Ricci flow and geometric applications

Lecture Notes in Mathematics 2166
CIME Foundation Subseries

Michel Boileau
Gerard Besson
Carlo Sinestrari
Gang Tian

Ricci Flow and
Geometric
Applications
Cetraro, Italy 2010

Riccardo Benedetti
Carlo Mantegazza Editors


Lecture Notes in Mathematics
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B. Teissier, Paris
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More information about this series at />
2166



Michel Boileau • Gerard Besson •
Carlo Sinestrari • Gang Tian

Ricci Flow and Geometric
Applications
Cetraro, Italy 2010
Riccardo Benedetti, Carlo Mantegazza
Editors

123


Authors
Michel Boileau
Aix-Marseille Université, CNRS, Central
Marseille
Institut de Mathematiques de Marseille
Marseille, France
Carlo Sinestrari
Dip. di Ingegneria Civile e Ingegneria
Informatica
Università di Roma “Tor Vergata”
Rome, Italy

Editors
Riccardo Benedetti
Department of Mathematics
University of Pisa
Pisa, Italy

ISSN 0075-8434
Lecture Notes in Mathematics
ISBN 978-3-319-42350-0
DOI 10.1007/978-3-319-42351-7

Gerard Besson
Institut Fourier
Université Grenoble Alpes
Grenoble, France

Gang Tian
Princeton University
Princeton, NJ
USA

Carlo Mantegazza
Department of Mathematics
University of Naples
Naples, Italy

ISSN 1617-9692 (electronic)
ISBN 978-3-319-42351-7 (eBook)

Library of Congress Control Number: 2016951889

Mathematics Subject Classification (2010): 53C44, 57M50, 57M40
© Springer International Publishing Switzerland 2016
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Preface

Our aim in organizing this CIME course was to present to young students and
researchers the impressive recent achievements in differential geometry and topology obtained by means of techniques based on the Ricci flow. We then invited some
of the leading researchers in the field of geometric analysis and low-dimensional
geometry/topology to introduce some of the central ideas in their work. Here is the
list of speakers together with the titles of their lectures:
• Gérard Besson (Grenoble) – The differentiable sphere theorem (after S. Brendle
and R. Schoen)
• Michel Boileau (Toulouse) – Thick/thin decomposition of three-manifolds and
the geometrization conjecture

• Carlo Sinestrari (Roma “Tor Vergata”) – Singularities of three-dimensional Ricci
flows
• Gang Tian (Princeton) – Kähler–Ricci flow and geometric applications.
The summer school had around 50 international attendees (mostly PhD students
and postdocs). Even though they were sometimes technically heavy, the lectures
were followed by all the students with interest. The participants were very satisfied
by the high quality of the courses. The not-so-intense scheduling of the lectures gave
the students many opportunities to interact with the speakers, who were always very
friendly and available for discussion. It should be mentioned that the wonderful
location and the careful CIME organization were also greatly appreciated.
We think that the fast-growing field of geometric flows and more generally of
geometric analysis, which has always received great attention in the international
community, but which is still relatively “young” in Italy, will benefit from its
diffusion by this CIME course.
We briefly describe the contents of the lectures collected in this volume.
Gérard Besson presented the impact of the Ricci flow technique on the theory of
positively curved manifolds, the central result being the differentiable 1/4-pinched
sphere theorem, proved by Brendle and Schoen. It says that a complete, simply

v


vi

Preface

connected Riemannian manifold whose sectional curvature varies in .1=4; 1 is
diffeomorphic to the standard sphere.
The problem was first proposed by H. Hopf, and then in 1951, H.E. Rauch
showed that a complete Riemannian manifold whose sectional curvature is positive

and varies between two numbers whose ratio is close to 1 has a universal cover
homeomorphic to a sphere. In the 1960s, M. Berger and W. Klingenberg obtained
the optimal result: a simply connected Riemannian manifold which is strictly 1=4pinched is homeomorphic to the sphere. The analogous diffeomorphic conclusion
remained open until S. Brendle and R. Schoen proved the following:
Theorem (S. Brendle and R. Schoen, 2008) Let M be a pointwise strictly 1=4pinched Riemannian manifold of positive sectional curvature. Then M carries a
metric of constant sectional curvature. Hence, it is diffeomorphic to the quotient of
a sphere by a finite subgroup of O.n/.
The proof relies on the use of the Ricci flow introduced by R. Hamilton and
culminating in the work of G. Perelman. The idea is to construct a deformation of
the Riemannian metric, evolving it by means of the Ricci flow toward a constant
curvature metric. We recall that this was the method that R. Hamilton used in his
seminal paper, proving the following theorem:
Theorem (R. Hamilton, 1982) Let M be a closed 3-dimensional Riemannian
manifold which carries a metric of positive Ricci curvature; then it also carries
a metric of positive constant curvature.
The lectures also focus on the extension to higher dimensions of the following
result, due to C. Böhm and B. Wilking. Recall that a curvature operator is 2-positive
if the sum of its two smallest eigenvalues is positive.
Theorem (C. Böhm and B. Wilking, 2008) Let M be a closed Riemannian manifold whose curvature operator is 2-positive; then M carries a constant curvature
metric.
In the lectures, the connection between this method and the algebraic properties
of the Riemann curvature operator is stressed, the main focus being the identification
of those properties of the curvature operator which are preserved under the Ricci
flow.
In his lectures, Michel Boileau gave an introduction to the geometrization of
3-manifolds. Sections 2.1 and 2.2 cover Thurston’s classification of the eight 3dimensional geometries and the characterization of geometric (and Seifert) closed
3-manifolds in terms of basic topological properties. This follows by combining
Thurston’s hyperbolization theorems (in particular the characterization of hyperbolic 3-manifolds that are fibered over S1 ), Perelman’s general geometrization
theorem, and Agol’s recent (2013) proof of a deep conjecture of Thurston that closed
hyperbolic 3-manifolds are “virtually fibered.”

