Global rigidity in CR geometry: the Schoen-Webster theorem.

HAL Id: hal-00171879

https://hal.archives-ouvertes.fr/hal-00171879

Submitted on 13 Sep 2007

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

theorem.

Benoît Kloeckner, Vincent Minerbe

To cite this version:

Benoît Kloeckner, Vincent Minerbe. Global rigidity in CR geometry: the Schoen-Webster theorem..

Differential Geometry and its Applications, Elsevier, 2009, 27 (3), pp.399-411. �hal-00171879�

hal-00171879, version 1 - 13 Sep 2007

SCHOENWEBSTER THEOREM

Benoît KLOECKNER

1

and Vin ent MINERBE 2

Abstra t. S hoen-Webster theoremassertsapseudo onvexCR

manifold whose automorphism group a ts non properly is either

thestandardsphereortheHeisenbergspa e. Thepurposeof this

paper is to survey su essive works around this result and then

provideashortgeometri proofin the ompa t ase.

Keywords: CRgeometry,rigidity.

MS lassi ation numbers: 32V05, 32V20, 53C24.

Among the many aspe ts of geometri rigidity, the vague prin iple

a ordingtowhi hagivengeometryisrigidwhenfewmanifoldsadmit

a large automorphism group has a fairly ri h history. In this survey

paper, wetrytoshowhowstri tlypseudo onvexCRgeometryts into

this on ept of rigidity.

André Li hnerowi z rst raised the question in the onformal ase.

It is well known that the isometry group of a ompa t Riemannian

manifold is ompa t, due to the ompa tness of the group

O(n)

(see Se tion 1.2). Sin e the orresponding group

CO(n)

of onformal ge-ometry is not ompa t, one might expe t some ompa t manifolds to

have non ompa t onformal groups. There is a simple example: the

Eu lidean sphere has onformal group

SO(1, n + 1)

. The Li hnerow-i z onje ture stating that there are no other examples was settled in

the early seventies by Ja queline Ferrand [LF71℄ and in a weak form

by Morio Obata 3

[Oba72℄; it was extended by Ferrand a while later

[Fer96℄.

A fewyears afterObataand Ferrand'sworks,itappearedthat

Li h-nerowi z onje ture was not spe i to onformal geometry: Sidney

Websterextended partsofthe proofofObatainthesettingof(stri tly

pseudo onvex) CR geometry[Web77℄. The question raised a lotof

in-terestagaininthenineties,severalmathemati ianstryingtoworktheir

way out from Webster's result to the full statement. Ri hard S hoen

1

bkloe knfourier.ujf-grenoble.fr, Institut Fourier, 100rue des Maths, BP 74,

38402StMartind'Hyères,Fran e. 2

minerbe-vuniv-nantes.fr, Laboratoirede mathématiques Jean Leray,

Univer-sitédeNantes,2ruedelaHoussinière,BP92208,44322Nantes, edex3,Fran e.

3

It appeared later that the proof was awed at some point, but Ja ques

gavetherst ompleteproof,usingoriginalanalyti methodsrelatedto

the Yamabeproblem[S h95 ℄. Infa t,he gaveaproofinthe onformal

ase that adaptsto CRgeometry and obtained the followingresult. 4

Theorem (S hoenWebster)  Let

M

bea stri tly pseudo onvexCR manifold, not ne essarily ompa t. If its automorphism group

Aut(M)

a ts non-properly, then

M

is either the standard CR sphere

S

or

S

with one point deleted.

Let us re all that ana tion of a topologi al group

G

is proper if for any ompa t subset

K

of

M

, the subset

G

K

= {g ∈ G; g(K) ∩ K 6= ∅}

of

G

is ompa t. In parti ular if

M

is ompa t,

Aut(M)

a ts properly if and onlyif it is ompa t.

Thepaperisorganisedasfollows. Therstse tionisdevotedto

pre-liminaries, in luding CR geometry, two properties that are important

in the sequel and

(G, X)

-stru tures. We then survey the su essive works on the S hoenWebster Theorem, trying to give for (almost)

ea hresulttheavoroftheproofwithoutgettingintotoomu hdetail.

The word proof will therefore often be followed by quite impre ise

arguments. Thelastse tion isdevoted toa leanedgeometri proofof

the theorem when

M

is ompa t, based onsome of the ideas exposed. Before getting started, let us point out that Bun Wong proved a

very lose theorem for domainsof

C

n+1

[Won77℄. Many developments

arose from his result and parts of the S hoenWebster Theorem an

be dedu ed from this work. Indeed, unless

n = 1

, a ompa t stri tly pseudo onvex CR manifold

M

2n+1

an always be embedded as the

boundaryofadomainof

C

n+1

anditsautomorphisms an beextended

to automorphismsof the domain. See [Lee96℄for detailsdue toDaniel

Burns.

However, wewillnotdis uss Wong's theoremand itsimprovements.

First, it annot be of any help for the least dimensional ase. Se ond,

weareinterestedinmoreintrinsi methodsofproof,independantofany

embedding. For further informations on this topi , the reader should

refer to[Won03℄.

Forthesakeof ompleteness, notethatin[Pan90 ℄PierrePansugave

a hint of howone ould try to adaptFerrand's proofto the CR ase.

1. Preliminaries

1.1. Basi s of CR geometry. We only give a glimpseon CR

geom-etry. The interested reader an referto [D'A93℄or [Ja 90℄.

Given a

2n + 1

-dimensional manifold

M

, a CR stru ture on

M

is a ouple

(ξ, J)

where:

4

(1)

ξ

is a

2n

-dimensionalsubbundle of

T M

, (2)

J

is a pseudo omplex operatoron

ξ

:

J

x

: ξ

x

→ ξ

x

,

J

x

2

= −

Id

∀x ∈ M,

(3) forallve torelds

X

,

Y

tangentto

ξ

,theve toreld

[JX, Y ]+

[X, JY ]

istangentto

ξ

and thefollowingintegrability ondition holds:

J([JX, Y ] + [X, JY ]) = [JX, JY ] − [X, Y ].

Anysmooth hypersurfa e

H

ina omplexmanifold

X

admitsa nat-ural CR-stru ture: denoting by

J

the omplexstru tureof

X

,one an dene

ξ = T H ∩ J(T H)

so that

J

a ts on

ξ

; note that the vanishing of the Nijenhuistensor implies the integrability ondition.

A dierentiable map between two CR manifolds is a CR map if it

onjugates the hyperplanes distributionsand the pseudo omplex

oper-ators. An automorphism of a CR manifold

M

is a dieomorphism of

M

that is a CR map. The group of those is denoted by

Aut(M)

and its identity omponent by

Aut

0

(M)

.

1.1.1. Calibrations,theLeviformandthe Webstermetri . GivenaCR

stru ture

(ξ, J)

on

M

, a (possibly lo al)

1

-form

θ

su h that

ξ = ker θ

is alled a(lo al) alibration. One an alwaysnd lo al alibrations. If

M

is orientable, one an always nd a global alibration. However, a alibration need not bepreserved by automorphisms.

From now on, all the manifolds under onsideration are assumed

to be onne ted and orientable; all the alibrations are assumed to be

global.

Given a alibration

θ

, one denes on

ξ

the Levi form:

L

θ

(·) = dθ(·, J·).

As a onsequen e of the integrability ondition, the Levi form is a

quadrati form.

A hange of alibrationindu es alinear hange in the Levi form:

(1)

L

λθ

= λL

θ

,

thusitssignature is,up toa hange of sign, aCR invariant.

