The first eigenvalue of Dirac and Laplace operators on surfaces

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The first eigenvalue of Dirac and Laplace operators on surfaces

Jean-Francois Grosjean, Emmanuel Humbert

To cite this version:

Jean-Francois Grosjean, Emmanuel Humbert. The first eigenvalue of Dirac and Laplace operators on

surfaces. 2006. �hal-00095771�

ccsd-00095771, version 1 - 18 Sep 2006

SURFACES

J.F. GROSJEAN ET E. HUMBERT

Abstract. Let (M, g, σ) be a compact Riemmannian surface equipped with a spin structureσ. For any metric ˜gonM, we denote byµ₁(˜g) (resp.λ₁(˜g)) the first positive eigenvalue of the Laplacian (resp. the Dirac operator) with respect to the metric ˜g. In this paper, we show that

infλ₁(˜g)² µ₁(˜g) 61

2.

where the infimum is taken over the metrics ˜gconformal tog. This answer a question asked by Agricola, Ammann and Friedrich in [AAF99].

1

MSC 2000: 34L15, 53C27, 58J05.

Contents

1. Introduction 1

2. Generalized metrics 2

3. The metrics (g

α,ε

)

α,ε

4

4. Proof of relation (5) 5

5. Proof of relation (6) 7

References 13

1. Introduction

Let (M, g, σ) be a compact Riemannian surface equipped with a spin structure σ. For any metric ¯ g on M , we denote by Σ

¯g

M the spinor bundle associated to ¯ g. We let ∆

¯g

be the Laplace-Beltrami operator acting on smooth functions of M and D

¯g

be the Dirac operator acting on smooth spinor fields with respect to the metric ¯ g. We also denote by µ

1

(¯ g) (resp. λ

1

(¯ g)) the smallest positive eigenvalue of ∆

¯g

(resp. D

g¯

).

Agricola, Ammann and Friedrich asked the following question in [AAF99]:

When M is a two dimensional torus, can we find on M a Riemannian metric g ˜ for which λ

1

(˜ g)

²

< µ

1

(˜ g) ? The main goal of this article is to answer this question. We prove the

Theorem 1.1. There exists a family of metrics (g

ε

)

ε

conformal to g for which lim sup

ε→0

λ

1

(g

ε

)

²

Vol

gε

(M ) 6 4π lim inf

ε→0

µ

1

(g

ε

)Vol

gε

(M ) > 8π.

Date: September 18, 2006.

1[email protected], [email protected] 1

Theorem 1.1 clearly answers the question of [AAF99] but says much more: first, the result is true on any compact Riemannian surface equipped with a spin structure and not only when M is a two-dimensional torus. In addition, the metric ˜ g can be chosen in a given conformal class. Finally, this metric ˜ g can be chosen such that (2 − δ)λ

1

(g)

²

< µ

1

(g) where δ > 0 is arbitrary small. More precisely Theorem 1.1 shows Corollary 1.2. On any compact Riemannian surface (M, g), we have

inf λ

1

(¯ g)

²

µ

1

(¯ g) 6 1

2 where the infimum is taken over the metric g ¯ conformal to g.

Theorem 1.1 has other interesting consequences. Indeed, it proves

Corollary 1.3. For any compact surface (M, g) equipped with a spin structure σ, we let λ

⁺_min

(M, g, σ) = inf λ

1

(¯ g)Vol

1 2

¯ g

(M )

where the infimum is taken over the metrics g ¯ conformal to g. Then, we have λ

⁺_min

(M, g, σ) 6 λ

⁺_min

( S

²

) where λ

⁺_min

( S

²

) is the same invariant computed on the standard sphere S

²

.

This corollary is an immediate consequence of the fact that λ

⁺_min

( S

²

) = 2 √

π (see [AHM03]). This result was announced in [AHM03]. The conformal invariant λ

⁺_min

has been studied in many papers (see for example [Hij86, Lot86, B¨ar92, Amm03, AHM03, AH06] ). Indeed, it has many relations with Yamabe problem (see [LP87]). Corollary 1.3 has been proved in all dimensions by Ammann in [Amm03] if either n > 3 or is D is invertible. Corollary 1.3 extends the result to the remaining case: n = 2 and Ker(D) 6 = { 0 } . In [AHM03], an alternative proof of the case n ≥ 3 is given and the proof of the case n = 2 is skectched.

In the same spirit, a consequence of Theorem 1.1 is Corollary 1.4. For any compact surface (M, g), we let

µ

sup

(M, g) = sup µ

1

(¯ g)Vol

1 2

¯ g

(M )

where the infimum is taken over the metrics ¯ g conformal to g. Then, we have µ

sup

(M, g) > µ

sup

( S

²

) where µ

sup

( S

²

) is the same invariant computed on the standard sphere S

²

.

