Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary

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operator on bounded domains with smooth boundary

Daniel Maerten

To cite this version:

Daniel Maerten. Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains

with smooth boundary. 2007. �hal-00199521�

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hal-00199521, version 1 - 19 Dec 2007

DANIELMAERTEN

Abstrat. Eigenvalue estimate for the DiraWitten operator is given on bounded

domains (with smooth boundary) of spaelike hypersurfaes satisfying the dominant

energy ondition,underfour natural boundaryonditions (MIT,APS, modiedAPS

andhiralonditions). Roughlyspeaking, anyeigenvalueoftheDiraWittenoperator

satises

|λ|²> n (n−1)R₀ ,

where R₀ is the innimumof (theoppositeof) the lorentziannormof the onstraints vetor. Equalityasesarealsoinvestigatedandleadtointerestinggeometrisituations.

1. Introdution

Theaimofspetralgeometryistoderivesomegeometripropertiesfromthestudyofthe

spetrumof a ertainellipti operator, whih is mostlya Laplae operator (Laplaian on

funtions, Hodge Laplaian on

p

^forms ^or DiraLaplaian on spinors). In that ontext, a lassial issue is to give lower bounds for the eigenvalues of the Dira operator on a

ompat manifold

(M

ⁿ

, g)

^,

(n > 2)

^. În ^{[5 ℄,} ^T. ^F^riedrih ^proved ^that âny êigenvalue

λ

^of

theDira operator satises

(1.1)

λ

²

> n

4(n − 1) inf

M

Scal

^g

,

where

Scal

^g ^denotes ^the ^salar ^urvature ^of

(M

ⁿ

, g)

^. ^A ^few ^years ^later, ^O. ^Hijazi ^{[9 ,} ^10℄

improvedthis result byshowing that, for any

n > 3

^,

(1.2)

λ

²

> n

4(n − 1) µ

₁

,

where

µ

₁ îs ^the ^rst êigenvâlue ôf ^the ônformal ^Laplaian. ^Clearly, êstimates ^{(1.1 )} ând

(1.2)areunderinterestonlyifthe salarurvatureispositive. Inareentwork,O. Hijazi

andX.Zhang [12℄establishedananalogous versionof(1.1 )for theDiraWittenoperator

underahiralboundaryondition, ofaompatspaelikehypersurfae

(M, g, k)

^satisfying

thedominant energy ondition, namely

(1.3)

λ

²

> n 4(n − 1) inf

M

n

Scal

^g

+ (tr

_g

k)

²

− |k|

²_g

− 2 |δ

_g

k + dtr

_g

k|

_g

o .

On the other hand, S. Raulot [16℄ proved an eigenvalue estimate for the Dira operator

ondomainswithboundary. Morepreisely,if

Ω

îsâ ômpat^domain ôfân

n

dimensional Riemannianspinmanifold

(M, g)

^,^whose^boundary

∂Ω

^has^positive^mean^urvature

H

^,^he

showedundera naturalboundaryondition (alled"MIT"boundary ondition), thatany

Date:19thDeember2007.

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eigenvalue

λ

ôf ^the ^spetrumôf ^the^Diraôperator ôn

Ω

^(whih îsân ûnbounded ^disrete

setofomplex numbers ofpositive imaginarypart)satises

(1.4)

|λ|

²

> n

4(n − 1) inf

Ω

Scal

^g

+n Im(λ) inf

∂Ω

H .

Themostinteresting fatinInequality(1.4) isthatthebound dependsonsomeboundary

geometriquantity,whihisnottheaseforInequality(1.3). Inaddition,S.Raulotshowed

thatequalityin (1.4) leads to the existene ofimaginary Killing spinor on

Ω

^,^and ^also ^to

theonlusion that the boundary

∂Ω

îs â ^totally ûmbilial ând ônstant ^mean ûrvature

hypersurfae.

The goal of this artile is to generalise Inequalities (1.3 ) and (1.4 ) in several dire-

tions. Indeed,weshallprove analogous versionsof(1.3) fortheDiraWittenoperator on

bounded domains, under four natural boundary onditions (see [11 ℄) and also generalise

(1.4)for the DiraWitten operator.