A CR stru ture is said to be stri tly pseudo onvex if its Levi form

is denite (and then we hoose our alibrations so that it is positive

denite). It implies that

dθ

is nondegenerate on

ker θ

, that is

θ

is a onta tform. IftheLeviformvanishesatea hpoint,theCRstru ture

issaidtobeLevi-at. Then

dθ

iszeroon

ξ

andtheFrobeniusTheorem shows that

ξ

denes a foliation.

One should therefore not think of CR geometry as one geometry

: ea h signature of the Levi form orresponds to a geometry of its

Ker dθ

(θ = 1)

ξ

(θ = 0)

π

X

Figure 1. The Reeb ve tor eld of a alibration

Given a alibration

θ

onastri tlypseudo onvexCRmanifold,there isasingleve toreld

X

, alledtheReeb ve tor eldof

θ

,thatsatises:

(2)

θ(X) = 1

and

X y dθ = 0.

See Figure1.

Denote by

π : T M → ξ

thelinearproje tionon

ξ

alongthedire tion of

X

. IftheLeviformispositivedenite,onegetsaRiemannianmetri on

M

alledthe Webstermetri :

(3)

W

θ

= L

θ

◦ π + θ

2

_.

A hange of alibration

θ

′

_{= λθ}

hangesthe metri by afa tor

λ

along

ξ

and by a fa tor

λ

2

transversally that is, on the quotient

T M/ξ

. Therefore, the Webster metri does not dene a anoni al onformal

stru ture ona CRmanifold.

Note that if

M

has dimension

2n + 1

, the alibration

θ

denes a volume form

θ ∧ dθ

n

whi his ompatible with the Webster metri .

1.1.2. The Webster s alar urvature and the pseudo onformal

Lapla- ian. There is also a natural metri onne tion

∇

θ

on

T M

, the so alled Tanaka-Webster onne tion; beware its torsion

Tor

θ

does not vanish in general. Contra ting the urvature

Rm

θ

of this onne tion along

ξ

, we obtain a s alar urvature

R

θ

. A subellipti Lapla ian

∆

θ

arises by taking (minus) the tra e over

ξ

of the Hessian orresponding to

∇

θ

; the following integration by parts formulaholds:

∀u, v ∈ C

c

∞

(M),

Z

M

(∆

θ

u)vθ ∧ dθ

n

=

Z

M

L

θ

(du|

ξ

, dv|

ξ

) θ ∧ dθ

n

,

where

ξ

and

ξ

∗

areidentiedthanksto

L

θ

. Tounderstandtherelevan e of this operator, onsider another alibration

θ

′

, whi h we write

θ

u

n

2

θ

for some smooth positive fun tion

u

. The s alar urvature then transforms a ording tothe following law:

R

θ

′

= b(n)

−1

u

−

n+2

n

L

θ

u

where

b(n) =

n+1

4n+2

and

L

θ

= ∆

θ

+ b(n) R

θ

.

This formula is pretty similar to a onformal one. Indeed, given

onformally equivalent metri s

g

and

h = u

4

n−2

g

, for some positive fun tion

u

, the Riemannian s alar urvatures of

g

and

h

are related the same kind of formula, where

b(n)

should be repla ed by

n−2

4(n−1)

and

L

θ

by the onformal Lapla ian (and

∆

θ

by the Lapla e-Beltrami operator). Thisanalogyturnsouttobeverye ient: itisthe keyidea

behindS hoen's proof (see Se tion4).

1.1.3. The at models. The standard CR sphere

S

2n+1

(we will often

omit the supers ript) is the unit sphere on

C

n+1

:

S =

n

(z

0

, . . . , z

n

) ∈ C

n+1

;

X

|z

k

|

2

= 1

o

endowed with the orresponding CRstru ture. It is a stri tly

pseudo- onvexCRmanifold;itsautomorphismgroupis

Aut(S) = PU(1, n+1)

, a nitequotientof

SU(1, n + 1)

. It is non ompa t, onne tedand a ts transitively on

S

.

TheHeisenberggroupistheCRnon ompa tmanifold

H

obtainedby removingonepointof

S

. Itisthereforedieomorphi totheEu lidean spa e

R

2n+1

. Its automorphism group is the stabilizerof the removed

pointin

Aut(S)

,ita tsnonproperlyandtransitivelyandis onne ted. These twoCRmanifoldsare homogeneousand obviouslylo ally

iso-morphi ; they are referred to as the at models. They play the role

of the Eu lidean spa e in Riemannian geometry, or of the sphere and

Eu lidean spa e in onformal geometry.

Forinstan e, there are lo al normal oordinates inany Riemannian

manifold, where the metri is very lose a Eu lidean one. There is an

analogous lo al model for alibrated stri tly-pseudo onvex manifolds:

[JL89℄ provides lo al normal oordinates in whi h the geometry is

lose tothat of the Heisenberg group

H

. In the Riemannian ase, the lo al model (i.e. the Eu lidean spa e) is global for simply onne ted

omplete at manifolds. The followingstatement is the CRanalogue.

Proposition 1.1  A simply onne ted omplete alibrated

stri tly-pseudo onvex CR manifoldwithvanishing urvature and torsionisCR

equivalent to the Heisenberg group.

Let us pre ise what  omplete means. The form

θ

being onta t implies that any two points in

M

an be onne ted by a urve that is everywhere tangent to the onta t distribution. By minimizingthe

M

, we mean balls with respe t to the Carnot distan e. A stri tly pseudo onvex CR manifold is said to be omplete if losed balls are

ompa t.

Definition 1.2  We saythat anopen subset

U

of astri tly pseudo- onvex CR manifold

M

is at if any

x

in

U

has a neighborhoodwhi h is CR isomorphi to an open subset of

S

.

1.2. Finite order rigidity. For a general referen e on lo alrigidity,

see [Kob95℄, Theorems 3.2and 5.1.

Let usstart with the well-known rigidityof Riemannian geometry.

Proposition1.3 ARiemannianmetri onamanifold

M

isrigidto order

1

,thatis: twoisometriesthathavethesamevalueanddierential at somepoint are the same.

In fa tthisresultfollowsfromastrongerstatement. Let

OM

bethe bundle of orthonormal frames on

M

and

Isom(M)

its isometry group. We look at its a tion on

OM

. For ea h element

F

of the total spa e

OM

(

F

is thus the data of a point

x ∈ M

and an orthonormal frame of

T

x

M

), one denes the map

Isom(M)

_{→ OM}

f 7−→ f(F).

Proposition 1.3 asserts that this map is inje tive. In fa t, it is an

embedding and its image is a losed submanifold of

OM

. The group

Isom(M)

, endowed with the orresponding dierential stru ture, is a Lie group.

As a onsequen e, sin ethe bers of

OM

are ompa t, theisometry group of a ompa t Riemannian manifold is ompa t. One even gets:

Corollary 1.4  Let

U

be an open set on a manifold

M

,

K ⊂ U

be a ompa t set with nonempty interior,

g

be a Riemannian metri denedon

U

and

G

be a Lie group a ting on

M

andpreserving

K

and

g

. Then

G

is ompa t.

Nowweturn to the rigidityof stri tly pseudo onvex CR geometry.

Proposition 1.5  Let

M

be a stri tly pseudo onvex CR manifold. Thegroup

Aut(M)

isaLie groupandisrigid toorder

2

, thatis: iftwo automorphisms

f

,

f

′

have the same

2

-jet (the data of their derivatives up to order

2

) at somepoint, then

f = f

′

.