The invariant µ

sup

has been studied in [CoES03] and Corollary 1.4 is a particular case of Theorem A in this paper. We obtain here another proof.

Acknowledgement: The authors are very grateful to Bernd Ammann for having drawn our attention to the question in [AAF99].

2. Generalized metrics

Let f be a smooth positive function and set ¯ g = f

²

g. Let also for u ∈ C

^∞

(M ) I

g¯

(u) =

R

M

|∇ u |

^¯g

dv

g¯

R

M

u

²

dv

¯g

.

It is well known that µ

1

(¯ g) = inf I

g¯

(u) where the infimum is taken over the smooth non-zero functions u for which R

M

udv

g¯

= 0. We now can write all these expressions in the metric g. We then see that for u ∈ C

^∞

(M ), we have

I

¯g

(u) = R

M

|∇ u |

²g

dv

g

R u

²

f

²

dv

g

and µ

1

(¯ g) = inf I

¯g

(u) where the infimum is taken over the smooth non-zero functions u for which R

M

uf

²

dv

g

= 0. Now if f is only of class C

^0,a

(M ) for some a > 0, we can define ¯ g = f

²

g. The 2-

form ¯ g is not really a metric since f is not smooth. We then say that g is a generalized metric. We can

define the first eigenvalue µ

1

(¯ g) of ∆

¯g

using the definition above. Now, by standard methods, one sees that there exists a function u ∈ C

²

(M ) with R

M

uf

²

dv

g

= 0 and such that I

¯g

(u) = µ

1

(¯ g). Writing the Euler equation for u, we see that

∆

g

u = µ

1

(¯ g)f

²

u. (1)

We prove the following result

Lemma 2.1. If (f

n

) is a sequence of smooth positive functions which converges uniformily to f , then µ

1

(f

_n²

g) tends to µ

1

(¯ g).

Proof. Let u

n

be a eigenfunction function associated to µ

1

(f

_n²

g). Without loss of generality, we can assume that

Z

M

u

²_n

f

_n²

= 1. We set v

n

= u

n

− Z

M

u

n

f

²

dv

g

Z

M

f

²

dv

g

. We then have Z

M

v

n

f

²

dv

g

= 0 and hence

µ

1

(¯ g) 6 I

¯g

(v

n

). (2)

We have

Z

M

|∇ v

n

|

²

dv

g

= Z

M

|∇ u

n

|

²

dv

g

= Z

M

u

n

∆

g

u

n

dv

g

. By equation (1), we get that

Z

M

|∇ v

n

|

²

dv

g

= µ

1

(f

_n²

g) Z

M

f

_n²

u

²_n

dv

g

= µ

1

(f

_n²

g). (3) We also have

Z

M

f

²

v

²_n

dv

g

= Z

M

f

²

u

²_n

− Z

M

u

n

f

²

dv

g

2

Z

M

f

²

dv

g

.

Now,

Z

M

u

n

f

²

dv

g

= Z

M

u

n

(f

²

− f

_n²

)dv

g

6 C Z

M

| u

n

| (f + f

n

)

²

k f − f

n

k

∞

. Since the sequence (f

n

)

n

tends uniformly to f and since

Z

M

f

_n²

u

²_n

dv

g

= 1, we get that lim

n

Z

M

u

n

f

²

dv

g

= 0. In the same way,

Z

M

f

²

u

²_n

dv

g

= Z

M

f

_n²

u

²_n

dv

g

+ o(1) = 1 + o(1).

Finally, we obtain

Z

M

f

²

v

_n²

dv

g

= 1 + o(1).

Together with (2) and (3), we obtain that µ

1

(¯ g) 6 lim inf

n

µ

1

(f

_n²

g). Now, let u be associated to µ

1

(¯ g) and set v = u −

Z

M

uf

_n²

dv

g

Z

M

f

n

dv

g

. We have Z

M

v

²

f

_n²

dv

g

= 0 and hence

µ

1

(f

_n²

g) 6 I

_fn²_g

(v).

It is easy to see that lim

n

I

f_n²g

(v) = I

g¯

(u) = µ

1

(¯ g). We then obtain that µ

1

(¯ g) > lim sup

_n

µ

1

(f

_n²

g). This

proves Lemma 2.1.