Section 2.3 discusses the following: (1) A central result of classical 3dimensional geometric topology, that is, the canonical decomposition of a


Preface

vii

3-manifold by splitting it along spheres and tori. (2) Thurston’s geometrization
conjecture. This roughly says that every piece of a canonical decomposition is
geometric together with a prediction on the geometry carried by the piece in
terms of basic topological properties. It includes as a particular case the celebrated
Poincaré conjecture. (3) Thurston’s fundamental hyperbolization theorem for Haken
manifolds.
Perelman’s proof of the general geometrization theorem deals with all of these
topics and also allows us to recover, as a by-product, the canonical decomposition
itself. This is done by completing the program based on the Ricci flow with surgeries,
first proposed by R. Hamilton. This is the subject of Boileau’s notes from Sect. 2.4.
Since the appearance of Perelman’s three celebrated preprints, several simplifications and variants of the original proofs have been developed by various authors.
At the end of the day, we can say that the Poincaré conjecture (i.e., the case
when the Ricci flow with surgery becomes extinct in finite time) is in a sense the
“simplest” case. The general case (when the Ricci flow with surgery exists at all
times, which includes the complete hyperbolization theorem) requires nontrivial
extra arguments, in particular, to obtain a key non-collapsing theorem. In Perelman’s
original work, these come from the theory of Alexandrov spaces. Bessières, Besson,
Boileau, Maillot, and Porti developed instead an alternative approach where the
basic tools are Thurston’s hyperbolization theorem for Haken manifolds and some
well-established properties of Gromov’s simplicial volume, allowing one to bypass
the need for the (somewhat more exotic) theory of Alexandrov spaces. Boileau’s
notes are largely based on the monograph by
L. Bessières, G. Besson, M. Boileau, S. Maillot, and J. Porti, Geometrisation of

3-Manifolds, EMS Tracts in Mathematics 13, 2010.
In this tract, the authors developed a slightly different notion of surgery by defining
the so-called Ricci flow with bubbling-off. Actually, one might roughly say that
the Ricci flow with bubbling-off reduces the general hyperbolization theorem to
Thurston’s hyperbolization theorem for Haken manifolds.
Carlo Sinestrari provided an extensive introduction to the Ricci flow by first
giving a survey of the basic results and examples, then concentrating on the analysis
of the singularities of the flow in the three-dimensional case, which is needed in
Hamilton and Perelman’s surgery construction. After reviewing the properties of the
Ricci flow and the fundamental estimates of the theory, such as Hamilton’s Harnack
differential inequality, the Hamilton–Ivey pinching estimate, and Perelman’s no
collapsing result, he presented Perelman’s analysis of kappa-solutions and the
canonical neighborhood property which gives a full description of the singular
behavior of the solutions in dimension 3. All these results are central to the proof of
the Poincaré and geometrization conjectures.
The exposition is quite accessible to nonexperts. Indeed, the presentation is
often informal, and the proofs are omitted except in some simple and significant
cases, focusing more on the description of the results and their applications and
consequences. A final detailed bibliographical section gives to the interested reader
all the references needed for an advanced study of these topics.


viii

Preface

Gang Tian’s expository notes, based on his lectures, discuss some aspects of
the Analytic Minimal Model Program through the Kähler–Ricci flow, developed
in collaboration with other authors, particularly, J. Song and Z. Zhang. Very
stimulating open problems and conjectures are also presented.

Section 4.2 contains a detailed account of the sharp version of the Hamilton–
DeTurck local existence theorem, which holds in the Kähler case: the maximal
time Tmax 2 .0; C1 such that the flow exists on the interval Œ0; Tmax / is precisely
determined in terms of a cohomological property of the initial Kähler metric. As a
corollary, one deduces that Tmax D C1 for every initial metric on a compact Kähler
manifold with a numerically positive canonical bundle.
In Sect. 4.3, the limit singularities that can arise when t ! Tmax < C1 are
analyzed. After having established a general convergence theorem (Theorem 4.3.1),
one faces questions concerning regularity and the geometric properties of the limit.
A combination of partial results (particularly in the case of projective varieties
and when one can apply deep results of algebraic geometry) and well-motivated
conjectures outlines a pregnant scenario.
Sections 4.4 and 4.5 discuss the construction of a Kähler–Ricci flow with surgery
(assuming the truth of a conjecture stated in Sect. 4.3) and its asymptotic behavior.
Numerous conjectures arise throughout this discussion such as: the characterization
(up to birational isomorphism) of “Fano-like” manifolds as those whose flow
becomes extinct at a finite time; the characterization of uniruled manifolds (up
to birational isomorphism) as those whose flow collapses in finite time; and the
existence of a flow with surgery globally defined in time and with only finitely many
surgery times.
In Sect. 4.6, algebraic surfaces are considered, showing how most of the program
is carried out in this case.