As before, there is a prin ipal bundle on

M

in whi h

Aut(M)

em-beds, but the bers are no longer ompa t and

Aut(M)

an thus be non ompa t even when

M

is ompa t.

As a dire t onsequen e of Proposition 1.5, two CRautomorphisms

of

M

that oin ideon anopen set are the same.

The stri t pseudo onvexity ondition is of primary importan e. For

example, the produ t

S

1

_{× Σ}

is a Leviat CR manifold, and the a tion of the innite-dimensional

dieomorphism group of

S

1

preserves the CRstru ture.

1.3. North-south dynami s. The following result isa ommon

fea-ture of allrank

1

paraboli geometries,that isof boundaries of nega-tively urved symmetri spa es. The standard CRsphere

S

2n+1

isone

of them: itbounds the omplex hyperboli spa e, seenas the unitball

of

C

n+1

. Note that by an unbounded sequen e in a topologi al spa e,

wemean a sequen e that is not ontained inany ompa t set.

Proposition 1.6  Let

(φ

k

)

k

be an unbounded sequen e in

Aut(S)

. There existsa subsequen e,stilldenoted by

(φ

k

)

k

, andtwo points (that may be the same)

p

+

and

p

−

on

S

su h that:

lim φ

k

(p) = p

+

∀p 6= p

−

(4)

lim φ

−1

_k

(p) = p

−

∀p 6= p

+

(5)

and the onvergen es are uniform on ompa t subsets of

S − {p

−

}

,

S − {p

+

}

respe tively.

Moreover if the

φ

k

's are powers of a single automorphism

φ

, then

p

±

are xed point of

φ

. The same result holds for a non ompa t ow, whi h has thus either one or two xed points.

Anunbounded oworautomorphism 5

issaidtobeparaboli ifithas

one xed point, hyperboli if it has two of them. A bounded ow or

automorphism issaid to be ellipti .

Proof. The prin ipleistolook atthe a tionof

Aut(S)

not onlyonthe sphere

S

, but also in the omplex hyperboli spa e it bounds and on the proje tive spa e

C

P

n+1

itis embedded in.

The asewhenthe

φ

k

'sarepowersofanautomorphism

φ

,orthe ase of aow, are simplelinearalgebraresults. They areroughly des ribed

by Figure 2, whi h shows the link between negative urvature of the

hyperboli spa es and north-south dynami s: when a geodesi

γ

is translated, any other geodesi isshrinked toward one of the ends of

γ

. The general ase an bededu edfromthe

KAK

de omposition: ev-eryelement

φ

ofthegroup

Aut(S)

writesdownasaprodu t

φ = k

1

ak

2

where

k

1

,

k

2

areelementsofamaximal ompa tsubgroup

K ⊂ Aut(S)

and

a

isanelementofamaximal non ompa t losedabeliansubgroup

A

. The dimension of

A

is the real rank of

Aut(S)

, namely

1

. More pre isely,

A

orrespondstoahyperboli ow, thatisanon ompa tow with two xed pointon

S

(one attra tive,one repulsive). The general result then follows fromthe ompa tnessof

K

.

5

S

p

−

p

+

γ

φ

φ

Figure 2. North-south dynami s.

1.4.

(G, X)

-stru tures. The notion of

(G, X)

-stru ture is a formali-sationofKlein'sgeometry. Inoursetting,theyariseasades riptionof

at CRstru turesintermofthemodelsphere

S

anditsautomorphism group.

Let

X

be a manifold and

G

a Lie group a ting transitively on

X

. Assume that the a tion is analyti in the following sense: an element

that a ts trivially on an open subset of

X

a ts trivially on the whole of

X

. A

(G, X)

-stru ture on a manifold

M

is an atlas whose harts taketheirvaluesin

X

andwhose hangesof oordinatesarerestri tions of elements of

G

. A dieomorphism of

M

is an automorphism of its

(G, X)

-stru ture if itreads in harts as restri tions of elementsof

G

. Let us onsider the ase when

G = PU(1, n + 1)

and

X = S

. A at stri tly pseudo onvex manifold

M

arries a

(G, X)

-stru ture and its CR automorphisms oin ide with its

(G, X)

automorphisms. Indeed, the atness of

M

means that it is lo ally equivalent to

S

, thus it is su ient to prove that any lo al automorphismof

S

an be extended into a global automorphism. This, in turn, follows from the order

2

rigidity and the followingfa t: any

2

-jet of alo alautomorphismof

S

an berealizedasthe

2

-jetofaglobalautomorphism(seee.g. [Spi97℄). The main tool we will need to study

(G, X)

-stru tures is the so- alled developping map. Let

M

be amanifold endowed with a

(G, X)

-stru ture and

M

˜

be its universal overing. Then there exists a dier-entiable map

D : ˜

M → X

that isalo aldieomorphismandsu hthat forallautomorphism

f

of

˜

M

, there exists some

φ ∈ G

satisfying

Note that in general, this developping map need not be a

dieomor-phism onto hisimage, nor a overing map. It is unique, up to

ompo-sition with anelementof

G

. More details on

(G, X)

-stru tures anfor example be found in the lassi al [Thu97 ℄.

2. Webster: a lo al Theorem

In1977,SydneyWebsterpublishedtherstworktowardtheS hoen

Webster Theorem, [Web77℄. Until the end of the paper,

M

denotes a stri tly pseudo onvex CR manifold of dimension

2n + 1

.

Theorem 2.1  If

M

is ompa t and

Aut

0

(M)

is non ompa t, then

M

isat.

There are several reasons why this result has raised a lot of eorts

to be improved. First, it is a lo al statement though Webster gave in

the same paper a very spe i global result:

Theorem 2.2  If

M

is ompa t and has nite fundamental group and

Aut

0

(M)

isnon ompa t,then

M

isglobally equivalent tothe stan-dard sphere.

Se ond, he assumes that

M

is ompa t and that the identity om-ponent

Aut

0

(M)

is non ompa t. We refer to these hypotheses as the ompa tness assumption and the onne tedness assumption.

His paper also ontains a result on onne ted groups of CR

auto-morphisms having a xed point we shall dis uss briey.

Theorem 2.3 If

M

is ompa tand

Aut

0

(M)

admitsanon ompa t one-parameter Lie subgroup

G

1

that has a xed point

p

0

, then

M

is globally equivalent to the standard sphere

S

.

Let usturn to the proofs of these three results.

2.1. Canoni al alibration. The following result is the key to the

lo alstatement.

Lemma 2.4  For ea h alibration

θ

on

M

there isa ontinuous non-negative fun tion

F

θ

on

M

su h that:

(1)

F

θ

vanishes on a given open set

U

if and only if

U

is at, (2) on the open set where

F

θ

is positive, it is smooth,

(3) the family

(F

θ

)

θ

is homogeneousof degre

−1

:

(7)

F

λθ

= |λ|

−1

F

θ

.

Su h a family of fun tions

(F

θ

)

θ

is alled a relative invariant after Cartan's one (see the proof below). Mostof the time, theare given by

the norm of a urvature tensor.

Let usshow the interestof su h fun tions. Pi k any alibration

θ

of

M

whose Levi formis positive and dene

(8)

θ

∗

_{= F}

θ

θ.