The artile is organised as follows: In Setion 2, we give our geometri onventions and

preliminary results; Setion 3 is devoted to the statements of the main results and their

proves.

2. Geometri Bakground

2.1. Notations. We onsider

(N

ⁿ⁺¹

, γ)

^a^Lorentzian ^manifold^of^signature

(−, +, · · · , +)

whih ontains

(M

ⁿ

, g, k)

^,^a ^spin⁽ⁱⁿ^dimension ³^this ^only ^meansorientable) Riemannian hypersurfae(notneessarilyompat)whoseinduedmetriis

g

^and^seondfundamental form (extrinsi urvature) is

k

^. ^Let

Ω

^be ^a ^ompat ^domain ⁱⁿ

(M

ⁿ

, g, k)

^satisfying ^the

dominant energy ondition, whih readsasthe following inequality along

Ω Scal

^g

+ (tr

_g

k)

²

− |k|

²_g

> 2 |δ

_g

k + dtr

_g

k|

_g

.

We willworkwiththe omplexspinor bundleof

N

^restrited ^to ^thehypersurfae domain

Ω

^,^that^is^to ^say

Σ := Σ(N )

_|Ω ^whih îs^given^by^the^hoieôfâ ûnit^normal

e

₀ ^of

M

ⁱⁿ

N

^,

along

Ω

^. ^More ^preisely^, ^if^one ^denotes ^by

P

Spin(n,1)

(N )

^the^bundle ^of ^Spin

(n, 1)

^frames

on

N

^,^and ^by

ρ

n,1 ^the ^standardrepresentation ofSpin

(n, 1)

^then

Σ(N ) := P

Spin(n,1)

(N ) ×

ρn,1

C

^[(n+1)/2]

.

Now the hoie of unit normal

e

₀ ^of

M

ⁱⁿ

N

^, ^along

Ω

^, îndues â ^natural înlusion

Spin

(n) ⊂

^Spin

(n, 1)

^and^so^we ^an ^dene

Σ := P

Spin(n,1)

(N )

_|Ω

×

_(ρ_n,1₎_|

Spin(n)

C

^[(n+1)/2]

.

Σ

^naturally ^arries ^two sesquilinear inner produts: the rst one denoted by

(∗, ∗)

^is

Spin

(n, 1)

^-invariant ^(it îs ^not ^neessary ^positive); ^the ^seond ône ^whih îs ^denoted ^by

h∗, ∗i := (e

₀

· ∗, ∗)

^is ^Spin

(n)

^-inv^ariant ^and ^Hermitian ^denite ^positive ⁽

·

^is ^the ^Cliord

ation with respet to the metri

γ

^). ^The ^Hermitian ^or anti-Hermitian harater of the Cliord multipliation by vetors diers if we onsider

(∗, ∗)

^or

h∗, ∗i

^and ^is ^desribed

in [13 ℄, for instane.

Σ

îs âlso êndowed ^with ^two ^dierent ônnetions

∇, ∇

^whih ^are

respetively the Levi-Civitaonnetions of

γ

^and

g

^. ^Letûs ^take â ^spinor êld

ψ ∈ Γ(Σ)

anda vetor eld

X ∈ Γ(T Ω)

^,^thenôur ônventions âre

∇

_X

ψ = ∇

_X

ψ − 1

2 k(X) · e

₀

· ψ

hk(X), Y i

_γ

= h∇

X

Y, e

0

i

_γ

.

(4)

Inthese formulae

·

^denotes ^the^Cliord ^ation ^with^respet ^to ^the^metri

γ

^. ^The^indued

metri, Levi-Civita onnetion and seond fundamental form of the boundary

∂Ω

^, ^are

respetivelydenotedby

ℓ, ∇, θ e

^. Ôurônventionsâre,^forâny

X, Y ∈ Γ(T ∂Ω)

^and

ψ ∈ Σ

_|∂Ω^,

∇

X

Y = ∇ e

X

Y + θ(X, Y )ν

∇

_X

ψ = ∇ e

_X

ψ + 1

2 θ(X) · ν · ψ,

where

ν

^is^the ^unit^normal^to

∂Ω

^pointing^inside

Ω

^,^and

·

^still ^denotes ^the^Cliord^ation

withrespetto the metri

γ

^. ^Finally^we ^dene ^the ^following ^geometri ^quantity

R

₀

:= 1

4 inf

Ω

n

Scal

^g

+ (tr

_g

k)

²

− |k|

²_g

− 2 |δ

_g

k + dtr

_g

k|

_g

o .

