Laplacian eigenvalues functionals and metric deformations on compact manifolds

0.27 MB·English

by  Ahmad El Soufi

LAPLACIAN EIGENVALUES FUNCTIONALS AND
METRIC DEFORMATIONS ON COMPACT
MANIFOLDS
AHMAD EL SOUFI AND SA¨ID ILIAS
7
0
0
Abstract. In this paper, we investigate critical points of the
2
Laplacian’s eigenvalues considered as functionals on the space of

n
Riemmannian metrics or a conformal class of metrics on a com-
a
pact manifold. We obtain necessaryand sufficient conditionsfor a
J
metric to be a critical point of such a functional. We derive spe-
6
2 cific consequences concerning possible locally maximizing metrics.
Wealsocharacterizecriticalmetricsoftheratiooftwoconsecutive

] eigenvalues.
G
M
.
h
t 1. Introduction
a
m
The eigenvalues of the Laplace-Beltrami operator associated with a
[
Riemannian metric on a closed manifold are among the most natural

1 global Riemannian invariants defined independently from curvature.
v
One of the main topics in Spectral Geometry is the study of uniform
7
7 boundedness of eigenvalues under some constraints and the finding out
7 of eventual extremal metrics. Let us start by recalling some impor-
1
tant results in this direction, where the eigenvalues are considered as
0
7 functionals on the set of Riemannian metrics of fixed volume.
0
In all the sequel, we will denote by M a compact smooth manifold
/
h of dimension n 2 and, for any Riemannian metric g on M, by
t ≥
a
m 0 < λ (g) λ (g)
1 k
≤ ··· ≤ ≤ ··· → ∞
:
v the sequence of eigenvalues of the Laplacian ∆ associated with g, re-
g
i peated according to their multiplicities. Notice that λ is not invariant
X k
under scaling (i.e., c > 0, λ (cg) = 1λ (g)). Hence, a normalization is
ar ∀ k c k
needed and it is common to restrict the functional λ to the set (M)
k
R
of Riemannian metrics of fixed volume. We will denote by C(g) the set
of Riemannian metrics conformal to g and having the same volume as
g.
The result that, in dimension 2, k 1, the functional λ is uni-
k
∀ ≥
formly bounded on the set of metrics of fixed area is due to Korevaar
[23], after been proved for k = 1, by Hersch [20] in the case of the
2-sphere S2, and by Yang and Yau [33] and Li and Yau [24] for all
2000 Mathematics Subject Classification. 49R50, 35P99, 58J50.
Key words and phrases. Eigenvalues, Laplacian, critical metrics.
1
2 AHMAD EL SOUFIANDSA¨ID ILIAS
compact surfaces (see also [10] for an improvement of the Yang-Yau
upper bound of λ in terms of the genus).
1
The situation differs in dimension n 3. Indeed, based on earlier
≥
results on spheres obtained by many authors [30, 25, 2, 32, 26], Colbois
andDodziuk[4]proved that, foranycompact manifoldM ofdimension
n 3, the functional λ is unbounded on (M).
1
≥ R
However, k 1, the functional λ becomes uniformly bounded
k
∀ ≥
when restricted to a conformal class of metrics of fixed volume C(g).
This result was first proved for k = 1 by the authors [11] (see also [16])
and, for any k 1, by Korevaar [23] (see also [19]).
≥
Existence results for maximizing metrics are available in only few
situations and concern exclusively the first eigenvalue functional. Her-
sch [20] proved that the standard metric is the only maximizing metric
for λ on the 2-sphere S2. The same result holds for the standard met-
1
ric of the real projective plane RP2 (Li and Yau [24]). Nadirashvili
[27] outlined a proof for the existence of maximizing metrics for λ on
1
the 2-Torus and the Klein bottle (see [18, 17] for additional details on
Nadirashvili’s paper).
In higher dimensions, the authors [11] gave a sufficient condition
for a metric g on M to maximize λ in its the conformal class C(g).