Then

θ

∗

isa ontinuous 1-formthat vanishesonthe atpart of

M

and is a smooth alibration everywhere else. It is anoni al, for if

θ

′

_{= λθ}

isanother alibrationwith

λ > 0

(thatis,whoseLeviformispositive),

θ

′∗

= F

λθ

λθ

= λ

−1

F

θ

λθ

= θ

∗

.

We all

θ

∗

the anoni al alibration of

M

although it is not a genuine alibration unless

M

ontainsnoumbili points. If

M

isat,

θ

∗

iszero

and, therefore,useless.

Proof. Lemma2.4followsfromthestudyofinvariantsof alibratedCR

manifolds.

If

n > 1

, one an derive from the Chern-Mother urvature a tensor

S

onsomebundle

T

over

M

thatonlydependsuponthe CRstru ture and vanishes on an open set

U

if and only if

U

is at. A alibration

θ

indu es, via the Levi form, a metri on

T

. The orresponding norm

kSk

θ

of

S

yieldsthe desired fun tion. See [BS76,page201℄ or[Web78 , page 35℄ for details.

If

n = 1

,

S

is always zero even when

M

is not at so that Cartan's relative invariant is needed. It is a fun tion

r

θ

on

M

, asso iated to a alibration

θ

, that vanishes on anopen set if and onlyif it isat; the family

(r

θ

)

is homogeneous of order

−2

, thus

F

θ

=

√

r

does the job. For details, one an look at Élie Cartan's work [Car32a, Car32b℄ or,

for a more modern presentation, at the book of Howard Ja obowitz

[Ja 90℄.

2.2. The lo al theorem. Letusgiveanoutlineof theproofof

Theo-rem 2.1given byWebster. Weshallsee laterthatastronger statement

an beproved with the same tools.

Proof. Assume

M

is not at; we will show that any one-parameter subgroup of

Aut

0

(M)

has a ompa t losure, whi h implies the om-pa tness of

Aut

0

(M)

by a theorem of Deane Montgomery and Leo Zippin [MZ51℄.

Let

G

1

be a nontrivial one-parameter subgroup of

Aut

0

(M)

with innitesimal generator

Y

on

M

. Choose some alibration

θ

; by as-sumption

F

θ

ispositiveon anopen set

U

. Sin e the vanishing of

F

θ

is independent of

θ

,

U

is invariantunder the ow of

Y

.

Consider the fun tion

η = θ

∗

_{(Y )}

, on

U

. Assume it vanishes identi- ally. Then

Y

lies in the onta t distribution. Moreover, sin e

θ

∗

CR invariant form,

L

Y

θ

∗

vanishes. Cartan's magi formulayields:

0 = L

Y

θ

∗

= Y y dθ

∗

+ dη = Y y dθ

∗

,

so

Y

is identi allyzero, whi h ontradi ts the order two rigidity. Wemaythereforeassumethat

η > 0

somewhere,repla ing

Y

by

−Y

if ne essary. Choosing

ε

su iently small, the set

U

ε

dened by the inequation

η(p) > ε

has non empty interior. It is losed in

M

, thus is ompa t, and isinvariantunder the owof

Y

.

The losure

G

1

of

G

1

in

Aut

0

(M)

is a Lie group that preserves the ompa t

U

ε

andtheWebstermetri of

θ

∗

onit,thusis ompa t

(Corol-lary 1.4).

2.3. The global result. Webster derives Theorem 2.2 from a weak

formofProposition1.6anda(now) standarduse of

(G, X)

-stru tures. Proof. ByTheorem2.1,

M

anditsuniversal overing

M

˜

areat. There-fore they an be developped as

(SU(1, n + 1), S)

-stru tures. Sin e

M

has nitefundamentalgroup,

M

˜

is ompa t and the developpingmap

D : ˜

M → S

is a overing map. But

S

admits no nontrivial overing and

M

˜

is globally equivalent to

S

. By Montgomery-Zippin Theorem [MZ51℄, there exists some losed non ompa tone-parameter subgroup

G

1

of

Aut

0

(M)

. Thisgroupliftstoaoneparametersubgroup

G

˜

1

a ting on

M = S

˜

.

>From Proposition1.6, weknowthat

G

˜

1

has eitherone ortwoxed points. In both ases

G

1

has atleast a xed point.

Let

p

be a xed point of

G

1

. The lifts of

p

are xed points of

G

˜

1

of the same type (attra tive, repulsive or both). But

G

˜

1

has at most one xed point of a given type, thus

M → M

˜

must be a one-sheeted overing.

2.4. One-parameter subgroups with a xed point. Theorem 2.3

is based on a prin iple of extension of lo al onjuga y, making use of

the dynami s on the model spa e. We detail a similar argument at

the end of the paper, using it in the proof of the ompa t ase of the

S hoenWebster Theorem.

Proof. Let

Y

be an innitesimal generator of

G

1

. A ording to Theo-rem 2.1,

M

is at sothere isan isomorphismbetween aneighborhood

U

of

p

0

and an open set

U

′

of

S

. Denote by

Y

′

the ve tor eld on

U

′

orrespondingto the restri tion of

Y

to

U

. Then

Y

′

extends uniquely

to aCR ve tor eld on

S

, whi h has a xed point

p

′

0

.

If

Y

′

isellipti ,thenitfollowsfromtheniteorderrigiditythat

G

1

is ompa t,in ontradi tionwiththeassumptions. If

Y

′

isparaboli ,then

one anuse ittoextendthe onjuga ybetween

U

and

U

′

tothe basins

of attra tion and repulsion of

p

′

0

, therefore

M

is globally equivalent to

S

. If

Y

′

set

V ⊂ M

that is onjugatetoeither

S

or

S

with apoint(namelythe se ond xed point of

Y

′

) deleted. Sin e

Y

is a omplete ve tor eld with isolatedzeros,

M

itselfmust beglobally equivalent toeither

S

or

S

with a point deleted.

3. Kamishima and Lee: two ways from lo al flatness to

global rigidity

Yoshinobu Kamishima seems to be the rst to prove the lo al to

global statement (under both the ompa tness and onne tedness

as-sumptions) in a workshop in honor of Obata held at Keio University

in 1991. He announ ed the result in the pro eedings [Kam93℄ and the

omplete proofappeared a whileafter [Kam96℄.

Theorem 3.1  If

M

is at, ompa t and

Aut

0

(M)

is non ompa t then

M

isglobally equivalent to the standard sphere.

We will not detail his proof at all, but let us quote an interesting

orollary he gave inrelationwith theso- alledSeifert onje ture. This

elebrated onje ture states that any non singular ve tor eld on the

3

-dimensionalspherehasatleastone losedorbit. It wasdisprovedfor

C

1

ve tor elds by PaulS hweitzer [S h74 ℄, then in

C

∞

regularity by

Krystyna Kuperberg [Kup94℄. The question was then raised for

ve -tor elds preserving some geometri stru ture. Kamishima'sfollowing

result givesan answer for ve tor elds preserving aCR stru ture.

Corollary 3.2  If

M

is a rational homology sphere endowed with a stri tly pseudo onvex CR stru ture, then any nonsingularCR ve tor

eld on

M

has a losed orbit.

In [Lee96℄,John Leeproved Theorem 3.1independently of

Kamishi-ma. His methodrelies on Webster's Theorem 2.3: he proves

Theorem 3.3  If

M

is ompa t and

Aut

0

(M)

admit a losed non- ompa t one-parameter subgroup

G

1

, then

G

1

has a xed point.

On eagain,theMontgomeryZippinTheoremisusedtodedu e

The-orem 3.1 fromTheorems 3.3, 2.1 and 2.3.

