Universit¨ at Regensburg Mathematik

Universit¨ at Regensburg Mathematik

Hebey-Vaugon conjecture II

Farid Madani

Preprint Nr. 14/2011

FARIDMADANI

Abstrat. Inthis paper we onsider the remaining ases of Hebey

Vaugononjeture. Wegiveapositiveanswertotheonjeture.

1. Introdution

Let

(M, g)

^{be aompat Riemannianmanifold of dimension}

n ≥ 3

^{. Denote}

by

I (M, g)

^,

C(M, g)

^and

R _g

^{theisometrygroup, theonformal} transforma- tions group and thesalar urvature, respetively. Let

G

^{be a subgroup of}

the isometry group

I(M, g)

^{. The} equivariant Yamabe problem an be for- mulated asfollows: in the onformal lass of

g

^{, there exists a}

G−

^invariant

metri withonstant salar urvature. Assuming thepositive masstheorem

and the Weyl vanishing onjeture(for moredetails on the subjet, see [7℄,

[10℄andthereferenestherein),E.HebeyandM.Vaugon[4℄provedthatthis

problem hassolutions. Moreover, theyproved thattheinmumof Yamabe

funtional

(1)

I _g (ϕ) =

R

M |∇ϕ| ² + _4(n−1) ⁿ⁻² R _g ϕ ² dv kϕk ² 2n

n−2

over

G−

^invariant nonnegative funtions is ahieved by a smooth positive

G−

^{invariant funtion. Thisfuntion is a solution of theYamabe equation,}

whih isthe EulerLagrange equation of

I _g

^:

∆ _g ϕ + n − 2

4(n − 1) R _g ϕ = µϕ ^{n+2 n−2}

Oneof theonsequenesoftheseresults isthatthefollowingonjeturedue

to Lihnerowiz[6℄ istrue.

Lihnerowiz onjeture. For every ompat Riemannian manifold

(M, g)

^{whih is not onformal to the unit sphere}

S ⁿ

^{endowed with its stan-}

dard metri

g _s

^{, there exists a metri}

g ˜

^{onformal to}

g

^{for whih}

I (M, g) = ˜ C(M, g)

^{, and the salarurvature}

R _{g ˜}

^{is onstant.}

The lassialYamabe problem,whih onsistsof ndingaonformalmetri

withonstantsalarurvatureonaompatRiemannianmanifold, isapar-

tiularaseoftheequivariantYamabeproblem(itorrespondsto

G = {id}

^).

This problem was ompletely solved byH. Yamabe [13 ℄, N. Trudinger [12℄,

T. Aubin [1 ℄ and R. Shoen [11 ℄. The main idea to prove the existene of

positive minimizers for

I _g

^{is to show that if}

(M, g)

^{is not onformal to the}

sphere endowed withits standard metri,then

(2)

µ(g) := inf

C ^∞ (M) I _g (ϕ) < 1

4 n(n − 2)ω _n ^2/n

where

ω _n

^{isthe volume of theunit sphere}

S ⁿ

^.

T. Aubin [1℄ proved (2) in some ases by onstruting a test funtion

u _ε

satisfying:

I g (u ε ) < 1

4 n(n − 2)ω _n ^2/n

He onjeturedthat(2) always holdsexept for thesphere. R.Shoen on-

strutedanother testfuntionwhihinvolvestheGreen funtionoftheon-

formalLaplaian

∆ g + _4(n−1) ⁿ⁻² R g

^{. Usingthepositivemasstheorem,R.Shoen}

proved(2)forallompatmanifoldswhiharenotonformalto

(S ⁿ , g _s )

^{. The}

solution of the Yamabeproblem follows.