1
This condition is fulfilled in particular by the standard metric of the
sphere Sn (which enabled us to answer Berger’s problem concerning
the maximization of λ restricted to the standard conformal class of
1
Sn), and more generally, by the standard metric of any compact rank-
one symmetric space. The flat metrics g and g on the 2-Torus
sq eq
T2 associated with the square lattice Z2 and the equilateral lattice
Z(1,0) Z(1/2,√3/2) respectively are also maximizing metrics of λ
1
⊕
in their conformal classes (see [11, 24]).
In this paper we address the following natural questions:
(1) What about critical points of the functional g λ (g) ?
k
7→
(2) How to deform a Riemannian metric g in order to increase, or
decrease, the k-th eigenvalue λ ?
k
Despite the non-differentiability of the functional λ with respect
k
to metric deformations, perturbation theory enables us to prove that,
for any analytic deformation g of a metric g, the function t λ (g )
t k t
7→
always admits left and right derivatives at t = 0 (see Theorem 2.1 (i)
below). Moreover, these derivatives can be expressed in terms of the
eigenvalues of the following quadratic form
1
Q (u) = du du+ ∆ u2g,h v ,
h g g
− ⊗ 4
ZM (cid:18) (cid:19)
with h = dg , restricted to the eigenspace E (g) (Theorem 2.1 and
dt t t=0 k
Lemma 2.1). This enables us to give partial answers to Question (2)
(cid:12)
(cid:12)
CRITICAL METRICS OF THE LAPLACIAN EIGENVALUES 3
above (Corollary 2.1). In particular, if Q is positive definite on E (g),
h k
then λ (g ) < λ (g) < λ (g ) for all t (0,ε), for some positive ε.
k −t k k t
∈
Concerning Question (1), the existence of left and right derivatives
for t λ (g ), suggests the following natural notion of criticality. In-
k t
7→
deed, a metric g will be termed critical for the functional λ if, for any
k
volume-preserving deformation g of g, one has
t
d d
λ (g ) λ (g ) 0;
dt k t t=0− × dt k t t=0+ ≤
this means that either (cid:12) (cid:12)
(cid:12) (cid:12)
λ (g ) λ (g)+o(t) or λ (g ) λ (g)+o(t).
k t k k t k
≤ ≥
It is clear that if g is a locally maximizing or a locally minimizing
metric of λ , then g is a critical metric for λ in the previous sense.
k k
Inanearlier work([12]and[14]), wetreatedtheparticularcasek = 1
and gave a necessary condition ( [12, Theorem 1.1] and [14, Theorem
2.1]) as well as a sufficient condition ( [12, Proposition 1.1] and [14,
Theorem 2.2]) for a metric g to be critical for the functional λ or for
1
the restriction of the λ to a conformal class.
1
In section 3 and 4 below, we extend the results of [12] and [14] to
higher order eigenvalues, and weaken the sufficient conditions given
there. Actually, as we will see, in many cases, the necessary condition
of criticality is also sufficient (Theorem 3.1 and Theorem 4.1). Given
a metric g on M, we will prove that
Necessary conditions (Theorem 3.1(i) and Theorem 4.1(i)):
•
i) If g is critical for the functional λ , then there exists a
k
finite family u , ,u of eigenfunctions associated with
1 d
{ ··· }
λ such that
k
du du = g.
i i
⊗
i≤d
X
ii) If g is a critical metric of the functional λ restricted to
k
the conformal class of g, then there exists a finite family
u , ,u of eigenfunctions associated with λ such that
1 d k
{ ··· }
u2 = 1.
i
i≤d
X
Sufficient conditions (Theorem 3.1(ii) and Theorem 4.1(ii)):
•
If λ (g) > λ (g) or λ (g) < λ (g) (which means that λ (g)
k k−1 k k+1 k
corresponds to the first one or the last one of a cluster of equal
eigenvalues), then the necessary conditions above are also suf-
ficient.