Proof. Let

Y

be an innitesimal generator of

G

1

and assume by on-tradi tion that

Y

has nozero on

M

.

Note rst that

Y

must be somewhere tangent to the onta t dis-tribution

ξ

: otherwise

Y

would be the Reeb ve tor eld of a unique alibration, thus would preserve the asso iated Webster metri .

The rstpartoftheproof onsistsinunderstandingthesetof points

where

Y ∈ ξ

;itisa lassi al omputationthatinvolvesonlythe onta t stru tureon

M

: pi kany alibration

θ

of

M

anddene

η = θ(Y )

. Then one an show, using that

Y

has no zero, that

0

is a regular value of

η

. Therefore

H = {η = 0} ⊂ M

is a nonempty, ompa t, embedded hypersurfa e along whi h

Y

is tangentto both

H

and

ξ

.

The nextstep onsistsinprovingthatone an ndanew alibration

su h that

L

Y

θ = 0

and

L

Y

dθ = 0

atevery pointof

H

. It iseasytoget the rst ondition by res aling

θ

; then a rather tedious omputation allows Lee to renethe res alinginorder to get the se ond ondition.

These two onditions implythat the Webster metri on

T M

is pre-served by the ow of

Y

along

H

. It follows for any sequen e

(f

i

)

in

G

1

,

(f

i

|

H

)

onvergesin

C

∞

topology. Usingthe omplexoperator

J

,it is then possible to prove that the

2

-jets of the sequen e

(f

i

)

onverge at all pointsof

H

. By the order

2

rigidity,

(f

i

)

is onvergent in

G

1

, a ontradi tion.

4. S hoen: Yamabe problem methods

The aim of this se tion is to survey the proof of S hoenWebster

theorem by R. S hoen in [S h95 ℄. For onvenien e, we only deal with

the ompa t ase, even though[S h95 ℄ also onsiders the non- ompa t

ase with the same kind of te hniques, based on global analysis. R.

S hoen rstprovesthat the onformalgroup is ompa tfor any losed

Riemannian manifold whi h is not onformallyequivalent tothe

stan-dard sphere. Thenhe explainshowtoadapttheproofinaCRsetting,

whi h is what we want to develop below. Another proof of the

on-formal group ompa tness is given in [Heb97 ℄ : it is a bit shorter but

relies on the positive mass theorem, whi h makes it less elementary

than what follows.

4.1. Yamabe theorem. The elebrated Yamabe problem is basi in

onformal geometry: is there a metri with onstant s alar urvature

in ea h onformal lass of agiven losed manifold? This question was

the beginning of a long story: see the ex ellent [Heb97℄ or [LP87℄ for

anexhaustive a ount. Theanswertotheproblemisyesandthe proof

relies on a areful study of the onformal Lapla ian.

As explained in [JL87℄, there is a deep analogy between onformal

and CR geometry. In parti ular, Yamabe theory has a ounterpart

in the CR realm, whi h enables R. S hoen to extend his onformal

geometry argumentsto the CR ase.

In order todevelop aYamabetheoryin the CRsetting,one needs a

Sobolev-likeanalysis. Inthe onformal ase, the natural onformal

op-eratorisellipti ,sothatitsanalysisisratherstandard. IntheCR ase,

the orresponding natural operator

L

θ

is only subellipti . G. Folland and E. Stein [FS74℄ (see also paragraph 5 of [JL87℄) have nonetheless

As in onformal geometry, given a alibration

θ

, we dene the CR Yamabeinvariant

Q(M, θ)

asthe inmum of the fun tional

Z

M

φ L

θ

φ θ ∧ dθ

n

overtheelements

φ

oftheunitsphereintheLebesguespa e

L

2n+2

n

(M)

.

The hoi eofthisexponentisrelatedtothe transformationlawforthe

volume form: if

θ

′

_{= u}

2

n

θ

forsome positivefun tion

u

, then

θ

′

_{∧ (dθ}

′

)

n

= u

2n+2

n

θ ∧ dθ

n

.

It turnsout that

Q(M, θ)

is a CRinvariant.

D. Jerison, J. M. Lee [JL87℄, N. Gamara and R. Ya oub [Gam01℄,

[GY01℄ adapted the proof of the onformal Yamabe theorem to prove

the

Theorem 4.1  A losed stri tly pseudo onvex CR manifold admits

a alibration with onstant s alar urvature

1

(resp.

0

and

−1

) if its CR Yamabe invariant is positive (resp. zero and negative).

We will only need the nonpositive (and easiest) ase, whi h was

settled by [JL87℄.

4.2. The proof. Theorem 4.1leads tothe

Proposition 4.2  When the CR Yamabe invariant is nonpositive,

the CR automorphism group is ompa t.

Proof. WeprovethatCRautomorphismsareisometriesfortheWebster

metri ofa alibration;sin etheisometrygroupofa losedRiemannian

manifoldis ompa t,theresultwillfollow. Endow

M

witha alibration

θ

.

If

Q(M) = 0

,we anassume

θ

hasvanishings alar urvature (Y am-abe). A CR automorphism

F

of

M

then obeys

F

∗

_{θ = u}

n

2

θ

with

L

θ

u = ∆

θ

u = 0

(

F

∗

_θ

has s alar urvature

F

∗

_R

θ

= 0

). An integra-tion by parts yields

0 =

Z

u∆

θ

u =

Z

L

θ

(du|

ξ

, du|

ξ

) θ ∧ dθ

n

.

So we an write

du = f θ

, whi h implies

0 = df ∧ θ + fdθ

. Sin e

dθ

is denite onthe kernel

ξ

of

θ

,

f

vanishes,so

u

is onstant. Sin e

vol

θ

(M) = vol

θ

(F (M)) = vol

F

∗

_θ

(M) = u

2n+2

n

vol

θ

(M),

u

is onstant to

1

:

F

preserves

θ

hen e

W

θ

.

If

Q(M) < 0

, we an make a similar argument: we are left to show that a solution

u

of

∆

θ

u = b(n)



u − u

n+2

n



is onstant to

1

. It follows from a weak maximum prin iple. At a maximumpoint,

∆

θ

u

isnonnegativesothattheequationensures

u ≤ 1

.

At a minimum point, one nds

u ≥ 1

for the same reason. Therefore

u

is onstant to

1

.

The following lemma is the key to omplete the proof. We denote

by

D

r

the ballof radius

r

in

R

2n+1

. To avoid te hni al details, we do

not give the pre ise statement ( f. [S h95 ℄).

Lemma 4.3  Let

F : (D

1

, θ) → (N, σ)

be a CR dieomorphism. We assume

θ

is lose to the Heisenberg alibration and

σ

has vanishing s alar urvature. If

λ :=

p(F

∗

_σ/θ)(0)

denotes the dilation fa tor at

0

, then:

•

the dilation fa tor is almost onstant to

λ

, i.e.

F

∗

_{σ/θ ≈ λ}

on

D

1/2

;

•

images of balls have moderate e entri ity, i.e.

F (D

1/2

) ≈

B(F (0), λ/2)

;

•

the total urvature and the torsion are small when the dila-tion fa tor is large, i.e.

|Rm

σ

| . λ

−2

and

|Tor

σ

| . λ

−2

on

B(F (0), λ/2)

.