2.2. Dira-Witten operators and BohnerLihnerowiz formulæ. From now on,

(e

_k

)

ⁿ_k=0 ^denotes^anorthonormalbasisatthepoint,withrespetto themetri

γ

^,^and^where

ν = e

₁^. ^Wê ân ^then^dene ^the^DiraWittenôperator ôf

∇

D ϕ = X

n k=1

e

_k

· ∇

ek

ϕ ,

where

·

^is^the^Cliord^ation^with^respet^to^the^metri

γ

^. ^Notie^that

D

anbeonsidered asa deformation of

D

the usual Dira operator of

∇

^sine ^we ^reover

D = D

as soon as

k ≡ 0

^. ^The Dira-Witten operator

D

islearly formally selfadjoint in

L

² ^with^respet ^to

h∗, ∗i

ⁱⁿ^the^lassôfômpatly^supported^spinor êlds,ând^we^have ^the^lassial^Bohner-

Lihnerowiz-Weitzenbök formula (f.[1, 7 ,15℄for instane)

(2.1)

D

^∗

D = DD = ∇

^∗

∇ + R ,

where

R := 1 4

^Sal

γ

+ 4

^Ri^γ

(e

0

, e

0

) + 2e

0

·

^Ri^γ

(e

0

)

= 1 4

n

Scal

^g

+ (tr

_g

k)

²

− |k|

²_g

+ 2(δ

_g

k + dtr

_g

k) · e

₀

o .

As usual, we derive from (2.1 ) an integration formula. The idea is to onsider a ertain

spinoreld

ϕ

^and^to ^dene ^the ^1form

ω

_ϕ

∈ Γ(T

^∗

Ω)

^by^the^relation

ω

ϕ

(X) = h∇

X

ϕ + X · D ϕ, ϕi .

Then, omputingthe

g

^divergene ^of

ω

_ϕ^,^we ^obtain

div

_g

ω

_ϕ

= | D ϕ|

²

− |∇ϕ|

²

− h R ϕ, ϕi ,

whih gives,applying Stokes' theorem

(2.2)

Z

Ω

| D ϕ|

²

= Z

Ω

|∇ϕ|

²

+ Z

Ω

h R ϕ, ϕi + Z

∂Ω

h∇

_ν

ϕ + ν · D ϕ, ϕi .

We now have to introdue

P

^the ^twistor ^operator ^with ^respet ^to ^the ^onnetion

∇

^,

whih isdened bytherelation

P

_X

ϕ := ∇

_X

ϕ + 1

n X · D ϕ ,

for every

X ∈ Γ(T Ω)

ând êvery ^spinor êld

ϕ ∈ Γ(Σ)

^. ^W^e ^derive ^a ^seond integration

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2.1. Proposition. Forany spinor eld

ϕ ∈ Γ(Σ)

^, ^we ^have

(2.3)

Z

Ω

|P ϕ|

²

=

n − 1 n

Z

Ω

| D ϕ|

²

− Z

Ω

h R ϕ, ϕi − Z

∂Ω

ω

ϕ

(ν) .

Proof: We rst provethe BohnerLihnerowiz formulafor

P

^,^namely

|P ϕ|

²

= X

n k=1

∇

_e_k

ϕ + 1

n e

_k

· D ϕ, ∇

_e_k

ϕ + 1

n e

_k

· D ϕ

= |∇ϕ|

²

+ 1

n − 2 n

| D ϕ|

²

= |∇ϕ|

²

− 1

n | D ϕ|

²

.

Weintegrate this formulaon

Ω

^so^that

Z

Ω

|P ϕ|

²

= Z

Ω

|∇ϕ|

²

− 1 n

Z

Ω

| D ϕ|

²

=

n − 1 n

Z

Ω

| D ϕ|

²

− Z

Ω

h R ϕ, ϕi − Z

∂Ω

ω

_ϕ

(ν) ,

where theseondline isobtained thanksto (2.2).