The condition (i) above means that the map u := (u , ,u ) :
1 d
(M,g) Rd, is an isometric immersion, whose image is a m··in·imally
−→
immersed submanifold of the Euclidean sphere Sd−1( n ) of radius
λk(g)
q
4 AHMAD EL SOUFIANDSA¨ID ILIAS
n (see[29]). Inotherwords, ametricg iscriticalforthefunctional
λk(g)
qλk, for some k 1, if and only if g is induced on M by a minimal
≥
immersion of M into a sphere. Therefore, the classification of critical
metrics of eigenvalue functionals on M reduces to the classification
of minimal immersions of M into spheres. The many existence and
classification results (see for instance [15, 21, 8, 31] and the references
therein) of minimal immersions into spheres give examples of critical
metrics for the eigenvalue functionals.
Notice that, for the first eigenvalue functional, the critical metrics
are classified on surfaces of genus 0 and 1. Indeed, on S2 and RP2,
the standard metrics are the only critical ones ([11]). On the torus
T2, the flat metrics g and g mentioned above are, up to dilatations,
eq sq
the only critical metrics for λ ([12]). The metric g corresponds to a
1 eq
maximizer for λ ([27]), while g is a saddle point. For the Klein bottle
1 sq
K, Jacobson, Nadirashvili and Polterovitch [22] showed the existence
of a critical metric and El Soufi, Giacomini and Jazar [9] proved that
this metric is, up to dilatations, the unique critical metric for λ on K.
1
Now, the condition (ii), concerning the criticality for λ restricted
k
to a conformal class, is equivalent to the fact that the map u :=
(u , ,u ) : (M,g) Sd−1 is a harmonic map with energy den-
1 d
··· −→
sity e(u) = λk(g) (see for instance [7]). Thus, a metric g is critical
2
for some λ restricted to the conformal class of g if and only if (M,g)
k
admits a harmonic map of constant energy density in a sphere. In par-
ticular, the metric of any homogeneous compact Riemannian space is
critical for λ restricted to its conformal class (for other examples see
k
[28, 8, 31] and the references therein).
A consequence of the necessary condition (ii) is that, if g is a critical
metric of λ restricted to C(g), then the multiplicity of λ (g) is at least
k k
2 (Corollary 4.2). This means thatλ (g) = λ (g) orλ (g) = λ (g).
k k−1 k k+1
In the case where the metric g is a local maximizer of λ restricted to
k
C(g), we prove that one necessarily has: λ (g) = λ (g) (Corollary
k k+1
4.2). For a local minimizer, one has λ (g) = λ (g). Together with a
k k−1
recent result of Colbois and the first author [5], this result tells us that
a Riemannian metric can never maximize two consecutive eigenvalues
simultaneously on its conformal class (Corollary 4.3). In fact, if g
maximizes λ on C(g), then
k
λk+1(g)n2 sup λk+1(g′)n2 nn2ωn,
≤ g′∈C(g) −
where ω is the volume of the unit Euclidean n-sphere.
n
As an application of the results above, one can derive characteriza-
tions of the metrics which are critical for various functions of eigenval-
ues. To illustrate this, we treat in the last section of this paper, the
CRITICAL METRICS OF THE LAPLACIAN EIGENVALUES 5
case of the ratio functional λk+1 of two consecutive eigenvalues and give
λk
characterizations of critical metrics for these functionals.
2. Derivatives of eigenvalues with respect to metric
deformations
Let M be a compact smooth manifold of dimension n 2. For any
≥
Riemannian metric g on M, we denote by 0 < λ (g) λ (g)
1 2
≤ ≤ ···
the eigenvalues of the Laplace-Beltrami operator ∆ associated with g.
g
For any k N, we denote by E (g) = Ker(∆ λ (g)I) the eigenspace
k g k
∈ −
corresponding to λ (g) and by Π : L2(M,g) E (g) the orthogonal
k k k
−→
projection on E (g).
k
Let us fix a positive integer k and consider the functional g λ (g).
k
7−→
This functional is continuous but not differentiable in general. How-
ever, perturbation theory tells us that λ is left and right differentiable
k
along any analytic curve of metrics. The main purpose of this section
is to express the derivatives of λ with respect to analytic metric de-
k
formations, in terms of the eigenvalues of an explicit quadratic form on
E (g). Indeed, we have the following
k
Theorem 2.1. Let g be a Riemannian metric on M and let (g ) be a
t t
family of Riemannian metrics analytically indexed by t ( ǫ,ǫ), such
∈ −
that g = g. The following hold
0
i) The function t ( ǫ,ǫ) λ (g ) admits a left and a right
k t
∈ − 7−→
derivatives at t = 0.