Proof. By s aling

σ

, we an assume

λ = 1

. Write

F

∗

_{σ = u}

2

n

θ

and

observethat

R

σ

= 0

implies

L

θ

u = 0

. Sin e

θ

is losetotheHeisenberg alibration,

L

θ

is lose totheHeisenberg subellipti Lapla ian,sothat

u

satisesaHarna k inequality ([JL87℄,5.12) :

sup u ≤ C inf u

,with a ontrolled onstant. The rstand se ondassertions follow. Subellipti

regularity ([JL87℄, 5.7) also yields a

C

2

bound on

u

, hen e the third assertion.

Nowwe an nish the proofof the

Theorem 4.4 (S hoenWebster)  The CR automorphism group of

a losedstri tlypseudo- onvexCR manifoldwhi hisnotCRequivalent

to a standard sphere is ompa t.

Here, we only deal with

C

0

topology. Thanks to a bootstrap

argu-ment,[S h95 ℄ provesthat all

C

k

topologies,

k ≥ 0

,are thesame. They also oin idewith the Lie grouptopology.

Proof. Assume

M

2n+1

is a losed stri tly pseudo- onvex CR manifold

with non- ompa t onformal group and hoose a alibration

θ

. As- oli theorem yields CRautomorphisms

F

i

and points

x

i

su h that the dilation fa tors

λ

i

:=

p(F

i

∗

θ/θ)(x

i

) = max

p(F

i

∗

θ/θ)

go toinnity.

The rough idea of the proof onsists in multiplying the alibration

θ

by suitable Green fun tions so as to build a sequen e of onformal s alar at blow ups; then lemma 4.3 willenable us to nd a sequen e

smaller urvature and torsion: taking a limit,wewill realize

M

minus a point as a Heisenberg group; a last eort will seal the fate of the

missing point.

Tobegin with, we an hoose a small

ǫ > 0

su h that the geometry ofalltheballsofradius

ǫ

in

M

is losetothatoftheHeisenberg group. Then we hoose points

y

i

outside

F

i

(B(x

i

, ǫ))

and use a standardtri k in Yamabe theory. Sin e the CR Yamabe invariant is positive (4.2),

the operator

L

θ

is positive. Therefore, there are Green fun tions

G

i

, i.e. preimages of Dira distributions

δ

y

i

( f. [Gam01℄ for instan e):

outside

y

i

, they are smooth, satisfy

L

θ

G

i

= 0

and we an normalize them sothat their minimum value is

1

. Put

z

i

:= F

i

(x

i

)

and onsider the alibration

θ

i

:=



G

i

G

i

(z

i

)



n

2

θ,

dened outside

y

i

. It has vanishing s alar urvature.

We anassume

y

i

onvergesto

y

,

z

i

onvergesto

z

and

G

i

onverges to

G

on ompa tsetsof

M −{y}

. Besides,one anshowthat

G

i

(z

i

)

re-mainsbounded,thatis

y 6= z

: itstemsfroma onvenientuse oflemma 4.3and fromaHarna kinequalityfor thedilationfa tor between

F

∗

i

θ

i

and

θ

. Sowe an assume

G

i

(z

i

)

onverges. Therefore

θ

i

tends to a alibration

θ

∞

= cG

2

n

θ

on the ompa t sets

of

M − {y}

. Now lemma 4.3 ensures that, roughly,

θ

i

has urvature and torsion of magnitude

λ

−2

i

on

F

i

(B

θ

(x

i

, ǫ/2)) ≈ B

θ

i

(z

i

, λ

i

ǫ/2)

, so that letting

i

gotoinnity,we on ludeour manifold,outside

y

,isCR equivalent to a alibrated stri tly pseudo- onvex CR manifold with

vanishing urvature and torsion; and it happens to be omplete and

simply onne ted (it is a nonde reasing union of topologi al balls), so

that it is

H

2n+1

.

Thus there is a CR dieomorphism

F

between

M

minus

y

and the standard sphere minus some point,

∞

. In the neighbourhood of

∞

in

S

2n+1

, onsider a CR equivalent Heisenberg alibration

σ

. Writing

F

∗

_{σ = u}

2

n

θ

, we obtain

L

θ

u = 0

outside

y

, sin e

σ

has vanishings alar urvature. Extending

F

at

y

amountstoshow that

u

has aremovable singularityat

y

. Butthe integralof

u

2n+2

n

oversomeballisexa tlythe

volume of the image of this ball through

F

, whi h is bounded by the volume of the standard sphere; it follows (proposition 5.17 in [JL87℄)

that

u

isaweaksolutionof theequation

L

θ

u = 0

overaneighborhood of

y

so that it extends as a smooth fun tion in the neighborhood of

y

(5.10, 5.15 in [JL87℄). Thus

(M, θ)

is CR equivalent to the standard sphere.

5. Fran es: a unified dynami al proof

Charles Fran es re ently gave a unied proof of the Ferrand-Obata

and S hoenWebsterTheorems [Fra06℄;infa the alsoproves

analogu-ous resultsforquaternioni - onta tando tonioni - onta tgeometries.

To obtain these results, he uses the setting of Cartan geometries

(see [Sha97℄ for a detailed a ount on this topi ). Given a model

ho-mogeneous spa e

X = G/P

, a Cartan geometry modelled on

X

on a manifold

M

onsistsof:

•

a

P

-prin ipalbundle

B → M

and

•

a

1

-form

ω

onthe total spa e

B

with values in the Lie algebra

g

.

The form

ω

is alled the Cartan onne tion of the stru ture and is supposed to satisfy some ompatibility onditions we donot detail.

The pointsof

B

play the role of adapted frames (like orthonormal frames for Riemannian geometry). The Cartan onne tion is used to

identify innitesimally

B

with

G

: inparti ular,itisasked thatatea h point

p ∈ B

,

ω

p

is anisomorphismbetween

T

p

B

and

g

.

The geometries Fran es is on erned with are modelled on the

ho-mogeneous spa es

∂K

H

d

_{= G/P}

where

K

H

d

is the hyperboli spa e

based on

K

= R

,

C

,

H

or

O

,

G

is the isometry group of

K

H and

P

is the stabilizerof a boundary point. Note that when

K

= C

,

X = S

.

Forea h of these Cartangeometries, the equivalen e problem has

been solved, thatis: thereexists a onstru tionthat givesfor any

on-formal,stri tlypseudo onvexCR,et . stru tureon

M

a orresponding Cartanstru ture

B, ω

su hthat isomorphismsoftheoriginalstru ture indu e isomorphisms of the Cartan stru ture and re ipro ally. The

Cartan stru ture is not unique, one an impose further assumptions.

Inparti ulartheCartan onne tion anbe hosenregular(ate hni al

ondition involving the urvature of

ω

) for the geometries onsidered here.

We an nowstate the result of Fran es.

Theorem 5.1  Let

(M, B, ω)

be a Cartan geometry modelled on

X = ∂K

H

d

, withregular onne tion. If

Aut(M, ω)

a tsnonproperly on

M

, then

M

is isomorphi to either

X

or

X

with a point deleted. Proof. The rst and main step is to prove that any sequen e

(f

k

)

of automorphisms of

M

that a ts nonproperlyadmit a subsequen e that shrinks an open set

U ⊂ M

onto a point

p

. The prin iple is to use some sort of developping map from the spa e of urves on

M

passing through

p

to the spa e of urves on

X

passing through a given base point

o

. Then, hoosing an appropriate family of urves inthe model and itsnorth-south dynami s,one gets the desiredproperty on

M

.

the anoni al alibration (see Se tion 2). As a onsequen e, one an

hoose

U

tobeof the form

U = Γ\(X − {o})

where

Γ

is adis rete subgroup of the stabilizer

P

of

o ∈ X

.