In the same way, if ¯ g = f

²

g is a metric conformal to g where f is positive and smooth, we define

J

_¯g^′

(ψ) = Z

M

| D

g¯

ψ |

²¯g

f

⁻¹

dv

g¯

Z

M

h D

¯g

ψ, ψ i

^¯g

dv

¯g

.

The first eigenvalue of the Dirac operator D

¯g

is then given by λ

⁺₁

(¯ g) = inf J

_g¯^′

(ψ) where the infimum is taken over the smooth spinor fields ψ for which R

M

h D

g

ψ, ψ i dv

g

> 0. Now, it is well known (see [Hit74, Hij01]) that where can identify isometrically on each fiber spinor fields for the metric g and spinor fields for the metric ¯ g. Moreover, we have for all smooth spinor field:

D

g¯

(f

⁻¹²

ψ) = f

⁻³²

D

g

ψ.

This implies that if we set ϕ = f

⁻¹²

ψ, we have

J

g¯

(ϕ) :=

Z

M

| D

g

ϕ |

²

f

⁻¹

dv

g

Z

M

h D

g

ϕ, ϕ i dv

g

= J

_g¯^′

(ψ)

and the first eigenvalue of the Dirac operator D

¯g

is given by λ

⁺₁

(¯ g) = inf J

¯g

(ϕ) where the infimum is taken over the smooth spinor fields ϕ for which

Z

M

h Dϕ, ϕ i dv

g

> 0. With the definition above, we can extend the definition of λ

1

(¯ g) when ¯ g is a generalised metric. By standard methods, there exists a spinor fields ϕ ∈ C

¹

(M ) such that λ

⁺₁

(¯ g) = J

¯g

(ϕ) and such that

D

g

ϕ = λ

⁺₁

(¯ g)f ϕ. (4)

We then have a result similar to Lemma 2.1:

Lemma 2.2. If (f

n

) is a sequence of smooth positive functions which converges uniformly to f , then λ

1

(f

_n²

g) tends to λ

1

(¯ g).

The proof is similar to the one of Lemma 2.1 and we omit it here.

3. The metrics (g

α,ε

)

α,ε

In this paragraph, we construct the metrics (g

α,ε

)

α,ε

which will satisfy:

lim sup

ε→0

λ

1

(g

α,ε

)

²

Vol

gα,ε

(M ) 6 4π + C(α) (5) where C(α) is a positive constant which goes to 0 with α and

lim inf

ε→0

µ

1

(g

α,ε

)Vol

gα,ε

(M ) > 8π. (6) Clearly this implies Theorem 1.1. By Lemmas 2.1 and 2.2, one can assume that the metrics (g

α,ε

)

α,ε

are generalized metrics. We just have to define the volume of M for generalized metric by Vol

f²g

(M ) = R

M

f

²

dv

g

. At first, without loss of generality, we can assume that g is flat near a point p ∈ M . Let α > 0 be a small number to be fixed later such that g is flat on B

p

(α). We set for all x ∈ M and ε > 0,

f

α,ε

(x) = (

ε²

ε²+r²

if r 6 α

ε²

ε²+α²

if r > α

where r = d

g

(., p). The function f

α,ε

is of class C

^0,a

for all a ∈ ]0, 1[ and is positive on M . We then define for all ε > 0, g

α,ε

= f

_α,ε²

g. The 2-forms (g

α,ε

)

α,ep

will be the desired generalized metrics. For these metrics, we have

Vol

gα,ε

(M ) = Z

M

f

_α,ε²

dv

g

= Z

Bp(α)

f

_α,ε²

dv

g

+ Z

M\Bp(α)

f

_α,ε²

dv

g

.

Since g is flat on B

p

(α), we have Z

Bp(α)

f

_α,ε²

dv

g

= Z

2π

0

Z

α

0

ε

⁴

r

(ε

²

+ r

²

)

²

drdΘ.

Setting y =

^x_ε

we obtain:

Z

Bp(α)

f

_α,ε²

dv

g

= 2πε

²

Z

^α_ε

0

r

(1 + r

²

)

²

dr = 2πε

²

Z

+∞

0

r

(1 + r

²

)

²

dr + o(1)

= πε

²

+ o(ε

²

).

Since f

_α,ε²

6

^ε4

α⁴

on M \ B

p

(α), we have R

M\Bp(α)

f

_α,ε²

dv

g

= o(ε

²

). We obtain

Vol

gε

(M ) = πε

²

+ o(ε

²

). (7)

In the whole paper, the notation “o(.)” must be understood as ε tends to 0.