We are pleased to express our thanks to the speakers for their excellent lectures
and to the participants for contributing with their enthusiasm to the success of the
Summer School.
The speakers, the participants, and the CIME organizers collectively created a
stimulating, rich, pleasant, and friendly atmosphere at Cetraro. For this reason, we
would finally like to thank the Scientific Committee of CIME and, in particular,
Pietro Zecca and Elvira Mascolo.

Pisa, Italy
Naples, Italy

Riccardo Benedetti
Carlo Mantegazza


Acknowledgements

CIME activity is carried out with the collaboration and financial support of: INdAM
(Istituto Nazionale di Alta Matematica) and MIUR (Ministero dell’Istruzione,
dell’Università e della Ricerca).

ix



Contents

1

The Differentiable Sphere Theorem
(After S. Brendle and R. Schoen).. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Gérard Besson

1

2 Thick/Thin Decomposition of Three-Manifolds
and the Geometrisation Conjecture .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Michel Boileau

21

3 Singularities of Three-Dimensional Ricci Flows . . . . . .. . . . . . . . . . . . . . . . . . . .
Carlo Sinestrari

71

4 Notes on Kähler-Ricci Flow .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105
Gang Tian

xi


Chapter 1

The Differentiable Sphere Theorem (After
S. Brendle and R. Schoen)
Gérard Besson

Abstract In these notes we describe a major result obtained recently using the
Ricci flow technique in the context of positive curvature. It is due to S. Brendle and
R. Schoen and states that a strictly 1/4-pinched closed manifold carries a metric
of constant (positive) sectional curvature. It relies on a technique developed by
C. Böhm and B. Wilking who obtained the same conclusion assuming that the
manifold has positive curvature operator. The maximum principle applied to the
Ricci flow equation leads to studying an ordinary differential equation on the space
of curvature operators.

1.1 Introduction

In 1951, Rauch [38] showed that a complete Riemannian manifold whose sectional
curvature is positive and varies between two numbers whose ratio is close to 1 has
a universal cover homeomorphic to a sphere. Precisely, for ı > 0, we say that a
closed Riemannian manifold M is pointwise ı-pinched if at each point the ratio
between the smallest sectional curvature and the largest one is greater or equal to
ı. We say that it is ı-pinched if this inequality is satisfied by the ratio between
the global minimum (on M) of the sectional curvature and its global maximum.
Finally, it is said to be strictly ı-pinched (pointwise or globally) if this inequality
is strict. These notions make sense whether the curvature is positive or negative.
In the sequel all Riemannian manifolds will be positively curved; for results in the
negative setting the following articles [26, 40] can be consulted. Rauch’s Theorem
then asserts that a ı-pinched simply connected Riemannian manifold, for an explicit
value of ı close to 1, is homeomorphic to a sphere. On the other hand a standard
computation (see [20, Sect. 3.D.2, p. 149]) shows that the complex projective spaces,
of complex dimension greater than 1, have a sectional curvature varying between 1
and 4, for a suitable normalization; this is also true for the other closed Riemannian
symmetric spaces of rank one. Rauch’s Theorem then cannot be true for a (non
strictly) 1=4-pinched Riemannian manifold of positive curvature. It was Berger [1]
G. Besson ( )
Université Grenoble Alpes – Institut Fourier, CS40700, 38058 Grenoble Cedex 9, France
e-mail:
© Springer International Publishing Switzerland 2016
R. Benedetti, C. Mantegazza (eds.), Ricci Flow and Geometric Applications,
Lecture Notes in Mathematics 2166, DOI 10.1007/978-3-319-42351-7_1