ii) The derivatives dλ (g ) and dλ (g ) are eigenvalues
dt k t t=0− dt k t t=0+
of the operator Π ∆′ : E (g) E (g), where ∆′ = d∆ .
k◦ (cid:12)k −→ k (cid:12) dt gt t=0
iii) If λ (g) > λ (g), then(cid:12)dλ (g ) and(cid:12)dλ (g ) are the
k k−1 dt k t t=0− dt k t t=0+ (cid:12)
greatest and the least eigenvalues of Π ∆′ on E (g), res(cid:12)pec-
(cid:12) k ◦ (cid:12)k
tively. (cid:12) (cid:12)
iv) If λ (g) < λ (g), then dλ (g ) and dλ (g ) are the
k k+1 dt k t t=0− dt k t t=0+
least and the greatest eigenvalues of Π ∆′ on E (g), respec-
(cid:12) k ◦ (cid:12)k
tively. (cid:12) (cid:12)
Proof. The family of operators ∆ := ∆ depends analytically on t
t gt
and, t, ∆ is self-adjoint with respect to the L2 inner product induced
t
∀
by g (but not necessarily to that induced by g). However, as done in
t
[1], after a conjugation by the unitary isomorphism
U : L2(M,g) L2(M,g )
t t
→
1/4
g
u | | u,
7→ g
(cid:18)| t|(cid:19)
where g is the Riemannian volume density of g , we obtain an ana-
t t
lytic fa|mi|ly P = U−1 ∆ U of operators such that, t ( ǫ,ǫ), P is
t t ◦ t◦ t ∀ ∈ − t
self-adjoint with respect to the L2 inner product induced by g. More-
over, P and ∆ have the same spectrum. In particular, λ (g ) is an
t t k t
6 AHMAD EL SOUFIANDSA¨ID ILIAS
eigenvalue of P . The Rellich-Kato perturbation theory of unbounded
t
self-adjointoperatorsappliestotheanalyticfamilyofoperatorst P .
t
7→
Therefore, if we denote by m the dimension of E (g), then there exist,
k
t ( ǫ,ǫ), m eigenvalues Λ (t), ,Λ (t) of P associated with an
1 1 t
∀ ∈ − ···
L2(M,g)-orthonormal family of eigenfunctions v (t), ,v (t) of P ,
1 m t
···
that is P v (t) = Λ (t)v (t), so that Λ(0) = = Λ (0) = λ (g), and
t i i i m k
···
i m, both Λ (t) and v (t) depend analytically on t. Setting, i m
i i
∀ ≤ ∀ ≤
and t ( ǫ,ǫ), u (t) = U v (t), we get, i m,
i t i
∀ ∈ − ∀ ≤
(1) ∆ u (t) = Λ (t)u (t)
t i i i
and the family u (t), ,u (t) is orthonormal in L2(M,g ). Since
1 m t
{ ··· }
t λ (t) is continuous and, i m, t Λ (t) is analytic with
k i
7→ ∀ ≤ 7→
Λ (0) = λ (g), there exist δ > 0 and two integers p, q m such that
i k
≤
Λ (t) for t ( δ,0)
λ (g ) = p ∈ −
k t Λ (t) for t (0,δ).
q
(cid:26) ∈
Assertion (i) follows immediately. Moreover, one has
d
λ (t) = Λ′(0)
dt k t=0− p
and (cid:12)
(cid:12)
d
λ (t) = Λ′(0).
dt k t=0+ q
Differentiating both sides of (1)(cid:12)at t = 0, we get
(cid:12)
∆′u +∆u′ = Λ′(0)u +λ (g)u′
i i i i k i
with u′ = du (t) and u := u (0). Multiplying this last equation
i dt i t=0 i i
by u and integrating by parts with respect to the Riemannian volume
j
(cid:12)
element v of g, w(cid:12)e obtain
g
Λ′(0) if j = i
(2) u ∆′u v = i
j i g 0 otherwise.
