sup |S| ≤ C ′ Diam(M ) kDSk p

C^′ 1 + (p − 2)CΛ¹21/γ_{−2 1+}1

γ

alors,

DimE_I(m) = DimE_I

, pourtout point

m

M

, et la famille

(S_i)_i∈I

est presqu'or-thonormée entout point de

M

(i.e.

k < Si, S_j > −δijk∞< _(1−η)^2η 2

Démonstration. Dans le as

n ≥ 4

, le orollaire 3.14 dé oule de laproposition 3.13 en posant

p₁ = p

p₂ = ^p₂

q^′ = p

. Dans le asoù

n = 2, 3

, le orollaire 3.14 dé oule du orollaire 3.8et de l'inégalité de Sobolev de lase tion1.3.2, quidonne :

1 − ^{inf |S|}

sup |S| ^{≤ C}^′^{Diam(M )}

kDSkp

kSk∞

.

Remarque. Ce orollaire estunegénéralisation aux asdes ombinaisons linéairesde se tionspropresduthéorèmedenon-annulationdeM.LeCouturieretG.Robert(théorème 1.4.1. de [65 ℄, valable pour les se tions harmoniques). Dans leur résultat, l'inégalité de Sobolev est une onséquen e du théorème de S. Gallot ité dans la se tion 1.3.2, et la ondition

q ≥ max(p, 4)

a été omise dans leur énon é, mais elle semble né essaire pour que leurdémonstration soit orre te. Le orollaire3.14 permet d'obtenir l'existen e d'une trivialisationd'unsous-bré,nonplusseulementpardesse tionsharmoniquesrelativement à un opérateur (lapla ien+potentiel), mais aussi par des se tions propres asso iées à de petites valeurs propres de et opérateur. On peut espérer qu'une telle généralisation des te hniques "à la Bo hner" donne une méthode générale pour lassier les variétés qui réalisent ertaines hypothèses de pin ement de la ourbure et des valeursspe trales d'un opérateur (lapla ien+potentiel). Deux exemples de résultats de lassi ation de e type sontdonnés danslespartie IIet III de ette thèse.

Cara térisation spe trale des

Curvature, Harna k's Inequality, and a Spe tral Chara terization of Nilmanifolds

Published in Ann. of Glob. Anal. and Geom. 23 (2003) p.227-246

Erwann Aubry, BrunoColbois,Patri k Ghanaat,Ernst A.Ruh

Abstra t. For losed

n

dimensional Riemannian manifolds

M

with almost positive Ri i urvature, the Lapla ian on oneforms is known to admit at most

n

small eigenvalues. If there are

n

small eigenvalues, or if

M

is orientable and has

n − 1

small eigenvalues, then

M

is dieomorphi to a nilmanifold,andthemetri isalmostleftinvariant.Weshowthatourresults areoptimal for

n ≥ 4

1.Introdu tion

A lassi al theorem of Bo hner states that the rst real Betti number of a losed

n

dimensional Riemannian manifold

M

with positive semidenite Ri i urvature tensor

Ric

satisesthe inequality

b₁(M ) ≤ n

,with equalityonlyif

M

isisometri toa at torus. This resultis a onsequen e of Weitzenbö k'sformulafor the Hodgede RhamLapla ian

∆ = dδ + δd

on oneforms

α

∆α = ∇^∗∇α + Ric(α^♯, ·). (1.1)

The formula implies that all harmoni oneforms on

M

are parallel with respe t to the LeviCivita onne tionof the metri . Sin ethe spa e ofparallel oneformshasdimension at most

n

, Bo hner's Betti number estimate is a onsequen e of the Hodge theorem on harmoni forms. And if

b₁(M ) = n

, then the Albanese map obtained by integrating an

L²

-orthonormal basis of the spa e of harmoni forms yields an isometry of

M

with its Albanese torus.

Bo hner'sinequalityfor

b₁(M )

hasbeenextendedbyGallot([49 ℄Cor.3.2)andGromov ([57 ℄ p. 73) to in ludemanifolds whose Ri i tensor and diameter satisfy

Ric Diam²(M ) ≥ −ǫ(n) (1.2)

for suitablysmallpositive

ǫ(n)

dependingonlyon

n

. The aseofequalitywassettledonly re ently by Cheeger and Colding ([29℄ p. 459) to the ee t that (1.2) and

b₁(M ) = n

stillimply that

M

isdieomorphi to the torus.Butit appears tobe unknownwhethera dieomorphismis given bythe Albanesemap.

Gallot and Meyer ([53 ℄) extendedBo hner's theoremin a dierent dire tion by giving an expli it bound for the number of small eigenvalues of the Lapla ian, instead of only the multipli ity

b₁(M )

of the zero eigenvalue. Consider a ompa t onne ted Riemannian manifold

(M, g)

withoutboundary, ofdimension

n

and diameter

Diam(M, g) ≤ d

. Let

0 ≤ λ1≤ λ2 ≤ . . .

denote the spe trum of

∆

on oneforms, with ea h eigenvalue repeated a ording to its multipli ity.AssumingaRi i urvaturebound

Ric d2 ≥ −ǫ

forarealnumber

ǫ

,theresult ofGallot and Meyer ([53 ℄p. 574,see also[47℄), whenspe ialized to

λ_n+1

, statesthat

λ_n+1d² ≥ ^λ^∗^d

2 8(n + 1)2 − ǫ. (1.3)

Here

λ^∗

denotes the smallest positive eigenvalue of the Lapla ian on fun tions. Lower bounds for

λ∗

in terms of Ri i urvature and diameter were obtained by Liand Yau. In parti ular,Theorem 10 in [67℄states that

λ^∗d²+ max{0, ǫ} ≥ π²/4. (1.4)

Combined with (1.3), this yields a positive lower bound on

λ_n+1

, provided

ǫ

is not too positive. So

∆

an haveat most

n

smalleigenvalues.