1


2

G. Besson

and Klingenberg [28], in the 1960’s, who got the optimal result: a simply connected
Riemannian manifold which is strictly 1=4-pinched is homeomorphic to a sphere.
The reader is referred to [4] (Sect. 12.2.2.1, p. 552) for a description of the technique
and some historical notes.
Let us remark that these results do not exclude the exotic spheres. It is indeed an
interesting question to investigate whether an exotic sphere can carry a Riemannian
metric of positive curvature. This question is still open. This text grew out of a series
of lectures given at a CIME conference in Cetraro, Italy. The goal is to describe the
following remarkable result due to S. Brendle and R. Schoen.
Theorem 1.1 (Brendle and Schoen [9, 10]) Let M be a Riemannian n-manifold of
positive sectional curvature and pointwise strictly 1=4-pinched, then M carries a
metric of constant sectional curvature. Hence it is diffeomorphic to the quotient of
a sphere by a finite subgroup of O.n/.
In particular a consequence is that no exotic sphere can carry a strictly 1=4pinched metric. The method relies on the use of the Ricci flow introduced by
Hamilton in [21]. The idea is to construct a deformation of the Riemannian metric
which evolves towards a constant curvature metric. This ideal behavior only occurs
when the metric one starts with has nice properties. Let us recall that the seminal
article by R. Hamilton has had extraordinary developments which culminated in
Perelman’s works ([34–36], and also [5]) proving the geometrization conjecture. In
[21] Hamilton shows, using the same method, the following theorem.
Theorem 1.2 (Hamilton [21]) Let M be a closed 3-dimensional Riemannian
manifold which carries a metric of positive Ricci curvature, then it also carries
a metric of positive constant curvature.
Later on, the same technique extended to 4-manifolds was used to get the results
proved in [22, 25].
It is a remarkable extension to higher dimensions which is at the core of the
works we intend to describe and it is due to Böhm and Wilking [6]. We shall say that
a curvature operator is 2-positive if the sum of its smallest eigenvalues is positive.

This notion first appeared in the article [12] of Chen.
Theorem 1.3 (Böhm and Wilking [6]) Let M be a closed Riemannian manifold
whose curvature operator is 2-positive, then M carries a constant curvature metric.
Using analytical methods to prove results such as Theorem 1.1 is not new. The
theory of harmonic maps is used in [30] whereas the Ricci flow is used in [22,
27, 29, 33]. Note that in [29], Margerin studies a natural curvature condition and
proves an optimal result. All this is part of the so-called Geometric Analysis and the
maximum principle for parabolic systems is a crucial tool which allows to reduce
the problem to an algebraic question and, in general, answers the question: what are
the properties of the curvature operator which are preserved under the Ricci flow?
In this text we shall limit ourselves to the dimensions greater than 3. After
recalling the different curvature notions, we shall describe the main steps of the
proof. The first one consists in using the above-mentioned maximum principle in


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

3

order to reduce the problem to the study of an ordinary differential equation on the
space of curvature endomorphisms. In the second step we shall exhibit inequalities
satisfied by the curvature operator which are invariant under this dynamical system.
Finally, standard geometric arguments allow to conclude. General references for the
basics of Riemannian Geometry are [4, 20] for an overview and [14–16] for the Ricci
flow. These last references are exhaustive and C. Böhm and B. Wilking’s method is
described in details in [16, Chap. 11]. This text is then to be used as a guide for the
reading of the original articles as well as these references. We certainly advise the
reader to study the recent survey by S. Brendle and R. Schoen [11].
I wish to warmly thank C. Böhm, S. Brendle, L. Ni and H. Seshadri for their
answers to my very naive questions as well as M. Berger, P. Bérard, L. Bessières,

J-.P. Bourguignon, Z. Djadli, S. Gallot, H. Nguyen and T. Richard for fruitful
discussions.

1.2 Basics of Riemannian Geometry
In what follows .M; g/ is a closed Riemannian manifold. We shall denote the metric
either by g or < :; : >. The Levi-Civita covariant derivative is denoted by r. Let X,
Y, Z and T be vector fields, we define the .0; 4/-curvature tensor by (see [20])
R.X; Y; Z; T/ D< rY rX Z

rX rY Z

rŒY;X Z; T > :

It is known that this tensor is skew symmetric in X and Y and in Z and T and
satisfies R.X; Y; Z; T/ D R.Z; T; X; Y/. It defines a symmetric endomorphism of
2
.TM/ that we shall also denote by R and which is called the curvature operator.
The convention used in this text are similar to those in [20], the reader is welcome
to compare them to those in [16]. The curvature tensor also satisfies the first Bianchi
identity, which is analogous to the Jacobi identity satisfied by the Lie bracket of a
Lie algebra. Precisely, it reads
R.X; Y; Z; T/ C R.Y; Z; X; T/ C R.Z; X; Y; T/ D 0 :
The sectional curvature of a 2-plane P

Tm M tangent at m to M is

K.P/ D< R.x ^ y/; x ^ y > ;
where .x; y/ is an orthonormal basis of P. We note that K is the value of R computed
on the decomposed vector whereas R is defined on the whole 2 .TM/ .
If fei g is an orthonormal basis of Tm M, we endow 2 .Tm M/ with the scalar

product such that fei ^ ej ginotations Rijkl D< R.ei ^ ej /; ek ^ el >. The Ricci curvature at m is a symmetric
bilinear form on Tm M that we also think of as a symmetric endomorphism of Rn


4

G. Besson

(we shall give the same name to these two objects). It is defined by
Ricij D Ric.ei ; ej / D< Ric.ei /; ej >D

X

Rikjk :

k

It is a metric-like tensor. Finally, the scalar curvature is the trace of the Ricci
curvature, that is, for m 2 M
X
X
scal.m/ D
Ricii D
Rikik D 2 trace R :
i

i;k

With these conventions the curvature operator of the standard sphere is the identity,

its sectional curvatures are all equal to 1, its Ricci curvature is .n 1/g and its scalar
curvature is constant equal to n.n 1/.