The nal step isa result ongeometri al rigidity of embeddings: if a

at manifold

Γ\(X − {o})

embedsin

M

,theneither

M = Γ\(X − {o})

or

Γ = {

Id

}

. In the latter ase,

M = X

or

M = X − {o}

.

The on lusionfollowssin etheautomorphismgroupof

Γ\(X −{o})

a ts properly when

Γ

isnot trivial.

6. Gathering a geometri proof in the ompa t ase

Inthis lastse tion,wegiveageometri proofofthe S hoenWebster

Theoremunderthe ompa tnessassumption. Itisnotelementary,asit

makesuseofLemma2.4. However: itisageometri proof,thusgivesan

alternativeto S hoen's te hniques; it do not rely onthe

Montgomery-Zippin Theorem, holds without the onne tedness assumption and is

quite short, whi h makes it an improvement of those of Webster,

Ka-mishima and Lee together.

It does not pretend to originality, sin e it relies on arguments of

Webster [Web77℄ and Fran es and Tarquini [FT02℄, rephrased.

6.1. The lo al statement.

Theorem 6.1  If

M

is ompa t and

Aut(M)

is non ompa t, then

M

isat.

Proof. Suppose that

M

is not at; then the anoni al alibration

θ

∗

dened thankstoLemma2.4doesnotvanishidenti ally. Denoteby

W

the Webster metri asso iated with

θ

∗

: it is ontinuous on

M

, smooth and positivedenite on the open set

U

of nonumbili points and zero on its omplementary

F

. Forall

x

and

y

in

M

let

d(x, y) = inf

γ

Z

γ

pW ( ˙γ)

denethenaturalsemimetri asso iatedto

W

(nottobe onfusedwith the Carnot metri of Se tion 1.1.3 : here the inmum is taken on all

urves onne ting

x

to

y

). We have

d(x, y) = 0

if and only if

x

and

y

are in

F

, inparti ular

d

is a genuine metri on

U

.

If

U = M

, then

Aut(M)

preserves a Riemannian metri , thus is ompa t. Otherwise,

F

is nonempty, the distan e

d(x, F )

is nite for every

x ∈ M

and we an dene the set

U

ε

= {x ∈ U; d(x, F ) > ε}

for any positive

ε

. This set is ompa t and has nonempty interior for

ε

smallenough.

6.2. The lo al-to-global statement.

Theorem 6.2  If

M

is at and

Aut(M)

a ts nonproperly, then

M

is globally equivalent to the standard CR sphere

S

or to

S

witha point deleted.

This result follows, by a prin iple of extension of lo al onjuga y,

from the dynami s of unbounded sequen es of

Aut(S)

. Note that we do not use the ompa tnessassumption for this part.

The end of the se tion is dedi ated to the proof of Theorem 6.2.

Notethatitholdsasitisforanyrank-oneparaboli 

(G, X)

-stru ture (namely

X = ∂K

H

n

with

K

= R

,

C

,

H

or

O

).

6.2.1. Set up: developping the dynami s. We assume that

M

is at, thus arries a

(SU(1, n + 1), S)

-stru ture, and that

Aut(M)

a ts non-properly: there is a onvergent sequen e

x

i

∈ M

and a sequen e

f

i

∈ Aut(M)

going to innity (that is, having no onvergent subse-quen e), su hthat

y

i

= f

i

(x

i

)

onvergesin

M

. Weset

x

∞

= lim x

i

and

y

∞

= lim y

i

.

Let

M

˜

be the universal over of

M

. There are lifts

(˜

x

i

)

i∈N∪{∞}

,

(˜

y

i

)

i∈N∪{∞}

and

f

˜

i

su h that

lim ˜

x

i

= ˜

x

∞

,

lim ˜

y

i

= ˜

y

∞

and

y

˜

i

= ˜

f

i

( ˜

x

i

)

.

Moreover,thesequen e

( ˜

f

i

)

hasno onvergentsubsequen ein

Aut( ˜

M )

. Let

D : ˜

M → S

bethedeveloppingmapof

M

and

φ

i

beasequen eof

Aut(S)

su hthat

D ˜

f

i

= φ

i

D

. If

(φ

i

)

had a onvergentsubsequen e, by the order

2

rigidityandsin e

φ

i

and

f

˜

i

arelo ally onjugated,sowould

( ˜

f

i

)

. Thus

(φ

i

)

is unbounded and admit a North-South dynami s, whose poles are denoted by

p

+

and

p

−

.

Sin e

D(˜y

i

) = φ

i

D(˜x

i

)

, we have either

D(˜y

∞

) = p

+

or

D(˜x

∞

) = p

−

. Uptoinvertingthe

f

i

'sandex hangingthe

x

i

'sandthe

y

i

's,weassume that

D(˜y

∞

) = p

+

.

6.2.2. Stret hing inje tivity domains. A subset of

M

˜

is said to be an inje tivity domainif the developping map isone-to-one on its losure.

We denote by

U

0

an open onne ted inje tivity domain ontaining

˜

y

∞

and we let

V

0

= D(U

0

)

. We hoose an open onne ted inje tivity domain

Ω

ontaining

x

˜

∞

and having onne tedboundary

Bd Ω

whose image

D(Bd Ω)

does not ontain

p

−

. Up to extra tinga subsequen e, we an assume that for all

i

,

x

˜

i

∈ Ω

and

y

˜

i

∈ U

0

.

A ordingtoProposition1.6,thereisanin reasingsequen e ofopen

sets

V

i

⊂ S

(

i > 0

) su hthat, extra tingasubsequen e if ne essary: (1) for all

i

,

D(Bd Ω) ⊂ V

i

,

(2)

S V

i

= S − {p

−

}

, (3) for all

i

,

φ

i

(V

i

) ⊂ V

0

.

Let

δ : U

0

→ V

0

be the restri tion of

D

and dene the following open onne ted inje tivity domains:

U

i

= ˜

f

−1

assumed

x

˜

i

∈ Ω

and

y

˜

i

∈ U

0

, we get

(9)

U

i

∩ Ω 6= ∅ ∀i

and by onstru tion wehave

(10)

D(Bd Ω) ⊂ D(U

i

) = V

i

⊂ D(U

i+1

) = V

i+1

∀i.

6.2.3. Monotony and onsequen es. We prove that

(U

i

)

(or a subse-quen e) is anin reasing sequen e.

Ifwe anextra t asubsequen esu hthat

U

i

⊂ Ω

forall

i

,sin e

Ω

is aninje tivity domainand

(DU

i

)

isin reasing,

(U

i

)

must bein reasing.

Otherwise weuse the following Lemma.

Lemma 6.3  Let

A

,

B

be two inje tivity domains su h that

B

is open,

A

is onne ted and

A ∩ B 6= ∅

. If

D(A) ⊂ D(B)

, then

A ⊂ B

. Proof. Sin e

A

is onne ted,we onlyhaveto prove that

A ∩ B

isopen and losedin

A

.

First,

B

isopen sothat

A ∩ B

isopen in

A

. Se ond, let

y

bea point in

A∩B

. Sin e

D(A) ⊂ D(B)

,thereisa

z ∈ B

su hthat

D(z) = D(y)

. But sin e

B

isaninje tivity domainand

y

belongs tothe losureof

B

,

z = y

and

y ∈ B

. Therefore,

A ∩ B

is losedin

A

.