4. Proof of relation (5)

Let f : R

²

→ R

²

be defined by f (x) =

_1+|x|² 2

. Let ψ

0

be a non-zero parallel spinor field on R

²

such that

| ψ

0

|

²

= 1. As in [AHM03], we set on R

²

ψ(x) = f

ⁿ²

(x)(1 − x) · ψ

0

. As easily computed, we have on R

²

Dψ = f ψ and | ψ | = f

¹²

. (8)

Now, we fixe a small number δ > 0 such that g is flat on B

p

(δ). Then, we take ε 6 α 6 δ. We will let ε goes to 0. We let also η be a smooth cut-off function defined on M such that 0 6 η 6 1, η(B

p

(δ)) = { 1 } , η(M \ B

p

(2δ)) = { 0 } . Identifying B

p

(δ) in M with B

0

(δ) in R

²

, we can define a smooth spinor field on M by ψ

ε

= η(x)ψ

^x_ε

. Using (8), we have D

g

(ψ

ε

) = ∇ η · ψ x

ε + η

ε f x ε

ψ x ε

. (9)

Since h∇ η · ψ(

^x_ε

), ψ(

^x_ε

) i ∈ i R and since | D

g

ψ

ε

|

²

∈ R , we have Z

M

| D

g

ψ

ε

|

²

f

_α,ε⁻¹

dv

g

= I

1

+ I

2

(10) where

I

1

= Z

M

|∇ η |

²

ψ x

ε

2

dx and I

2

= Z

M

η

²

ε

²

f

²

x

ε ψ x

ε

2

f

_α,ε⁻¹

dx.

At first, let us deal with I

1

. By (8), I

1

6 C

Z

M

f x ε

f

_α,ε⁻¹

dx = C Z

Bp(α)

f x ε

f

_α,ε⁻¹

dx + C Z

Bp(2δ)\Bp(α)

f x ε

f

_α,ε⁻¹

dx

where, as in the following, C denotes a constant independant of α and ε. On B

p

(α), f

^x_ε

f

_α,ε⁻¹

= 2.

Hence,

Z

Bp(α)

f x ε

f

_α,ε⁻¹

dx 6 Cα

²

.

On B

p

(2δ) \ B

p

(α), since ε 6 α, f x

ε

f

_α,ε⁻¹

6 4α

²

ε

²

+ r

²

= 4α

²

ε

²

(1 +

^r_ε

2

)

Hence,

Z

Bp(2δ)\Bp(α)

f x ε

f

_α,ε⁻¹

dx 6 4α

²

ε

²

Z

2π

0

Z

δ

α

r (1 +

^r_ε

2

) drdΘ 6 8πα

²

Z

^δ_ε

α ε

r (1 + r

²

) dr 6 8πα

²

ln

ε

²

+ δ

²

ε

²

+ α

²

. We get

Z

Bp(2δ)\Bp(α)

f x ε

f

_α,ε⁻¹

dx 6 Cα

²

ln 2δ

²

α

²

.

Finally, we obtain

I

1

6 Cα

²

+ C ln 2δ

²

α

²

= a(α) (11)

where a(α) goes to 0 with α. Now, by (8), I

2

6 C

Z

Bp(2δ)

f

³

( x

ε )f

_α,ε⁻¹

dx.

Since f

α,ε

>

¹₂

f (

^x_ε

), we have

I

2

6 2 ε

²

Z

Bp(2δ)

f

²

( x ε )dx.

Mimicking what we did to get (7), we obtain that

I

2

6 8π + o(1) when ε tends to 0. Together with (10) and (11), we obtain

Z

M

| D

g

ψ

ε

|

²

f

_α,ε⁻¹

dv

g

6 8π + a(α) + o(1). (12) In the same way, by (9), since

Z

M

h D

g

(ψ

ε

), ψ

ε

i dv

g

∈ R and since h∇ η · ψ(

^x_ε

), ψ(

^x_ε

) i ∈ i R , we have Z

M

h D

g

(ψ

ε

), ψ

ε

i dv

g

= Z

M

η

²

ε f x

ε ψ x

ε

2

dv

g

. By (8), this gives

Z

M

h D

g

(ψ

ε

), ψ

ε

i dv

g

= Z

M

η

²

ε f

²

x

ε dv

g

.

With the computations made above, it follows that Z

M

h D

g

(ψ

ε

), ψ

ε

i dv

g

= 4πε + o(ε).

Together with (12) and (7), we obtain λ

1

(g

α,ψ

)

²

Vol

gα,ψ

(M ) 6 J

gα,ψ

(ψ

ε

)

2

Vol

gα,ψ

(M ) 6

8π + a(α) + o(1) 4πε + o(ε)

2

(πε

²

+o(ε

²

)) = 1

ε (4π + a(α) + o(1)) .