ZM (cid:26)
Since u , ,u is an orthonormal basis of E (g) with respect to
1 m k
{ ··· }
the L2-inner product induced by g, we deduce that
(Π ∆′)u = Λ′(0)u .
k ◦ i i i
In particular, Λ′(0) and Λ′(0) are eigenvalues of Π ∆′, which proves
p q k◦
Assertion (ii).
Assume now λ (g) > λ (g). Hence, i m, Λ (0) = λ (g) >
k k−1 i k
∀ ≤
λ (g). By continuity, we have Λ (t) > λ (g ) for sufficiently small
k−1 i k−1 t
t. Hence, there exists η > 0 such that, t ( η,η) and i m,
∀ ∈ − ∀ ≤
Λ (t) λ (g ), which means that λ (g ) = min Λ (t), ,Λ (t) .
i k t k t 1 m
≥ { ··· }
This implies that
d
λ (g ) = max Λ′(0), ,Λ′ (0)
dt k t t=0− { 1 ··· m }
(cid:12)
(cid:12)
CRITICAL METRICS OF THE LAPLACIAN EIGENVALUES 7
and
d
λ (g ) = min Λ′(0), ,Λ′ (0) .
dt k t t=0+ { 1 ··· m }
Assertion (iii) is proved(cid:12).
(cid:12)
The proof of Assertion (iv) is similar. Indeed, if λ (g) < λ (g),
k k+1
one has, for sufficiently small t, λ (g ) = max Λ (t), ,Λ (t) and,
k t 1 m
{ ··· }
then,
d
λ (g ) = max Λ′(0), ,Λ′ (0)
dt k t t=0+ { 1 ··· m }
and (cid:12)
d (cid:12)
λ (g ) = min Λ′(0), ,Λ′ (0) .
dt k t t=0− { 1 ··· m }
(cid:3)
(cid:12)
(cid:12)
The quadratic form associated with the symmetric operator Π ∆′
k
◦
acting on E (g) can be expressed explicitely as follows:
k
Lemma 2.1. Let (g ) be an analytic deformation of the metric g and
t t
let h := dg . The operator P := Π ∆′ : E (g) E (g) is a
dt t t=0 k,h k ◦ k → k
symmetric with respect to the L2-norm induced by g ; the corresponding
(cid:12)
quadratic for(cid:12)m is given by, u E (g),
k
∀ ∈
1
Q (u) := uP uv = du du+ ∆ u2g,h v ,
h k,h g g g
− ⊗ 4
ZM ZM (cid:18) (cid:19)
where (, ) is the pointwise inner product induced by g on covariant 2-
tensors. Moreover, if , t ( ǫ,ǫ), g = α g is conformal to g, then
t t
∀ ∈ −
h = ϕg with ϕ = dα , and, u E (g),
dt t t=0 ∀ ∈ k
n n n 2
Qh(u) = ϕ( du 2(cid:12)(cid:12)+ ∆gu2)vg = ϕ(λk(g)u2 − du 2)vg.
− | | 4 −2 − n | |
ZM ZM
Proof. The derivative at t = 0 of t ∆ is given by the formula (see
7→ gt
[3]):
d 1
(3) ∆′u := ∆ u = (Ddu,h) (du,δh+ d(trace h)),
dt gt t=0 − 2 g
(cid:12)
where D is the canonical covariant derivative induced by g. Thus,
(cid:12)
(cid:12)
1 1
(4) u∆′u v = u(Ddu,h)v (du2,δh+ d(trace h))v .
g g g g
− 2 2
ZM ZM ZM
One has, u,
∀ 1
uDdu = Ddu2 du du.
2 − ⊗
Hence,
1
u(Ddu,h)v = (Ddu2,h)v (du du,h)v .
g g g
2 − ⊗
ZM ZM ZM
Since δ is the adjoint of D w.r.t. the L2(g)-inner product, we obtain
1
u(Ddu,h)v = (du2,δh)v (du du,h)v .
g g g
2 − ⊗
ZM ZM ZM
8 AHMAD EL SOUFIANDSA¨ID ILIAS
On the other hand
(du2,d(trace h))v = ∆ u2 trace hv = (∆ u2 g,h)v .
g g g g g g g
ZM ZM ZM
Replacing in (4) one immediately gets the desired identity.