1.2.1 Algebraic Curvature Operators
For a point m 2 M the choice of an orthonormal basis fei g of Tm M gives
an identification with the Euclidean space Rn . The scalar product also gives an
identification of 2 .Rn / with the Lie algebra so.n/ by associating to the unit vector
ei ^ ej the rank 2 endomorphism which is the rotation of angle =2 in the plane
generated by ei and ej . Via this identification one has < A; B >D 1=2 trace.AB/,
for A; B 2 2 .Rn /. The space of symmetric endomorphisms (which we identify
with the symmetric bilinear forms) of 2 .Rn / is denoted by S 2 .so.n//. This
space encodes the first three relations satisfied by the curvature tensor. We call
algebraic curvature tensor an element in S 2 .so.n// which furthermore satisfies
the first Bianchi identity; The space of algebraic curvature operators is denoted by
SB2 .so.n// (see [16, p. 81]).

1.2.2 Algebraic Products on S 2 .so.n//
Let A and B be two symmetric endomorphisms of Rn , we define a new symmetric
endomorphism of so.n/ ' 2 .Rn / by (see [16, p. 74]) :
.A ^ B/.v ^ w/ D

1
.A.v/ ^ B.w/ C B.v/ ^ A.w// :
2

It is easily checked that A ^ B 2 S 2 .so.n// and satisfies the following equality
A ^ B D B ^ A.


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

5

Let now R and S be two tensors in S 2 .so.n//, we define R]S 2 S 2 .so.n// by:
< .R]S/.h/; h >D

1X
< ŒR.!˛ /; S.!ˇ ; h > : < Œ!˛ ; !ˇ ; h > ;
2
˛;ˇ

where h 2 so.n/ and f!˛ g is an orthonormal basis of so.n/. It is easily checked that
this definition is independent of the choice of the basis and that R]S D S]R (see [6]
and [16, p. 72]). Finally we shall denote by R] D R]R. This product which has been
introduced by Hamilton (see for example [22]) will play a key role in the sequel.

1.2.3 Irreducible Components Under the Action of O.n/
The group O.n/ acts by changes of basis on the space of algebraic curvature
operators. One can decompose the space SB2 .so.n// in three irreducible components
under this action. For R a curvature operator we denote by Ric0 the traceless part of
its Ricci tensor, that is
< Ric.R/.ei /; ei >D

n
X

Rikik

and

Ric0 .R/ D Ric.R/

kD1

scal.R/
I Rn ;
n

where scal.R/ D 2 trace.R/. One then has
SB2 .so.n// D RIS 2 ˚ < Ric0 > ˚ < W > :
B

The first factor consists of algebraic curvature operators which are multiple of the
identity, the second of multiple of operators of the type A^IRn where A is a traceless
symmetric operator acting on Rn and the third is the kernel of the map R ! Ric.R/.
This last space contains the Weyl curvature tensors, which among the components
of R is the most difficult to understand. Let us recall that the Weyl component of
the curvature tensor of a Riemannian metric vanishes, if and only if it is locally
conformally flat. For a curvature tensor R we denote by RI , RRic0 and RW its various
components and by I the identity in SB2 .so.n// as well as in Rn . We then have (see
[16, p. 87] for the details),
RD

scal.R/
2
IC
Ric0 .R/ ^ I C W :
n.n 1/
n 2

6

G. Besson

1.2.4 A New Identity
Böhm and Wilking establish in [6] a new identity satisfied by any algebraic
curvature tensor. It is remarkably simple.
Proposition 1.4 ([6]) For all R 2 SB2 .so.n// one has
R C R]I D .n

1/RI C

2

n
2

RRic0 D Ric ^I :

Below we shall deduce from it the decomposition along the irreducible components
of the quadratic expression in R, which appears in the evolution equation (along the
Ricci flow) of the curvature operator. It is remarkable that new simple identities on
this well-studied space can still be discovered.

1.2.5

O.n/-invariant Endomorphisms of SB2 .so.n//

Let ` be a self adjoint linear map of SB2 .so.n// into itself which is invariant under

the action of O.n/. It is diagonalizable and its eigenspaces are O.n/ invariant; it
then preserves the above irreducible components. It is furthermore a multiple of the
identity on each of them (see [16, p. 89]). In [6] the authors consider such maps
which preserve the Weyl component. It can be written as follows,
`a;b .R/ D RC2.n 1/aRI C.n 2/bRRic0 D .1C2.n 1/a/RI C.1C.n 2/b/RRic0 CRW ;

where a and b are real numbers.