Relation (5) immediatly follows.

5. Proof of relation (6) First we need the following estimate

Lemma 5.1. For any ε > 0 and u ∈ C

_c^∞

(B

p

(α)), then Z

M

u

²

f

_α,ε²

dv

g

6 ε

²

8

Z

M

|∇ u |

²

dv

g

+ 1 πε

²

Z

M

uf

_α,ε²

dv

g

2

.

Proof. Let g

ε

= f

_α,ε²

g. Then (B

p

(α), g

ε

) is embedded in a canonical sphere of volume Z

R²

ε

²

ε

²

+ r

²

²

dx = πε

²

. Then from the Poincar´e-Sobolev inequality, we have

Z

M

u

²

dv

gε

6 1 µ

1,ε

Z

M

|∇

^ε

u |

²gε

dv

gε

+ 1 V

ε

Z

M

udv

gε

2

where µ

1,ε

=

_ε⁸²

is the first nonzero eigenvalue of the Laplacian on the sphere of volume V

ε

= πε

²

and ∇

^ε

u denotes the gradient of u with respect to the metric g

ε

. Now since |∇

^ε

u |

²gε

= f

_α,ε⁻²

|∇ u |

²g

and dv

gε

= f

_α,ε²

dv

g

, we get the desired result.

Lemma 5.2. For any u, v ∈ C

^∞

(M ), we have Z

M

(∆u)uv

²

dv

g

= Z

M

|∇ (uv) |

²g

dv

g

− Z

M

u

²

|∇ v |

²g

dv

g

. Proof. The proof is an elementary calculation.

Because of the relation (7), the inequality (6) is equivalent to the following lim inf

ε−→0

ε

²

µ

1

(g

ε

) > .8 (13)

In order to prove this inequality, we assume that for any ε small enough, there exists k, 0 < k < 1 so that

µ

1

(g

ε

) < 8

ε

²

k. (14)

Let u

ε

be an eigenfunction associated to µ

1

(g

ε

). Then u

ε

∈ C

²

(M ) and ∆

gε

u

ε

= µ

1

(g

ε

)u

ε

where ∆

gε

denotes the Laplacian associated to the metric g

ε

. Since the dimension is 2, ∆

gε

=

_f2¹

α,ε

∆ and

∆u

ε

= µ

1

(g

ε

)f

_α,ε²

u

ε

. (15)

We normalize u

ε

so that k u

ε

k

H1²

= 1. Up to a subsequence we can assume that Z

M

|∇ u

ε

|

²

dv

g

−→ l and Z

M

u

²_ε

dv

g

−→ l

^′

with l + l

^′

= 1. Since (u

ε

) is bounded in H

₁²

, there exists a subsequence so that u

ε

−→ u weakly in H

₁²

. In the following, all the convergences are up to subsequence. We sometimes omit to recall this fact.

Lemma 5.3. There exists a constant c

0

such that u = c

0

.

Proof. Let ϕ ∈ C

^∞

(M ) and

η

ρ

:=

1 on B

p

(ρ) 0 on M \ B

p

(2ρ) satisfying 0 6 η

ρ

6 1 and |∇ η

ρ

| 6

¹_ρ

. We have

Z

M

h∇ u, ∇ ϕ i = Z

M

h∇ u, ∇ (η

ρ

ϕ) i dv

g

+ Z

M

h∇ u, ∇ ((1 − η

ρ

)ϕ) i dv

g

. (16) Now we have

Z

M

h∇ u, ∇ (η

ρ

ϕ) i dv

g

= Z

M

h∇ u, ∇ η

ρ

i ϕdv

g

+ Z

M

h∇ u, ∇ ϕ i η

ρ

dv

g

6 C Z

Bp(2ρ)

|∇ u |

²

dv

g

!

1/2

Z

Bp(2ρ)

|∇ η

ρ

|

²

dv

g

!

1/2

+ Z

Bp(2ρ)

|∇ u |

²

dv

g

!

1/2

Z

Bp(2ρ)

|∇ ϕ |

²

dv

g

!

1/2

.

The limit of the last term is 0 when ρ −→ 0. Moreover from the definition of η

ρ

and from the fact that M is a 2-dimensional locally flat domain, the limit of

Z

Bp(2ρ)

|∇ η

ρ

|

²

dv

g

!

