings: A Course for Mathematicians,

vol. 2, Providence, RI: American Mathematical Society.

Friedan D (1985) Nonlinear models in 2 þ " dimensions. Annals

of Physics 163: 318–419.

Hamilton R (1982) Three manifolds of positive Ricci curvature.

Journal of Differential Geometry 17: 255–306.

Hamilton R (1993) The Harnack estimate for the Ricci flow.

Journal of Differential Geometry 37: 225–243.

Hamilton R (1995) Formation of singularities in the Ricci flow.

In: Yau ST (ed.) Surveys in Differential Geometry vol. 2,

pp. 7–136. Boston, MA: International Press.

Ivey T (1993) Ricci solitons on compact three-manifolds.

Differential Geometry and Its Applications 3: 301–307.

Perelman G (2002) The entropy formula for the Ricci flow and its

geometric applications, math.DG/0211159.

Perelman G (2003a) Ricci flow with surgery on three-manifolds,

math.DG/0303109.

Perelman G (2003b) Finite extinction time for the solutions to the

Ricci flow on certain three-manifolds, math.DG/0307245.

Thurston W (1982) Three dimensional manifolds, Kleinian groups

and hyperbolic geometry. Bulletin of American Mathematical

Society 6: 357–381.

Singularity and Bifurcation Theory

J-P Franc¸ oise and C Piquet, Universite

P.-M. Curie,

Paris VI, Paris, France

Introduction

Dynamical systems first developed from the geometry

of Newton’s equations (see Goodstein and Goodstein

(1997)) and the question of the stability of the solar

system motivated further researches inspired by

celestial mechanics (cf. Siegel and Moser (1971)).

Then dynamical systems developed intensively from

stability theory (Lyapunov’s theory) to generic proper-

ties (based on functional analysis techniques,) hyper-

bolic structures (Anosov’s flows, Smale axiom A) and

to perturbation theory (Pugh’s closing lemma, KAM

theorem). There are many links with ergodic theory

dating back to Birkhoff’s ergodic theorem (motivated

by Boltzmann–Gibbs contributions to thermody-

namics). These aspects have been developed in several

articles of the encyclopedia (see Generic Properties of

588 Singularity and Bifurcation Theory

Dynamical Systems; Ergodic Theory; Hyperbolic

Dynamical Systems). This article develops another

aspect of dynamical systems, namely bifurcation

theory. In contrast, the mathematics involved relates

more to local analytic geometry in the broad sense and

provides local models like normal forms, uses blow-up

techniques and asymptotic developments. This con-

tains the singularity theory of functions (related to

singularities of gradient flows). A recent development

of the whole subject deals with bifurcation theory of

fast-slow systems.

Singularity Theory of Functions

A singular point of a gradient dynamics

¼ grad VðxÞ

is a critical point of the function V. Assume that the

function V: U !R is defined and infinitely differ-

entiable on an open set U. Let x

2 U be a critical

point of V.

Definition 1 The critical point x

is said to be of

Morse type if the Hessian of V at x

: D

V(x

)isof

maximal rank n. The corank of a singular point x

the corank of the matrix D

V(x

Denote by O the local ring of germs of C

functions at point x

Definition 2 The Jacobian ideal of the function V

at x

, denoted as Jac(V), is the ideal generated in

the ring O by the partial derivatives of

V: @V=@x

, i = 1, ..., n, considered as elements of

the local ring O.

The singularity (or the singular point) is isolated if

dim

O=JacðVÞ < 1

In that case, the Milnor number is defined as the

dimension

 ¼ dim

O=JacðVÞ

Local models of singularities at a point are simple

expressions that germs of functions singular at this

point have in local coordinates.

R Thom proposed to focus more particularly on the

singularities whose Milnor number is less than or

equal to 4 and whose corank is less than or equal to 2.

The list of local models V



(x) of functions whose

singularities at 0 display a Milnor number les s than

or equal to 4 and a corank less than or equal to 2 is

the following:



(x) =

þ 

x, the fold,



(x) =



þ 

x,thecusp,



(x) =



þ 

x,theswallow tail,



(x) =



þ 

x,the

butterfly,



(x) = x

 3xy

þ 

þ y

) þ 

x þ 

y, the

elliptic umbilic,



(x) = x

þ y

þ 

xy þ 

x þ 

y, the hyperbol ic

umbilic,and



(x) = y

þ x

y þ 

þ 

x þ 

y, the

parabolic umbilic.

Consider more particularly the first four cases.

The ‘‘state equation’’ defines the critical points of V



¼ 0

which contains the subset of the stable equilibrium

points of the associated gradient dynamics. The

nature of these equilibrium states changes at points

contained in the set defined by the equation



¼ 0

The projection of this set on the space of parameters

contains the set of values of the parameters for which

the equilibrium position is susceptible to change of

topological type (in other terms to undergo a bifurca-

tion). This set is called the catastrophe set (see Figure 1).