1.3 The Ricci Flow
It is an ordinary differential equation on the space of Riemannian metric introduced
by Hamilton in the seminal article [21]. We consider a family g.t/ of smooth
Riemannian metrics, depending smoothly in t, and solving the following Cauchy
problem:
8
< @g D 2 Ric
g.t/
@t
:
g.0/ D g0


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

7

There is a normalised version for which the volume of the evolving metric is
constant; indeed, it suffices to replace the first line of the above equation by:
@g
1
2

D 2 Ricg.t/ C
@t
n vol.M; g.t//

Z

Á
scal.x; t/dvg.t/ .x/ g.t/:

M

The reader is referred to [5, 14] for details of this theory. Classical results show that
the solutions exist for small time for any smooth initial data.
The idea is now to show that under the hypothesis of Theorems 1.1 and 1.3 the
normalized Ricci flow converges towards a constant curvature metric.

1.3.1 The Evolution Equations for the Curvatures
A standard computation gives the evolution equation of the various curvature
tensors. We shall limit ourselves to the scalar curvature and the curvature operator.
We have,
@ scal
C
@t

g.t/

scal D 2j Ricg.t/ j2g.t/

were g.t/ is the Laplace operator acting on functions for the metric g.t/. The
convention is here the one used in geometry (in dimension 1, it is d2 =dx2 ). The

norm of the Ricci tensor is computed with the metric g.t/.
Similarly one has
@R
C
@t

R D 2.R2 C R] /:

We use here the rough Laplacian, that is the opposite of the trace of the second
covariant derivative of the tensor R. The notation R2 denotes the square of the
endomorphism R and R] the quadratic expression defined in the previous section.
The evolution of the curvature tensor is written in this simple way thanks to a trick
due to K. Uhlenbeck that we shall describe below.

1.3.2 The Maximum Principle
It is the key tool for the study of the solutions of the heat equation. Here, we
shall give a vector version that is adapted to our parabolic system. This is done
by Hamilton in [21, 22]. The reader is referred to [14, 16].
Let us consider a PDE of the type @s=@t C t s D f .s/ and the ODE ds=dt D f .s/.
The above PDE are so-called reaction-diffusion equations, the diffusion part is given


8

G. Besson

by the Laplacian; if f Á 0, it is a heat equation which “spreads” the initial condition.
The non-linear term f .s/ is the reaction term which, alone, leads to a blow up in finite
time (convergence of certain norms towards C1). The main question is to know
which term will win, reaction or diffusion. The maximum principle is a comparison

between the solutions of the PDE and the solutions of the ODE.
The equation satisfied by the scalar curvature can be written @R=@t C R 0.
The simplest maximum principle leads to the fact that the minimum of the scalar
curvature does not decrease along the Ricci flow.
We now describe a vector version. Let M be endowed with a smooth family of
metrics g.t/, for t 2 Œ0; T, and let W E ! M be a vector bundle endowed with a
fixed metric and a smooth family of compatible connections, rt . These datas allow
to define a Laplace operator acting on sections of E , depending on t that we simply
shall denote by . Let us now consider a smooth function f W E Œ0; T ! E
such that, for given t, f .:; t/ preserves the fibres. Let K be a closed subset of E that
is assumed to be invariant under the parallel transport of rt , for all t 2 Œ0; T, and
such that Km D K \ 1 .m/ is closed and convex. The key hypothesis is a relation
between K and the ODE du
dt D f .u/, defined on each fibre Em of E ; we assume that
any of its solution u such that u.0/ 2 Km remains in Km for all t 2 Œ0; T.
Theorem 1.5 ([21, 22] or [14, Theorem 4.8]) Under the above hypothesis, let s.t/
be a solution of the PDE,
@s
C
@t

s D f .s/

such that s.o/ 2 K , then, for all t 2 Œ0; T, s.t/ 2 K .
In order to apply this result to the curvature operator we notice that, although
the metric on the bundle 2 .T M/ depends on t, a trick, due to K. Uhlenbeck (see
below and [14, Sect. 6.1]), allows to convert the situation to a fixed metric on a fixed
bundle with however a time dependent connection.
The set K is thought of as a geometric version of the inequalities satisfied by
the curvature operator such as the positivity.

Example 1.6 Let us recall the 3-dimensional situation, for which the maximum
principle has played an important role in G. Perelman’s works; it indeed shows
that the scalar curvature controls the full curvature tensor. At each point m 2 M and
t 2 Œ0; T the endomorphism R diagonalises and has three eigenvalues denoted by
.m; t/
.m; t/
.m; t/. In dimension 3 these numbers are sectional curvatures.
One shows that the two tensors R2 and R] diagonalise in the same basis and
have eigenvalues respectively equal to . 2 ; 2 ; 2 / and . ; ; /. The ordinary


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

9

differential equation is thus,
8
d
ˆ
ˆ
< dt D
d
D
dt
ˆ
ˆ
:d D
dt

2

C

2

C

2

C

In general we have to consider the ODE
dR
D R2 C R] D Q.R/ ;
dt
defined on SB2 .so.n// (we omit the factor 2 which does not play any role).