^1/2

is bounded in a neighborhood of 0. Then we deduce that

Z

M

h∇ u, ∇ (η

ρ

ϕ) i dv

g

−→ 0 (17)

when ρ −→ 0. On the other hand

Z

M

h∇ u, ∇ ((1 − η

ρ

)ϕ) i dv

g

= lim

ε−→0

Z

M

h∇ u

ε

, ∇ ((1 − η

ρ

)ϕ) i dv

g

= lim

ε−→0

Z

M

(∆u

ε

)(1 − η

ρ

)ϕdv

g

= lim

ε−→0

µ

1

(g

ε

) Z

M

f

_α,ε²

u

ε

(1 − η

ρ

)ϕdv

g

.

Now from the definition of f

α,ε

and from (14) we get

µ

1

(g

ε

) Z

M

f

_α,ε²

u

ε

(1 − η

ρ

)ϕdv

g

6 8 ε

²

kCε

⁴

Z

M

u

²_ε

dv

g

1/2

Z

M

(1 − η

ρ

)ϕ

²

dv

g

1/2

where C is a constant depending on the compact support of (1 − η

ρ

)ϕ. Then making ε −→ 0, we deduce that

Z

M

h∇ u, ∇ ((1 − η

ρ

)ϕ) i dv

g

= 0.

Now, reporting this and (17) in (16) we obtain that Z

M

h∇ u, ∇ ϕ i dv

g

= 0 and ∆u = 0 on M in the sense of distributions. This implies that u ≡ c

0

on M for a constant c

0

.

Lemma 5.4. Let (c

ε

)

ε

be a bounded sequence of real numbers. Then

Z

M

f

_α,ε²

u

²_ε

dv

g

6 O(ε

²

k u

ε

− c

ε

k

²L²

+ ε

⁴

).

Proof. Let η be a C

^∞

function defined on M so that

η :=

1 on B

p

(α/2) 0 on M \ B

p

(α) satisfying 0 6 η 6 1 and |∇ η | 6 1.

From the lemma 5.1, we have Z

M

(u

ε

− c

ε

)

²

f

_α,ε²

η

²

dv

g

6 ε

²

8

Z

M

|∇ ((u

ε

− c

ε

)η) |

²

dv

g

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

and applying the lemma 5.2 to the first term of the right hand side, we get Z

M

(u

ε

− c

ε

)

²

f

_α,ε²

η

²

dv

g

6 ε

²

8 Z

M

(∆(u

ε

− c

ε

))(u

ε

− c

ε

)η

²

dv

g

+ ε

²

8

Z

M

(u

ε

− c

ε

)

²

|∇ η |

²

dv

g

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

.

From (15) we deduce that Z

M

(u

ε

− c

ε

)

²

f

_α,ε²

η

²

dv

g

6 ε

²

8 µ

1

(g

ε

) Z

M

u

ε

(u

ε

− c

ε

)η

²

f

α,ε²

dv

g

+ ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

α,ε²

dv

g

2

. First case : assume that

Z

M

u

ε

(u

ε

− c

ε

)η

²

f

_α,ε²

dv

g

> 0.

The relation (14) implies Z

M

(u

ε

− c

ε

)

²

f

_α,ε²

η

²

dv

g

6 k Z

M

u

ε

(u

ε

− c

ε

)η

²

f

_α,ε²

dv

g

+ ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

. A straightforward computation shows that

(1 − k) Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

+ c

²_ε

Z

M

f

_α,ε²

η

²

dv

g

6 (2 − k)c

ε

Z

M

u

ε

f

_α,ε²

η

²

dv

g

+ ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

. (18)

Now note that

Z

M

u

ε

f

_α,ε²

η

²

dv

g

= Z

M

u

ε

f

_α,ε²

(η

²

− 1)dv

g

+ Z

M

u

ε

f

_α,ε²

dv

g

= Z

M

u

ε

f

_α,ε²

(η

²

− 1)dv

g

+ 1 µ

1

(g

ε

)

Z

M

∆u

ε

dv

g

= Z

M

u

ε

f

_α,ε²

(η

²

− 1)dv

g

6 Z

M\Bp(α/2)

u

ε

f

_α,ε²

(η

²

− 1)dv

g

and from the definition of f

α,ε

and η and from the fact that u

ε

is bounded in L

²

, we deduce that Z

M

u

ε

f

_α,ε²

η

²

dv

g

= O(ε

⁴

).

Since c

ε

is bounded (18) becomes

(1 − k) Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

+ c

²_ε

Z

M

f

_α,ε²

η

²

dv

g

6 O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

= O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

f

_α,ε²

u

ε

(η − 1)dv

g

+ Z

M

f

_α,ε²

u

ε

dv

g

− c

ε

Z

M

f

_α,ε²

η

2

= O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

f

_α,ε²

u

ε

(η − 1)dv

g

− c

ε

Z

M

f

_α,ε²

η

2

(19)

where in the last equality we have used the fact that Z

M

f

_α,ε²

u

ε

dv

g

= 1 µ

1

(g

ε

)

Z

M

∆u

ε

dv

g

= 0.