Itislear thatthe valueof the1form

ω

_ϕ ^depends^upon^the^boundary ^ondition^that^will

be used. We will need the intermediateresult:

2.2. Lemma. For any spinor eld

ϕ ∈ Γ(Σ)

^, ^we ^have ^along ^the ^boundary

∂Ω

(2.4)

ω

_ϕ

(ν) =

ν · e

_j

· ∇ e

_e_j

ϕ + 1 2

(tr

_g

k)ν · e

₀

· −k(ν) · e

₀

· +(tr

_ℓ

θ) ϕ, ϕ

.

Proof: Just ompute, using the relations of ompatibility between thedierent onne-

tions

∇

ν

ϕ + ν · D ϕ = ν · X

n j=2

e

j

· ∇

ej

ϕ

= ν · e

_j

· ∇ e

_e_j

− 1

2 k(e

_j

) · e

₀

· + 1

2 θ(e

_j

) · ν·

ϕ

= ν · e

_j

· ∇ e

_e_j

ϕ + 1 2

(tr

_g

k)ν · e

₀

· −k(ν ) · e

₀

· +(tr

_ℓ

θ) ϕ .

3. Main Results

3.1. "MIT" boundary ondition. Remind that our aim isto nd an estimate for the

spetrum of the DiraWitten operator under a natural boundary ondition (that had

been used in order to obtain some blak hole version of the positive mass theorem for

asymptotiallyhyperboli manifolds [4 , 13℄). It onsistsonnding a lower bound for

|λ|

²

where

λ

îs âny^nonzero ômplex ^(a ^priori)^number ⁱⁿ^volvingⁱⁿ^the^following êllipti^rst

orderboundaryproblem

(

^MIT

)

D ϕ = λϕ

^on

Ω

F (ϕ) = ϕ

^on

∂Ω ,

(6)

where

ϕ

^is ^nonzero (eigen)spinor eld, and where

F ∈ End Σ

_|∂Ω ^is ^dened ^by ^the

relation

F (ψ) = iν · ψ

^. ^The ^boundary ^ondition

F(ψ) = ψ

^was ^originally ^introdued ^by

physiistsoftheMIT,andthisthereasonwhyitisoftenalled"MIT"boundaryondition.

Their idea is based on the fat that the Dira (and also the DiraWitten) operator on

manifoldwithboundaryis not formallyselfadjoint anymore, sine we have thefollowing

integrationbypartsformula

(3.1)

Z

Ω

h D ϕ, ψi = Z

Ω

hϕ, D ψi − Z

∂Ω

hν · ϕ, ψi ,

foranyspinorelds

ϕ, ψ

^. ^This^defet^ofselfadjointnesshasaonsequeneonthespetrum ofthe problem(MIT).

3.1. Lemma. The spetrum of (MIT) is a disrete set of omplex numbers with positive

imaginaryparts.

Proof: Let

λ

^be ânyêigenvâlue ôf

(MIT)

^with

ϕ

^a orresponding eigenspinor eld. Just take

ψ = iϕ

ⁱⁿ^(3.1) ^andonsequently get

2 Im(λ) Z

Ω

|ϕ|

²

= Z

∂Ω

|ϕ|

²

> 0 ,

and so

Im(λ) > 0

^. ^Assume ^now ^that

Im(λ) = 0

^, ^then

ϕ

^should ^vanish ^on

∂Ω

^and ^so ^on

thewhole

Ω

^, ^by ^the ontinuation priniple. Thisontradits the fatthat an eigenspinor

isbydenition nonidentiallyzero.

For later usewe introdue some geometri quantities.

3.2. Denition. We set

H

₀^MIT

:= inf

∂Ω

tr

_ℓ

θ −

k

^∂Ω

(ν)

g

, k

^∂Ω

(ν) :=

X

n j=2

k(ν, e

_j

)e

_j

.

The rst main result of this noteis a generalisation of the lower bound of [16 ℄ for the

DiraWittenoperator (we reover(1.4) when we set

k ≡ 0

^on

Ω

^).