1.3.3 Constructing K
It is now useful to recall K. Uhlenbeck’s trick which allows to reduce to a fixed
vector bundle. Let us consider a Ricci flow .M; g.t// defined on an interval Œ0; T/.
One can construct a time dependent vector bundle isometry
Ã.t/ W .TM; g.0// ! .TM; g.t//
solving the equation
dÃ
.t/ D Ricg.t/ ıÃ.t/
dt

and Ã.0/ D id ;

where Ric is here to be understood as an endomorphism. It allows to pull back the

whole situation to a fixed vector bundle endowed with a fixed metric. In particular,
the pulled-back Levi-Civita connection depends on t. This isometry extends to all
natural vector bundle such as the bundle of curvature tensor E whose typical fibre is
isometric to SB2 .so.n//. Then, considering Ã.t/ .Rg .t// we can write the evolution
equation of the curvature operator on this bundle and then apply the above maximum
principle.
In order to construct K in E we proceed as follows. Let us consider a closed
convex set F
SB2 .so.n//. We furthermore assume that F is O.n/-invariant. We
can then transport F on each fibre Em by the identification given by the choice
of an orthonormal basis of Tm M; the image set is denoted by Km . The O.n/invariance of F guaranties that this construction does not depend on the chosen
basis. Similarly the parallel transport associated to the time dependent connection
induces an isometries between the fibres of K ; the invariance of F by the O.n/
action ensures the invariance of K by the parallel transport of all these connections.
If furthermore F is invariant by the ODE, the set K obtained satisfies the hypothesis


10

G. Besson

of Theorem 1.5. Such a set transcribes curvature conditions which are preserved by
the flow.

1.4 C. Böhm and B. Wilking’s Method
The question is now reduced to the study of the quadratic term Q.R/, the right hand
side of the ODE, which we view as the value at R of a vector field on SB2 .so.n//.
A flow invariant curvature condition can be defined by a closed convex and O.n/invariant cone, for example the cone C of 2-nonnegative curvature operators (the
sum of their two lowest eigenvalues is nonnegative). In order to show the invariance
of the condition by the ODE we have to show that if a trajectory of this ODE starts

inside the cone and reaches its boundary, the vector field pushes it back inside, or at
least keep it on the boundary. We have then to show that this vector field Q.R/ is in
the tangent cone TR C to C at each point of @C. Let C be such a cone, invariant by the
ODE, one can built others by taking the image of C by the maps `a;b defined above,
for suitable choices of a and b. The linear maps `a;b sends boundary into boundary
and tangent cones into tangent cones, hence the set `a;b .C/ is invariant by the ODE,
that is C is invariant by `a;b1 ı Q ı `a;b , if
Xa;b .R/ D `a;b1 ..`a;b .R//2 C .`a;b .R//] / ;
is in the tangent cone TR C to C at each point of @C. As it is the case for Q.R/ it
suffices, by convexity of C, to prove it for Da;b .R/ D Xa;b .R/ Q.R/. The maps `a;b
preserves the Weyl component and consequently we show that Da;b .R/ does depend
only on the Ricci component of the curvature operator R. The formulae then become
much simpler and the computation is possible.
Thanks to this enlightening idea C. Böhm and B. Wilking built families of convex
cones C.s/s2Œ0;1 , O.n/-invariant, invariant by the ODE whose intersection is reduced
to the curvature operator I, where > 0, that is the curvature operator of the
round sphere up to normalisation. If the theory works, the trajectories of the PDE
are “trapped” in this family and constrained to converge towards the curvature
operator of the round sphere. The underlying idea is to try to obtain Lyapunov
functions associated to the ODE (projected on the unit sphere of SB2 .so.n//) whose
minimum would be the curvature operator of a round sphere in order to prove
that the trajectories converge towards this fixed point. Such a function is not really
necessary; indeed, only its level sets play the key role. and they are not necessarily
included in one another but may be used in the same way.


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

11

1.4.1 Properties of Q.R/ and Da;b .R/
Below is a list of properties satisfied by these quantities; for more details the reader
is referred to [6, 16].
For R 2 SB2 .so.n// one shows that Q.R/ 2 SB2 .so.n// (see [16, p. 88]). It was
observed by G. Huisken that the ODE under consideration is the gradient flow of
P.R/ D

1
trace.R3 C RR] / ;
3

which is unfortunately too difficult to study to play the role of a Lyapunov
function. Finally, one can compute the components of Q.R/ in the decomposition
in irreducible components. That is where the new identity is used. The following
theorem is the key tool for proving the invariance of the cones `a;b .C/.
Theorem 1.7 (Böhm and Wilking [6]) For a and b real numbers,
Da;b .R/ D ..n
C

2/b2

2.a

b// Ric0 ^ Ric0 C2a Ric ^ Ric C2b2 Ric20 ^I

trace.Ric20 /
.nb2 .1
n C 2n.n 1/a

2b/

2.a

b/.1

2b C nb2 //I :

The remarkable fact, to be expected, is that Da;b .R/ does not depend on the Weyl
component of R. For example, one can compute its eigenvalues in terms of those of
Ric. It suffices to choose and orthonormal basis of Rn diagonalising Ric, we then
deduce an orthonormal basis of so.n/ in which Da;b .R/ diagonalises. Its eigenvalues
then depend on the choice of a and b.