Consider now the case of umbilics where there are

two state equations:

¼ 0

The catastrophe set S is determined by one further

equation:

Hess V ¼



@x@y



¼ 0

In both cases of hyperbolic and elliptic umbilics, the set

S is a singular surface. For the last case of the parabolic

umbilic, the set S is of dimension 3 and again it is only

possible to represent it by a family of its sections by a

variable hyperplane (see Figure 2).

All possible deformations (in the space of func-

tions) of a function with an isolated singularity can

be induced by a single -dimensional family of

deformations named the ‘‘universal deformation.’’ In

general, the ‘‘codimension’’ of a bifurcation is the

minimal number of parameters needed to display all

possible phase diagrams of all possible unfoldings.

Several deep mathematical techniques, like the

Malgrange division theorem and preparation theo-

rem, allowed J Mather to prove the theorem (local,

then global) of existence of the universal unfolding.

Singularity and Bifurcation Theory 589

The theory of unfoldings of singularities can be

used, for instance, to provide asymptotic expression

of stationary phase integrals when critical points of

the phase are not of Morse type. This relates to

monodromy, Bernstein polynomials, Miln or fibra-

tion near a singular point, and simultaneous local

models of forms and functions (cf. Malgrange

(1974))andsee Feynman Path Integrals).

Singularity Theory of Vector Fields

Transcritical Bifurcation

The transcritical bifurcation is the standard mechan-

ism for changes in stability. The local model is given by

x ¼ rx  x

For r < 0, there is an unstable fixed point at x



= r

and a stable fixed point at x



= 0. As r increases, the

unstable and the stable fixed points coalesce when

r = 0 and when r > 0, they exchange their stability.

A simplified model of the essential physics of a laser

is due to Haken (1983). It is given by

n ¼ GnN  kn

were n is the number of photons in the laser field, N is

the number of excited atoms, and the gain term comes

from the process of stimulated emission which occurs

at a rate proportional to the product n.N. Further-

more, the number of excited atoms drops down by the

emission of photons N = N

 n.Thenweobtain

n ¼ðGN

 kÞn  Gn

This model displays a transcritical bifurcation, which

explains in elementary terms the laser threshold.

Pitchfork Bifurcation

The local model for sup ercritical pitchfork bifurca-

tion is

x ¼ rx  x

Swallow tail Cusp

Elliptic umbilicHyperbolic umbilic

Figure 1 Examples of catastrophe sets. Adapted with permission from Franc¸oise J-P (2005) Oscillations en Biologie: Analyse

Qualitative et Mode

les (Mathe

matiques et Applications, vol. 46). Heidelberg: Springer.

590 Singularity and Bifurcation Theory

When the parameter r < 0, it displays one stable

equilibrium position. As r increases, this equilibrium

bifurcates (for r > 0) into two stable equilibria and

an unstable equilibrium. Its drawing sugges ts ‘‘the

pitchfork.’’ In case of subcritical pitchfork

bifurcation

x ¼ rx þ x

there is a single stabl e state for r < 0 that bifurcates

into two stable states and one unstable as r > 0.

Normal Forms

Local analysis of vector fields proceeds with local

models called normal forms. A local vector field

near a singular point (zero) is seen as a deri vation of

the local ring of functions which preserves the

unique maximal ideal (of the functions which vanish

at the singular point). It yields a linear operator of

the finite-dimensional vector spaces of truncated

Taylor expansions of functions. This leads to

decomposition of the vector fields into semisimple

and nilpotent parts (at the level of formal series). A

normal form is a formal coordinate system in which

the semisim ple part is linear. If the vector field

preserves a structure (like volume form or symplec-

tic form) the change of coordinates which brings it

to its normal form is also (volume-preserving,

symplectic). The simplicity of the normal form

depends on the number of allowed resonances for

the eigenvalues of the first-order jet of the vector

field at the singular point. The best-known example

is the Birkhoff normal form of Hamiltonian vector

fields that we recall now, but we should also

mention the Sternberg normal form of volume-

preserving vector fields.

Local analysis of a Hamiltonian vector field under

symplectic changes of coordinates is the same as the

local analysis of functions (namely its associated

Hamiltonian). Birkhoff normal form deals with the

Butterfly

(section

Parabolic umbilic

(section

Butterfly

(section

Parabolic umbilic

(section λ

Figure 2 Sections of catastrophe sets. Adapted with permission from Franc¸oise J-P (2005) Oscillations en Biologie: Analyse

Qualitative et Mode

les (Mathe

matiques et Applications, vol. 46). Heidelberg: Springer.

Singularity and Bifurcation Theory 591