3.3.Theorem. Let

Ω

^beâômpat^domainôfâ^spaelike^spinhypersurfae

(M, g, k)

^whih

satises the dominant energy ondition along

Ω

^(so ^that

R

₀

> 0

^). ^The ^boundary

∂Ω

^is

assumed to verify

H

₀^MIT

> 0

^. ^Then ^under ^the ^(MIT) ^boundary ^ondition, ^the ^spetrum

of the DiraWittenoperator on

Ω

îs ân ûnbounded ^disrete ^set ôf ômplex ^numbers ^with

positive imaginary part, suh that any eigenvalue satises

(3.2)

|λ|

²

> n

(n − 1)

R

₀

+ H

₀^MIT

Im(λ) .

Proof: The fatthat, underthe (MIT) boundary ondition, thespetrum oftheDira

Witten operator on

Ω

îs ân ûnbounded ^disrete ^set ôf ômplex ^numbers ^with ^positive

imaginarypart, hasbeen proved intheprevious lemma.

As far as Inequality (3.2 ) is onerned, we onsider a 1parameter family of modied

spinorial LeviCivita onnetion. Indeed,for any

α ∈ R

,we dene theation of

∇

^α ^on

Σ

bytherelation

∇

^α_X

ϕ := ∇

_X

ϕ + iαX · ϕ ,

(7)

for every

X ∈ Γ(T Ω)

^. ^We^an ^then ^dene ^the^DiraWitten ^operator ^with ^respet ^to ^the

Killingonnetion

∇

^α

D

^α

ϕ = X

n k=1

e

_k

· ∇

^α_e

k

ϕ ,

where

·

^is^the ^Cliord^ation^with^respet^to^the ^metri

γ

^. Ân êasyômputation ^gives^the

relation

D

^α

= D − inα

^,^so^that

D

^α isnot formallyselfadjointin

L

² ^with^respet^to

h∗, ∗i

inthe lassofompatlysupported spinor eldsif

α 6= 0

^.

Weonsider aertain spinor eld

ϕ

^and ^dene ^the ^1form

ω

^α_ϕ

∈ Γ(T

^∗

Ω)

^by^the^relation

ω

_ϕ^α

(X) = h∇

^α_X

ϕ + X · D

^α

ϕ, ϕi .

We notie that

ω

^α_ϕ

(X) = ω

_ϕ

(X) − α(n − 1) hiX · ϕ, ϕi

^. ^Then, ^we ^only ^have ^to ^ompute

the

g

^divergene ^of^the^1form

ξ(X) := hiX · ϕ, ϕi

^. ^Tô^this ênd,^we âssume^thatôur^loal

basesatises

∇

_e_j

e

_m

= 0

ât^the^point ^where^the ômputation îs^made^(thisîsêquivalent ^to

∇

_e_j

e

_m

= −k(e

_j

, e

_m

)e

₀⁾ ^so^that

div

_g

ξ = − X

n j=1

e

_j

· ξ(e

_j

)

= −

* i ∇

_e_j

e

_j

| {z }

=0

·ϕ, ϕ +

−

ie

_j

· ∇

_e_j

ϕ, ϕ

−

ie

_j

· ϕ, ∇

_e_j

ϕ

= − hi D ϕ, ϕi − 1 2

*

i e

j

· k(e

j

)

| {z }

=−trgk

·e

0

· ϕ, ϕ +

− hϕ, i D ϕi + 1 2

*

i k(e

j

) · e

j

| {z }

=−trgk

·e

0

· ϕ, ϕ +

= − hi D ϕ, ϕi − hϕ, i D ϕi .

UsingStokes' theoremandintegration formula(2.2 ),we get

Z

∂Ω

ω

^α_ϕ

(ν ) = Z

Ω

| D ϕ|

²

− Z

Ω

|∇ϕ|

²

− Z

Ω

h R ϕ, ϕ, i + α(n − 1) Z

Ω

hi D ϕ, ϕi + hϕ, i D ϕi .

Easyomputations leadto the relations

|∇

^α

ϕ|

²

= |∇ϕ|

²

+ nα

²

|ϕ|

²

+ α hi D ϕ, ϕi + hϕ, i D ϕi

| D

^α

ϕ|

²

= | D ϕ|

²

+ n

²

α

²

|ϕ|

²

+ nα hi D ϕ, ϕi + hϕ, i D ϕi .