1.4.2 Construction of the Family of Cones
p

2/C4 2
Let us recall that n 4. We set bN D 2n.n
and a0 .b/ D b C .n 2 2/ b2 . These
n.n 2/
N The following propositions
numbers are such that Da0 .b/;b is positive for b 20; b.
are the starting point of the construction (see [6] and [16, p. 113]) :

Proposition 1.8 (Hamilton [24]) The cone C of 2-positive algebraic curvature
operators is preserved by the ODE.
This result is also true in dimension 3. In this case the hypothesis says that the
Ricci curvature is positive (see [22]). Let f!i g be an orthonormal basis of so.n/ of
eigenvectors R. In order to show that the cone is preserved we have to show that
< Q.R/!i ; !i > C < Q.R/!j ; !j > 0 for all i < j such that

< R!i ; !i > C < R!j ; !j >D 0I
that is, that the vector field Q.R/ forces the trajectories back inside the cone. We
here use the precise description of Q.R/.


12

G. Besson

1.4.2.1 First Step
Proposition 1.9 Let C be the cone of 2-nonnegative curvature operators for n
N then one has the following properties:
and b 20; b,

4

(i) Cb D `a0 .b/;b .C/ is preserved by the ODE.
(ii) For b > 0, the vector field Q.R/ is transverse to the boundary of `a0 .b/;b .C/ at
points R ¤ 0.
(iii) `a0 .bN /;bN .C n f0g/ is included in the cone of positive curvature operators.
We use the description of Da;b .R/ to prove this proposition following the scheme
described above. We grosso modo show that Da;b .R/ is nonnegative. This produces
a first family of cones that links C to a cone included in the set of positive curvature
operators.

1.4.2.2 Second Step
Following the same scheme and similar proofs we now construct the following
family. For b 2 Œ0; 1=2 set
.n 2/b2 C 2b
2 C 2.n 2/b2

a.b/ D

and p.b/ D

.n 2/b2
:
1 C .n 2/b2

Then,
C0b D `a.b/;b

˚
R 2 SB2 .so.n//j R

0; Ric

p.b/

trace.Ric/ «Á
n

is a O.n/-invariant closed convex cone which is furthermore ODE-invariant [6,
Lemma 3.4]. It links the cone of nonnegative operators to C01 ; 1 .
2 2

1.4.2.3 Third Step
Finally, for b D 1=2 and s
a.s/ D

0 we set
1Cs
2

and p.s/ D 1

4
:
n C 2 C 4s

We then define,
C00s D `a.s/; 1
2

˚
R 2 SB2 .so.n//j R

0; Ric

p.s/

trace.Ric/ «Á
n


1 The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

13

which has the same properties than before. Let us notice that

1
`a.s/; 1 .R/ D 2.n
2
s!C1 a.s/
lim

1/RI ;

consequently the family C00s converges towards RC I when s goes to infinity. These
two last steps consist in adding to the construction of convex invariant sets a
pinching of the Ricci curvature. Indeed, the inequalities which define these families
yield a control of the Ricci curvature when the curvature operator is normalised so
that its largest eigenvalue is 1, for example.
It then suffices to concatenate the three families: Cb , C0b \ CbN and C00s \ CbN to
obtain, after a suitable change in the parameters, a collection of O.n/-invariant
closed convex cones, invariant by the ODE which links the 2-nonnegative curvature
operators to the multiples of the identity. We shall denote this family by C.s/s2Œ0;1 .
Such a family is called a pinching family in [6].

1.4.3 Pinching Set
The above construction is not really sufficient to obtain the desired conclusion, that
is that the solution of the PDE converges, after rescaling, towards the curvature
operator of the round sphere. Indeed, it could happen that the curvature operator
goes to infinity in these cones without getting closer to the multiples of the identity.
Somehow these cones are too widely opened at infinity. We then construct from
C.s/ a set F which is called a pinching set. This notion was introduced by Hamilton
in [22] and C. Böhm and B. Wilking describe a suitable generalisation.
Theorem 1.10 (Böhm and Wilking [6]) For 20; 1Œ and h0 > 0, there exists a
O.n/-invariant closed convex set F SB2 .so.n// such that
(i) F is preserved by the ODE,

(ii) C. / \ fR W trace.R/ Ä h0 g F C. /,
(iii) the closure of F n C.s/ is compact for all s 2 Œ ; 1Œ.
This theorem is shown by proving that the intersection of all O.n/-invariant
closed convex sets which satisfy .i/ and .ii/ works. In [6] a more general result
is proved which can be applied to a wide variety of families of cones. Intuitively
one can think of the set F as a parabola that crosses all the cones centred at the
origin; the asymptotic cone of such a set is reduced to the vertical axis. Going at
infinity in this set thus forces to get closer and closer to it (after renormalisation).
This set can now be “copied” in the fibers of the bundle of curvature operators
on M as described before in order to built a set F invariant by parallel transport.