Later, E. Hebey and M. Vaugon [4 ℄ showed that we an generalize (2) for

the equivariant ase asfollows:

Denote by

O _G (P)

^{the orbit of}

P ∈ M

^under

G

^{and by}

card O _G (P )

^its

ardinal. Let

C _G ^∞ (M )

^{be thesetof smooth}

G

^{-invariant funtionsand}

µ _G (g) := inf

C _G ^∞ (M ) I _g (ϕ)

Following E. Hebey and M. Vaugon [3 , 4℄, we dene the integer

ω(P )

^{at a}

point

P

^as

ω(P) = inf{i ∈ N /k∇ ⁱ W _g (P )k 6= 0} (ω(P) = +∞

^if

∀i ∈ N , k∇ ⁱ W _g (P )k = 0)

HebeyVaugon onjeture. Let

(M, g)

^{be a ompat Riemannianman-}

ifold of dimension

n ≥ 3

^and

G

^{be a subgroup of}

I(M, g)

^{. If}

(M, g)

^{is not}

onformal to

(S ⁿ , g _s )

^{or if the ation of}

G

^{has no xed point, then the fol-}

lowing inequality holds

(3)

µ _G (g) < 1

4 n(n − 2)ω ^2/n _n ( inf

Q∈M card O _G (Q)) ^2/n

E. Hebey and M. Vaugon showed that if this onjeture holds, then it im-

pliesthattheequivariantYamabeproblemhasminimizingsolutionsandthe

Lihnerowizonjetureis alsotrue. Notiethatif

G = {id}

^{,thenthis on-}

jeture orrespondsto (2).

Letus realltheresultsalready knownabout thisonjeture. Assuming the

positive masstheorem,E. Hebey andM. Vaugon [4℄proved thefollowing:

Theorem 1.1 (E.Hebeyand M.Vaugon). Let

(M, g)

^{be a smooth ompat}

Riemannian manifold of dimension

n ≥ 3

^and

G

^{be a subgroup of}

I(M, g)

^.

We always have :

µ _G (g) ≤ 1

4 n(n − 2)ω ^2/n _n ( inf

Q∈M card O _G (Q)) ^2/n

andinequality (3)holdsifatleast oneofthefollowingonditionsissatised.

1. The ationof

G

^on

M

^isfree.

2.

3 ≤ dim M ≤ 11

^.

3. There exists a point

P ∈ M

^{with nite minimal orbit under}

G

^suh

that

ω(P ) > (n − 6)/2

^or

ω(P ) ∈ {0, 1, 2}

^.

We have also thefollowing result obtained bytheauthor in[9℄:

Theorem 1.2. HebeyVaugon onjeture holds for every smooth ompat

Riemannianmanifold

(M, g)

^{of dimension}

n ≤ 37

^.

The main resultofthis paperisthefollowing:

Theorem 1.3. If there existsa point

P ∈ M

^{suh that}

ω(P ) ≤ (n − 6)/2

^,

then

(4)

µ G (g) < 1

4 n(n − 2)ω _n ^2/n (card O G (P)) ^2/n

Note that if we assume the positive mass theorem, then Theorem 1.3 and

Theorem 1.1implies thatHebeyVaugononjeture holds.

The proof of Theorem 1.3 doesn't require the positive mass theorem. If

card O _G (Q) = +∞

^{for all}

Q ∈ M

^{, then (3) holds. So we have to onsider}

only the ase when there exists a point in

M

^{with nite orbit. From now}

until the end ofthis paper, we suppose that

P ∈ M

^{is ontained ina nite}

orbit and

ω(P ) ≤ ⁿ⁻⁶ ₂

^{. Theassumption}

ω(P) ≤ ⁿ⁻⁶ ₂

^{deletesthease}

(M, g)

is onformalto

(S ⁿ , g _s )

^.

2. G-invariant test funtion

In orderto proveTheorem 1.3and1.2, we onstrutfromthefuntion

ϕ _ε,P

dened below a

G

^{-invariant testfuntion}

φ _ε

^{suh that}

(5)

I _g (φ _ε ) < 1

4 n(n − 2)ω _n ^2/n (card O _G (P)) ^2/n

Letusrealltheonstrutionin[9 ℄of

ϕ _ε,P

^{. Let}

{x ^j }

^{bethegeodesinormal}

oordinates in the neighborhood of

P

^{and dene}

r = |x|

^and

ξ ^j = x ^j /r

^.

Without lossof generality,wesuppose that

det g = 1 + O(r ^N )

^{, with}

N > 0

suiently large (for the existene of suh oordinates for a

G−

^invariant

onformal lass, see[4℄,[5 ℄).