Using the same arguments as above we see that Z

M

f

_α,ε²

u

ε

(η − 1)dv

g

= O(ε

⁴

). Reporting this in (19) we get

(1 − k) Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

+c

²_ε

Z

M

f

_α,ε²

η

²

dv

g

6 O(ε

⁴

)+ ε

²

8 k u

ε

− c

ε

k

²L²

+ O(ε

⁴

) ε

²

Z

M

f

_α,ε²

ηdv

g

+ c

²_ε

πε

²

Z

M

f

_α,ε²

ηdv

g

2

.

Now

Z

M

f

α,ε²

ηdv

g

= Z

Bp(α)

f

α,ε²

dv

g

= Z

2π

0

Z

α

0

ε

⁴

r

(ε

²

+ r

²

)

²

drdΘ

= 2πε

²

Z

α/ε

0

t (1 + t

²

)

²

dt 6 2πε

²

Z

+∞

0

t (1 + t

²

)

²

dt

= πε

²

.

This gives

(1 − k) Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

+ c

²_ε

Z

M

f

_α,ε²

η

²

dv

g

6 O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ c

²_ε

πε

²

Z

M

f

_α,ε²

ηdv

g

2

= O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ c

²_ε

Z

M

f

_α,ε²

ηdv

g

. Finally we have

(1 − k) Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

6 O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ c

²_ε

Z

M

f

_α,ε²

(η − η

²

)dv

g

6 O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ c

²ε

Z

Bp(α)\Bp(α/2)

f

α,ε²

dv

g

= O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

. (20)

Second case : Assume that Z

M

u

ε

(u

ε

− c

ε

)η

²

f

_α,ε²

dv

g

6 0.

In this case, we have Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

− 2c

ε

Z

M

u

ε

f

_α,ε²

η

²

dv

g

+ c

²_ε

Z

M

f

_α,ε²

η

²

dv

g

6 O(ε

⁴

) + ε

²

8 k u

ε

− c

ε

k

²L²

+ 1 πε

²

Z

M

(u

ε

− c

ε

)ηf

_α,ε²

dv

g

2

(21) and we conclude as in the previous case.

Then we have proved that Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

= O(ε

⁴

+ ε

²

k u

ε

− c

ε

k

²L²

).

To finish the proof, we write Z

M

u

²_ε

f

_α,ε²

dv

g

= Z

M

u

²_ε

f

_α,ε²

η

²

dv

g

+ Z

M

u

²_ε

f

_α,ε²

(1 − η

²

)dv

g

and the last term is O(ε

⁴

) which completes the proof.

Proof. of Relation (13). First we apply the lemma 5.4 to c

ε

= c

0

and we see that c

0

6 = 0. Indeed, let us compute the L

²

-norm of the gradient of u

ε

.

Z

M

|∇ u

ε

|

²

dv

g

= Z

M

(∆u

ε

)u

ε

dv

g

6 8k ε

²

Z

M

f

_α,ε²

u

²_ε

dv

g

= 8k

ε

²

O(ε

²

k u

ε

− c

0

k

²L²

+ ε

⁴

)

= o(1).

Then we deduce that up to a subsequence

Z

M

|∇ u

ε

|

²

dv

g

−→ 0.

But we have chosen u

ε

so that k u

ε

k

H1²

= 1. Then k u

ε

k

L²

−→ 1 and c

0

6 = 0.

Now let us consider u

ε

=

_Vol(M¹ ₎

Z

M

u

ε

dv

g

and a

ε

= k u

ε

− u

ε

k

H1²

. Then u

ε

−→ c

0

and a

ε

−→ 0. It follows that the function v

ε

=

^uε_a^−uε

ε

satisfies k v

ε

k

H1²

= 1 and there exists v ∈ H

₁²

so that v

ε

−→ v weakly in H

₁²

and strongly in L

²

.

To prove (13) we will consider two cases.

First case : Assume that up to a subsequence a

ε

= O(ε).

We have

Z

M

(∆u

ε

)

²

dv

g

= µ

1

(g

ε

)

²

Z

M

f

_α,ε⁴

u

²_ε

dv

g

6 µ

1

(g

ε

)

²

Z

M

f

α,ε²

u

²ε

dv

g

6 64k

ε

⁴

O(ε

²

k u

ε

− u

ε

k

²L²

+ ε

⁴

) 6 64k

ε

⁴

O(ε

²

a

²_ε

+ ε

⁴

) 6 M.