When

U

i

6⊂ Ω

,

Bd Ω ∩ U

i

6= ∅

and we anapply Lemma6.3:

Bd Ω ⊂

U

i

. But then

U

i

∩ U

i+1

6= ∅

thusby the same argument:

U

i

⊂ U

i+1

. Nowlet

U

∞

=

S U

n

;

D

is adieomorphism from

U

∞

to

S − {p

−

}

. If

U

∞

6= ˜

M

, let

x

be apoint ofthe boundary of

U

∞

. If

D(x)

were in

S − {p

−

}

, for any neighborhood

W

of

x

,

D(W ∩ U

∞

)

would meet any neighborhoodofanyinverseimageof

D(x)

, ontradi tingtheinje tivity of

D

on

U

∞

. Therefore the boundary of

U

∞

onsists of

x

alone and

D

is a globaldieomorphism from

M = U

˜

∞

∪ {x}

to

S

.

Wethusproved that

M

˜

isequivalenttoeither

S

or

S − {p

−

}

. More-over, any inverse image of

y

∞

in

M

˜

is an attra ting point, thus is

y

˜

∞

:

˜

M

is one-sheetedand

M

is itself equivalent toeither

S

or

S − {p

−

}

. Referen es

[BS76℄ D. Burns, Jr.and S. Shnider. Spheri alhypersurfa es in omplex

mani-folds.Invent.Math., 33(3):223246,1976.

[Car32a℄ Élie Cartan. Sur la géométrie pseudo- onforme des hypersurfa es de

l'espa ededeuxvariables omplexes.Ann.Mat.PuraAppl.(4),11:1790,

1932.

[Car32b℄ Élie Cartan. Sur la géométrie pseudo- onforme des hypersurfa es de

l'espa e de deux variables omplexes ii. Ann. S uola Norm. Sup. Pisa.

(2),1:333354,1932.

[D'A93℄ JohnP.D'Angelo.Several omplex variablesandthe geometryofreal

hy-persurfa es.Studies inAdvan ed Mathemati s.CRCPress,Bo aRaton,

FL,1993.

Rie-[Fra06℄ CharlesFran es.AFerrand-Obatatheoremforrankoneparaboli

geome-tries,2006.arXiv:math.DG/0608537.

[FS74℄ G.B.FollandandE.M.Stein.Estimatesforthe

∂

¯

b

omplexandanalysis ontheHeisenberggroup.Comm.PureAppl.Math., 27:429522,1974.

[FT02℄ Charles Fran es and Cédri Tarquini. Autour duthéorème de

Ferrand-Obata.Ann. Global Anal. Geom.,21(1):5162,2002.

[Gam01℄ Najoua Gamara. TheCR Yamabe onje turethe ase

n

= 1

. J. Eur. Math.So .(JEMS),3(2):105137,2001.

[GY01℄ NajouaGamaraandRidhaYa oub.CRYamabe onje turethe

onfor-mallyat ase.Pa i J. Math.,201(1):121175,2001.

[Heb97℄ Emmanuel Hebey. Introdu tion à l'analyse non linéaire surles variétés.

Fondations.DiderotS ien es,1997.

[Ja 90℄ HowardJa obowitz.Anintrodu tiontoCRstru tures,volume32of

Math-emati alSurveysandMonographs.Ameri anMathemati alSo iety,

Prov-iden e,RI,1990.

[JL87℄ DavidJerisonandJohnM.Lee.TheYamabeproblemonCRmanifolds.

J.DierentialGeom., 25(2):167197,1987.

[JL89℄ DavidJerisonandJohnM.Lee.Intrinsi CRnormal oordinatesandthe

CRYamabeproblem. J.DierentialGeom., 29(2):303343,1989.

[Kam93℄ YoshinobuKamishima.ArigiditytheoremforCRmanifoldsanda

rene-mentofObataandLelong-Ferrand'sresult.InGeometry andits

appli a-tions(Yokohama, 1991),pages7383.WorldS i. Publ.,RiverEdge,NJ,

1993.

[Kam96℄ YoshinobuKamishima.Geometri owson ompa tmanifoldsandglobal

rigidity.Topology,35(2):439450,1996.

[Kob95℄ Shoshi hi Kobayashi. Transformation groups in dierential geometry.

Classi s in Mathemati s. Springer-Verlag, Berlin, 1995. Reprint of the

1972edition.

[Kup94℄ KrystynaKuperberg.Asmooth ounterexampletotheSeifert onje ture.

Ann. of Math.(2),140(3):723732,1994.

[Laf88℄ Ja quesLafontaine.ThetheoremofLelong-FerrandandObata.In

Con-formalgeometry (Bonn,1985/1986), Aspe tsMath.,E12, pages93103.

Vieweg,Brauns hweig,1988.

[Lee96℄ JohnM. Lee. CR manifoldswith non ompa t onne ted automorphism

groups.J. Geom. Anal.,6(1):7990,1996.

[LF71℄ Ja queline Lelong-Ferrand. Transformations onformes et

quasi- onformes des variétés riemanniennes ompa tes (démonstration de la

onje ture de A. Li hnerowi z). A ad. Roy. Belg. Cl. S i. Mém. Coll.

in8

deg

(2),39(5):44,1971.

[LP87℄ JohnM.LeeandThomasH.Parker.TheYamabeproblem.Bull. Amer.

Math.So .(N.S.),17(1):3791,1987.

[MZ51℄ DeaneMontgomeryand Leo Zippin. Existen e of subgroups isomorphi

totherealnumbers.Ann. ofMath. (2),53:298326,1951.

[Oba72℄ MorioObata.The onje tureson onformaltransformationsof

Riemann-ianmanifolds.J. DierentialGeometry,6:247258,1971/72.

[Pan90℄ PierrePansu.Distan es onformeset ohomologie

l

n

.Publ. Univ. Pierre

etMarieCurie, 92,1990.

[S h74℄ PaulA.S hweitzer.CounterexamplestotheSeifert onje tureand

open-ing losed leavesoffoliations.Ann.of Math. (2),100:386400,1974.

[S h95℄ R.S hoen.Onthe onformalandCRautomorphismgroups.Geom.Fun t.

[Sha97℄ R. W. Sharpe. Dierential geometry, volume 166 of Graduate Texts in

Mathemati s. Springer-Verlag, New York, 1997. Cartan's generalization

ofKlein'sErlangenprogram,WithaforewordbyS. S.Chern.

[Spi97℄ AndreaF. Spiro. Smoothreal hypersurfa es in

C

n

with a non- ompa t

isotropy group of CR transformations. Geom. Dedi ata, 67(2):199221,

1997.

[Thu97℄ WilliamP. Thurston.Three-dimensional geometry and topology. Vol. 1,

volume35ofPrin etonMathemati al Series.Prin etonUniversityPress,

Prin eton,NJ,1997.EditedbySilvioLevy.

[Web77℄ S.M.Webster.Onthetransformationgroupofarealhypersurfa e.Trans.

Amer. Math.So ., 231(1):179190,1977.

[Web78℄ S. M. Webster. Pseudo-Hermitian stru tures on areal hypersurfa e. J.

Dierential Geom., 13(1):2541,1978.

[Won77℄ BunWong.Chara terizationoftheunit ballin

C

n

byitsautomorphism

group.Invent.Math., 41(3):253257,1977.

[Won03℄ B.Wong.On omplexmanifoldswithnon ompa tautomorphismgroups.

InExplorationsin omplexandRiemanniangeometry,volume332of