ϕ _ε,P (Q) = (1 − r ^{ω(P )+2} f (ξ))u _ε,P (Q)

u _ε,P (Q) =



 

  ε

r ² + ε ^{2 n−2} ₂

− ε

δ ² + ε ^{2 n−2} ₂

if

Q ∈ B _P (δ)

0

^if

Q ∈ M − B _P (δ)

for all

Q ∈ M

^{, where}

r = d(Q, P )

^{is the distane between}

P

^and

Q

^{, and}

B _P (δ)

^{is the geodesi ball of enter}

P

^{and radius}

δ

^{xed suiently small.}

f

^{is a funtion depending only on}

ξ

^{(dened on}

S ⁿ⁻¹

^{), hosen suh that}

R

S ⁿ⁻¹ f dσ = 0

^.

Let

R ¯

^{betheleadingpartintheTaylorexpansionofthesalarurvature}

R _g

ina neighborhoodof

P

^and

µ(P)

^{is itsdegree. Hene,}

R g (Q) = ¯ R + O(r ^{µ(P )+1} ) R ¯ = r ^{µ(P )} X

|β|=µ(P )

∇ _β R _g (P)ξ ^β

We summarizesome propertiesof

R ¯

^{inthe following} proposition.

Proposition 2.1. 1.

R ¯

^{is a} homogeneous polynomial of degree

µ(P )

andis invariantunder the ation of the stabilizergroup of

P

^.

2. We always have

µ(P ) ≥ ω(P)

3. if

µ(P ) ≥ ω(P ) + 1

^{, then}

R

S ⁿ⁻¹ (r) Rdσ < 0

^for

r > 0

^suiently

small.

4. If

µ(P ) = ω(P )

^{, then there exist} eigenfuntions

ϕ _k

^{of the Laplaian}

on

S ⁿ⁻¹

^{suh that the restritionof}

R ¯

^{to the sphere isgiven by}

R| ¯ _S ⁿ⁻¹ =

q

X

k=1

ν _k ϕ _k

where

q ≤ [ω(P )/2]

^,

∆ _s ϕ _k = ν _k ϕ _k

^and

ν _k = (ω − 2k + 2)(n + ω − 2k)

are the eigenvalues of

∆ _s

^{with respet to the standard metri}

g _s

^of

S ⁿ⁻¹

^.

Sinethesalarurvatureisinvariantundertheationoftheisometrygroup

I (M, g)

^,

R ¯

^{is invariant under the ation of the stabilizer of}

P

^{. The seond}

statement of Proposition 2.1 is proven by E. Hebey and M. Vaugon ([4℄,

Setion 8) and the third one by T. Aubin ([2 ℄, Setion 3). So, in the ase

µ(P) ≥ ω(P ) + 1

^{, the onjeture holds} immediately, by hoosing

f = 0

^,

ϕ _ε,P = u _ε,P

^{(see[8, 9 ℄for more details).}

Fromnowwe supposethat

µ(P ) = ω(P )

^{. Using thefatthat}

R ¯

^{is homoge-}

neous polynomial ofdegree

ω(P )

^{andthefatthat for all}

j ≤ ω(P ) − 1

(6)

|∇ ^j R _g (P )| = 0, ∆ ^j+1 _g R _g (P) = 0

^and

|∇∆ ^j+1 _g R _g (P)| = 0

wededuethat

∆ ^[ω(P _E ^)/2] R ¯ = 0

^{. Hene,ifwe restrit}

R ¯

^{tothesphere,we get}

thedeompositionofitem4.inProposition 2.1. Theproofof (6)isgiven in

[4℄, Setion 8.

Using the split of

R ¯

^{given in} Proposition 2.1, we proved in [9 ℄ that if the ardinal of

O G (P )

^{isminimal and}

ω(P) ≤ 15

^{,thenthere exists}

c ∈ R

suh that for

f = c R| ¯ _S ⁿ⁻¹

^{,the funtion}

φ ε = X

P i ∈O G (P )

ϕ ε,P i

is

G

^{-invariant and satises (5) , whih proves Theorem 1.2. Moreover, we}

proved the following theorem:

Theorem 2.1. If

ω(P) ≤ (n − 6)/2

^{, then there exist}

c _k ∈ R

, suh that for

f = P q

k=1 c _k ϕ _k

^{, the funtion}

ϕ _ε,P

^satises

(7)

I _g (ϕ _ε,P ) < 1

4 n(n − 2)ω ^2/n _n

The proof of Theorem 2.1 is tehnial and uses Proposition 2.1. It isgiven

in[9 ℄ (see also[8℄ for adetailed proof).