Then k ∆u

ε

k

L²

, k∇ u

ε

k

L²

and k u

ε

k

L²

are bounded. It well known that the norms k v k := k ∆v k

L²

+ k ∇ v k

L²

+ k v k

L²

and k v k

H2²

are equivalent (it is a direct consequence of Bochner formula). Hence, this implies that (u

ε

)

ε

is bounded in H

₂²

which is embedded in C

⁰

. Then u

ε

−→ c

0

uniformily up to a subsequence. Since c

0

6 = 0 it follows that for ε small enough u

ε

has a constant sign, which is not possible because u

ε

is an eigenfunction in the metric g

ε

.

Second case : Assume that ε = a

ε

o(1). In this case we have the Lemma 5.5. v

ε

−→ c

1

in H

₁²

where c

1

is a constant.

Proof. The proof is similar to this of lemma 5.3. Indeed we consider ϕ ∈ C

^∞

(M ) and the function η

ρ

defined in this previous proof. Then Z

M

h∇ v, ∇ ϕ i = Z

M

h∇ v, ∇ (η

ρ

ϕ) i dv

g

+ Z

M

h∇ v, ∇ ((1 − η

ρ

)ϕ) i dv

g

. By the same arguments we have

Z

M

h∇ v, ∇ (η

ρ

ϕ) i dv

g

−→ 0 when ρ −→ 0. Moreover

Z

M

h∇ v, ∇ ((1 − η

ρ

)ϕ) i dv

g

= lim

ε−→0

Z

M

h∇ v

ε

, ∇ ((1 − η

ρ

)ϕ) i dv

g

= lim

ε−→0

Z

M

(∆v

ε

)(1 − η

ρ

)ϕdv

g

= lim

ε−→0

µ

1

(g

ε

) a

ε

Z

M

f

_α,ε²

v

ε

(1 − η

ρ

)ϕdv

g

.

Now

µ1(gε) aε

Z

M

f

_α,ε²

v

ε

(1 − η

ρ

)ϕdv

g

6

_a^8k

εε²

Cε

⁴

k v

ε

k

L²

Z

M

(1 − η

ρ

)ϕ

²

dv

g

1/2

. Since ε = a

ε

o(1), we de- duce that

Z

M

h∇ v, ∇ ((1 − η

ρ

)ϕ) i dv

g

= 0 and then Z

M

h∇ v, ∇ ϕ i = 0. Therefore ∆v = 0 in sense of distributions and v = c

1

on M .

Now let c

ε

= u

ε

+ a

ε

c

1

. Then c

ε

−→ c

0

. We denotes by µ(g) the smallest positive eigenvalue of the Laplacian with respect to the metric g. From the definition of a

ε

and the definition of µ(g), we have

a

²_ε

6 2 Z

M

|∇ u

ε

|

²

dv

g

+ Z

M

(u

ε

− u

ε

)

²

dv

g

6 2

1 + 1

µ(g) Z

M

|∇ u

ε

|

²

dv

g

= 2

1 + 1 µ(g)

Z

M

∆u

ε

u

ε

dv

g

= 2

1 + 1 µ(g)

µ

1

(g

ε

)

Z

M

f

_α,ε²

u

²_ε

dv

g

. (22) Applying lemma 5.4 we get

Z

M

f

_α,ε²

u

²_ε

dv

g

= O(ε

²

k u

ε

− c

ε

k

²L²

+ ε

⁴

)

= O(ε

²

k u

ε

− u

ε

− a

ε

c

1

k

²L²

+ ε

⁴

)

= O a

²_ε

ε

²

u

ε

− u

ε

a

ε

− c

1

2

L²

+ ε

⁴

!

= O(a

²_ε

ε

²

k v

ε

− c

1

k

²L²

+ ε

⁴

)

= O(ε

⁴

) + o(a

²_ε

ε

²

).

Now reporting this in (22) with the estimate (14) we find

a

²_ε

6 C 8k

ε

²

(O(ε

⁴

) + o(a

²_ε

ε

²

))

= O(ε

²

) + a

²_ε

o(1).

But ε = a

ε

o(1). Then a

²_ε

6 Ca

²_ε

o(1) and for ε small enough a

ε

= 0 and u

ε

is a constant which is impossible.

References

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Authors’ address:

Jean-Fran¸cois Grosjean and Emmanuel Humbert, Institut ´ Elie Cartan BP 239

Universit´e de Nancy 1

54506 Vandoeuvre-l`es -Nancy Cedex France

E-Mail:

[email protected], [email protected]