A straightforward computation gives the expression of Q for con-
h
(cid:3)
formal deformations.
In relation to the question (2) of the introduction, we give the fol-
lowing result which is a direct consequence of Theorem 2.1 and Lemma
2.1.
Corollary 2.1. Let (g ) be an analytic deformation of a Riemannian
t t
metric g on M and let Q be the associated quadratic form defined as
h
in Lemma 2.1, with h = dg .
dt t t=0
i) If Q is positive definite on E (g), then there exists ε > 0 such
h (cid:12) k
that λ (g ) < λ (g) <(cid:12) λ (g ) for all t (0,ε).
k −t k k t
∈
ii) Assume that λ (g) > λ (g). If there exists u E (g) such
k k−1 k
∈
that Q (u) < 0, then λ (g ) < λ (g) for all t (0,ε), for some
h k t k
∈
ε > 0.
iii) Assume that λ (g) < λ (g). If there exists u E (g) such
k k+1 k
∈
that Q (u) > 0, then λ (g ) > λ (g) for all t (0,ε), for some
h k t k
∈
ε > 0.
In particular, if Q (u) > 0 for a first eigenfunction u, then λ (g ) <
h 1 t
λ (g) for sufficiently small positive t.
1
3. Critical metrics of the eigenvalue functionals
Let M be a closed manifold of dimension n 2 and let k be a
≥
positive integer. Before introducing the notion of critical metric of
the functional λ , notice that this functional is not scaling invariant.
k
Therefore, we will restrict λ to the set of metrics of given volume. In
k
view of Theorem 2.1, a natural way to introduce the notion of critical
metric is the following:
Definition 3.1. A metric g on M is said to be “critical” for the func-
tional λ if, for any volume-preserving analytic deformation (g ) of g
k t t
with g = g, the left and the right derivatives of λ (g ) at t = 0 satisfy
0 k t
d d
λ (g ) λ (g ) 0.
dt k t t=0− × dt k t t=0+ ≤
It is easy to see that (cid:12) (cid:12)
(cid:12) (cid:12)
d d
λ (g ) 0 λ (g ) λ (t) λ (0)+o(t)
dt k t t=0+ ≤ ≤ dt k t t=0− ⇐⇒ k ≤ k
and (cid:12) (cid:12)
(cid:12) (cid:12)
d d
λ (g ) 0 λ (g ) λ (t) λ (0)+o(t).
dt k t t=0− ≤ ≤ dt k t t=0+ ⇐⇒ k ≥ k
(cid:12) (cid:12)
(cid:12) (cid:12)
CRITICAL METRICS OF THE LAPLACIAN EIGENVALUES 9
Therefore, g is critical for λ if, for any volume-preserving analytic
k
deformation (g ) of g, one of the following inequalities holds:
t t
λ (g ) λ (g)+o(t)
k t k
≤
or
λ (g ) λ (g)+o(t).
k t k
≥
Of course, if g is a local maximizer or a local minimizer of λ , then g
k
is critical in the sense of the previous definition.
In all the sequel, we will denote by S2(M,g) the space of covariant
0
2-tensors h satisfying trace hv = (g,h)v = 0, endowed with its
M g g M g
natural L2 norm induced by g.
R R
Proposition 3.1. If g is a critical metric for the functional λ on M,
k
then, h S2(M,g), the quadratic form
∀ ∈ 0
1
Q (u) = du du+ ∆ u2g,h v
h g g
− ⊗ 4
ZM (cid:18) (cid:19)
is indefinite on E (g).
k
Proof. Let h S2(M,g). The deformation of g defined for small t
∈ 0
2/n
vol(g)
by g = (g +th), where vol(g) is the Riemannian vol-
t
vol(g +th)
(cid:20) (cid:21)
ume of (M,g), is volume-preserving and depends analytically on t with
dg = h. Using Theorem 2.1, we see that, if g is critical, then the
dt t t=0
operator P admits a nonnegative and a nonpositive eigenvalues on
h,k
(cid:12)
E (g(cid:12)) which means that the quadratic form Q is indefinite (Lemma
k h
(cid:3)
2.1).