In [80 ℄ and [33 ℄, the authors onsidered what happens when

∆

a tually does have

n

small eigenvalues. Petersen and Sprouse showed in [80 ℄ that, under an additional bound on the urvature tensor

R

M

has to be dieomorphi to an infranilmanifold. In [37 ℄, under boundson

R

and its ovariant derivative

∇R

M

wasshownto be dieomorphi to anilmanifold. In thispaper,wegeneralize and sharpenboth results.

Re all that an infranilmanifold is a quotient

Λ\G

of a nilpotent Lie group

G

by a dis rete group

Λ

of isometries of some left invariant Riemannian metri . The indu ed metri on the quotient is alled left invariant by abuse of language. A nilmanifold is a quotient

Γ\G

of a nilpotent Lie group by a dis rete subgroup

Γ

G

. In parti ular, everynilmanifoldisaninfranilmanifold.Conversely,a ording toAuslander'sBieberba h Theorem([13 ℄,Theorem1),every ompa tinfranilmanifoldadmitsanite overingspa e thatisa nilmanifold.

For

m ∈ M

, let

Ric(m)

denote the lowest eigenvalue of the Ri i tensor

Ric(m)

, onsidered as a symmetri operator on

T_mM

. For a fun tion

f : M → R

, we denote by

f−(m) = max{0, −f(m)}

its negative part. Ourmain resultisthe following

Theorem 1.1. For every dimension

n

and real number

p > n

, there is a positive onstant

ǫ(n, p)

su h that the following istrue. Suppose

(Mⁿ, g)

isa ompa t Riemannian manifold satisfying

Diam(M, g) ≤ d

and

kRic⁻kp/2d² ≤ ǫ(n, p) 1 + kRkq/2d²_−β(n,p)

(1.5)

λ_nd² ≤ ǫ(n, p) 1 + kRkq/2d²_−β(n,p)

(1.6)

Dans le document Variétés de courbure de Ricci presque minorée: inégalités géométriques optimales et stabilité des variétés extrémales (Page 80-86)

C′ 1 + (p − 2)CΛ121/γ−2 1+1

γ

DimEI(m) = DimEI

m

M

(Si)i∈I

M

k < Si, Sj > −δijk∞< (1−η)2η 2

n ≥ 4

p1 = p

p2 = p2

q′ = p

n = 2, 3

1 − inf |S|

sup |S| ≤ C′Diam(M )

kDSkp

kSk∞

.

q ≥ max(p, 4)

n

M

n

n

M

n − 1

M

n ≥ 4

n

M

Ric

b1(M ) ≤ n

M

∆ = dδ + δd

α

∆α = ∇∗∇α + Ric(α♯, ·). (1.1)

M

n

b1(M ) = n

L2

M

b1(M )

Ric Diam2(M ) ≥ −ǫ(n) (1.2)

ǫ(n)

n

b1(M ) = n

M

b1(M )

(M, g)

n

Diam(M, g) ≤ d

0 ≤ λ1≤ λ2 ≤ . . .

∆

Ric d2 ≥ −ǫ

ǫ

λn+1

λn+1d2 ≥ λ∗d

2

8(n + 1)2 − ǫ. (1.3)

λ∗

λ∗

λ∗d2+ max{0, ǫ} ≥ π2/4. (1.4)

λn+1

ǫ

∆

n

∆

n

R

M

R

∇R

M

Λ\G

G

Λ

Γ\G

Γ

G

m ∈ M

C^′ 1 + (p − 2)CΛ¹21/γ_{−2 1+}1

DimE_I(m) = DimE_I

(S_i)_i∈I

k < Si, S_j > −δijk∞< _(1−η)^2η 2

p₁ = p

p₂ = ^p₂

q^′ = p

1 − ^{inf |S|}

sup |S| ^{≤ C}^′^{Diam(M )}

b₁(M ) ≤ n

∆α = ∇^∗∇α + Ric(α^♯, ·). (1.1)

b₁(M ) = n

L²

b₁(M )

Ric Diam²(M ) ≥ −ǫ(n) (1.2)

b₁(M ) = n

b₁(M )

λ_n+1

λ_n+1d² ≥ ^λ^∗^d

λ^∗

λ^∗d²+ max{0, ǫ} ≥ π²/4. (1.4)

λ_n+1

T_mM

(Mⁿ, g)

kRic⁻kp/2d² ≤ ǫ(n, p) 1 + kRkq/2d²_−β(n,p)

λ_nd² ≤ ǫ(n, p) 1 + kRkq/2d²_−β(n,p)