Below, we show that using Theorem 2.1, we an onstrut a

G−

^invariant

funtion

φ ε

^{whihsatises(5)for}

ω(P ) ≤ ⁿ⁻⁶ ₂

^{(theardinalof}

O G (P )

^isnot

neessarilyminimal). It impliesTheorem 1.3.

Proof of Theorem 1.3. Let

H ⊂ G

^{be the stabilizer of}

P

^{. We onsider the}

funtion

f = P q

k=1 c _k ϕ _k

^{of Theorem 2.1. Using the}exponential map on

P

as a loal hart, we an view

f

^and

ϕ _k

^{as funtions dened over the unit}

sphere of

T _P M

^{,the tangent spaeof}

M

^on

P

^{. Let}

h

^{bean isometryin}

H

^.

h _∗ (P) : (T _P M, g _P ) → (T _P M, g _P )

is the linear tangent map of

h

^on

P

^{. It is a linear isometry with respet}

to the inner produt

g _P

^{whih is Eulidean.}

h _∗ (P )

^{onserves the unit}

sphere

S ⁿ⁻¹ ⊂ T _P M

^{andthe Laplaian. Wealreadyknowthatthefuntion}

R ¯ = r ^ω(P) P q

k=1 ν _k ϕ _k

^is

H

-invariant. Notie that

ϕ _k

^and

ϕ j

^{belong to two}

dierent eigenspaes if

k 6= j

^{. Sine, isometries onserve the Laplaianand}

ϕ _k

^areeigenfuntionsoftheLaplaianon thesphereendowed withitsstan- dard metri, ityields that

ϕ _k

^and

f

^are

H

-invariant. Ontheotherhand,we have the following bijetivemap:

G/H −→ O _G (P ) σH 7−→ σ(P )

Sine

f

^is

H

^-invariant,

ϕ ε,P

^is

H−

^{invariant and thefuntion}

φ _ε = X

σ∈G/H

ϕ _ε,P ◦ σ ⁻¹

is

G−

^{invariant andsatises(5) .}

Aknowledgments

The author would like to thank Bernd Ammann for his remarks and the

helpful disussions. He also thanksEmmanuel Hebeyfor hissuggestions.

Referenes

1. T. Aubin, Équations diérentielles non linéaires et problème de Yamabe, J. Math.

Puresetappl55(1976),269296.

2. ,Surquelquesproblèmes deourburesalaire,J.Funt.Anal240(2006),269

289.

3. E. HebeyandM. Vaugon,Courbure salaire presrite pour des variétés nononfor-

mémentdiéomorphesàlasphère,C.R.Aad.Si.Paris316(1993),no.3,281282.

4. ,Leproblème deYamabeéquivariant,Bull.Si.Math.117(1993),241286.

5. J.M. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. So. 17 (1987),

3791.

6. A.Lihnerowiz,Surlestransformationsonformesd'unevariétériemannienneom-

pate,C.R.Aad.Si.Paris 259(1964).

7. F.MarquesM.KhuriandR.Shoen,AompatnesstheoremfortheYamabeproblem,

J.Di.Geom.81(2009),143196.

8. F. Madani, Le problème de Yamabe equivariant et la onjeture de HebeyVaugon,

Ph.D.thesis,UniversitéPierreetMarieCurie,2009.

9. ,Equivariant Yamabe problemandHebeyVaugononjeture, J.Fun. Anal.

258(2010),241254.

10. F.Marques,Blow-upexamplesfortheYamabeproblem,Cal.Var.36(2009),377397.

11. R.Shoen,Conformaldeformationof ariemannian metritoonstant salar urva-

ture,J.Dier.Geom.20(1984),479495.

12. N. Trudinger, Remarks onerning the onformal deformation of riemannian stru-

turesonompatmanifolds,Ann.SuolaNorm.Sup.Pisa 22(1968),265274.

13. H.Yamabe,Onadeformationofriemannianstruturesonompatmanifolds,Osaka

Math.J.12(1960),2137.

NWFI-Mathematik, UniversitätRegensburg,93040Regensburg, Germany.

E-mailaddress: Farid.Madanimathematik.uni-r egen sburg .de