In the case where λ (g) > λ (g) or λ (g) < λ (g), one can show
k k−1 k k+1
that the converse of Proposition 3.1 is also true. Indeed, we have the
following
Proposition 3.2. Let g be a Riemannian metric on M such that
λ (g) > λ (g) or λ (g) < λ (g). Then g is critical for the func-
k k−1 k k+1
tional λ if and only if, h S2(M,g), the quadratic form Q is in-
k ∀ ∈ 0 h
definite on E (g).
k
Proof. Let (g ) be an analytic volume-preserving deformation of g and
t t
let h = dg . Since vol(g ) is constant with respect to t, the tensor
dt t t=0 t
h belongs to S2(M,g) (indeed, (g,h)v = dvol(g ) = 0). The
(cid:12) 0 M g dt t t=0
indefinitenes(cid:12)s of Q implies that the operator P = Π ∆′ admits
h R k,h (cid:12) k ◦
both non-negative and non-positive eigenvalues on E ((cid:12)g) (see Lemma
k
2.1). The result follows immediately from Theorem 2.1 (iii) and (iv).
(cid:3)
The indefiniteness of Q on E (g) for all h S2(M,g), can be inter-
h k ∈ 0
preted intrinsically in terms of the eigenfunctions of λ (g) as follows.
k
10 AHMAD EL SOUFIANDSA¨ID ILIAS
Lemma 3.1. Let g be a Riemannian metric on M. The two following
conditions are equivalent:
i) h S2(M,g), the quadratic form Q is indefinite on E (g).
∀ ∈ 0 h k
ii) There exists a finite family u , ,u E (g) of eigenfunc-
1 d k
{ ··· } ⊂
tions associated with λ (g) such that
k
du du = g.
i i
⊗
i≤d
X
Proof. The proof of “(i) implies (ii)” uses the same arguments as in
the proof of Theorem 1.1 of [12]. For the sake of completeness, we will
recall the main steps. First, we introduce the convex set K S2(M,g)
⊂
given by
1
K = du du + ∆ u2g ;u E (g), J N, J finite .
j ⊗ j 4 g j j ∈ k ⊂
( )
j∈J (cid:20) (cid:21)
X
Let us first show that g K. Indeed, if g / K, then, applying classi-
∈ ∈
cal separation theorem in the finite dimensional subspace of S2(M,g)
generated by K and g, endowed with the L2 inner product induced
by g, we deduce the existence of a 2-tensor h S2(M,g) such that
∈
(g,h)v > 0 and, T K, (T,h)v 0. The tensor
M g ∀ ∈ M g ≤
1
R R
h = h (g,h)v g
0 g
− n vol(g)
(cid:18) ZM (cid:19)
belongs to S2(M,g) and we have, u E (g), u = 0,
0 ∀ ∈ k 6
1 (g,h)v
Q (u) = (du du+ ∆ u2g,h)v + M g du 2v
h0 − ⊗ 4 g g nvol(g) | | g
ZM R ZM
λ (g)
k (g,h)v u2v .
g g
≥ nvol(g)
ZM ZM
Since (g,h)v > 0, the quadratic form Q is positive definite, which
M g h0
contradicts the assumption (i). Now, g K means that there exists
R ∈
u , ,u E (g) such that
1 m k
··· ∈
1
(5) (du du + ∆ u2)g = g.
i ⊗ i 4 g i
i≤d
X
Hence, since ∆u2 = 2(λ (g)u2 du 2), we obtainafter taking thetrace
i k i−| i|
in (5),
λ (g) n 2
k u2 = 1+ − du 2.
2 i 2n | i|
i≤d i≤d
X X
For n = 2, we immediately get u2 = 2 and, for n 3, we
i≤d i λk(g) ≥
consider the function f := u2 n and observe that it satisfies
i≤d Pi − λk(g)
(n 2)∆ f = 2(n 2)P(λ (g) u2 du 2) = 4λ (g)f.
− g − k i − | i| − k
i≤d i≤d
X X