Plugging thisinto our integration byparts formula,we obtain

(3.3)

Z

∂Ω

ω

^α_ϕ

(ν ) = Z

Ω

| D

^α

ϕ|

²

− Z

Ω

|∇

^α

ϕ|

²

− Z

Ω

h R

^α

ϕ, ϕ, i ,

where

R

^α

:= R + n(n − 1)α

²^. ^The^twistor ^operator ^with^respet^to ^the ^onnetion

∇

^α ^is

dened by

P

_X^α

ϕ := ∇

^α_X

ϕ + 1

n X · D

^α

ϕ ,

forevery

X ∈ Γ(T Ω)

ând êvery ^spinor êld

ϕ ∈ Γ(Σ)

^. ^W^e ^straightly ^have

|P

^α

ϕ|

²

= |∇

^α

ϕ|

²

− 1

n | D

^α

ϕ|

²

> 0 ,

and,afterintegration on

Ω

^we ^get,^using ^{(3.3 )}

(3.4)

0 6 Z

Ω

|P

^α

ϕ|

²

=

n − 1 n

Z

Ω

| D

^α

ϕ|

²

− Z

Ω

h R

^α

ϕ, ϕ, i − Z

∂Ω

ω

^α_ϕ

(ν) .

(8)

We now need to give some elementary properties of the boundary spinorial endomor-

phism

F

^(the^proof ^is^left^to ^the^reader).

3.4. Proposition. The endomorphism

F

^is ^symmetri, ^isometri ^with ^respet ^to

h∗, ∗i

^,

ommutes to theation of

ν·

^and antiommutes to eah

e

_k

·, (k 6= 1)

^.

The important fat is that the boundary ondition of (MIT) allows us to express the

boundary integrand of(3.4) intermsof theextrinsiurvaturetensors

k

^and

θ

^.

3.5. Lemma. If

F (ϕ) = ϕ

^then ^along ^the ^boundary

∂Ω

^we ^have

(3.5)

ω

^α_ϕ

(ν ) = 1 2

D e

₀

· n

tr

_ℓ

θ − 2(n − 1)α

e

₀

+ k

^∂Ω

(ν ) o

· ϕ, ϕ E .

Proof of Lemma 3.5: We useEquation(2.4 )

ω

^α_ϕ

(ν ) =

ν · e

_j

· ∇ e

_e_j

ϕ + αiν · e

_j

· e

_j

· ϕ + 1 2

(tr

_g

k)ν · e

₀

· −k(ν ) · e

₀

· +(tr

_ℓ

θ) ϕ, ϕ

=

ν · e

_j

· ∇ e

_e_j

ϕ − (n − 1)αiν · ϕ + 1 2

(tr

_ℓ

k)ν · e

₀

· −k

^∂Ω

(ν) · e

₀

· +(tr

_ℓ

θ) ϕ, ϕ

.

Taking Proposition3.4into aount andalso the assumption

F (ϕ) = ϕ

^,îtômes ôut

ω

_ϕ^α

(ν) = 1

2 D

(tr

_ℓ

θ − 2α(n − 1)) − k

^∂Ω

(ν) · e

0

· ϕ, ϕ E

= 1 2

D

e

₀

· (tr

_ℓ

θ − 2α(n − 1))e

₀

+ k

^∂Ω

(ν)

· ϕ, ϕ E

= 1

2 he

₀

· K

^αMIT

· ϕ, ϕi ,

where wehave dened the vetor eld

K

^α_MIT

∈ Γ(T N

_|∂Ω

)

^as

K

^αMIT

= tr

_ℓ

θ − 2α(n − 1)

e

₀

+ k

^∂Ω

(ν) .

Then, it is well known that the boundary integrand

ω

^α_ϕ

(ν )

^is nonnegative ifand only if

K

^α_MIT îsâusal ând ^futureôriented,^whih ^readsâs

(3.6)

tr

_ℓ

θ − 2α(n − 1) >

k

^∂Ω

(ν)

g

.

Asaonsequene,ifone assumes that(3.6) holds andthat

D ϕ = λϕ

^,^then^(3.4) ^implies

0 6

n − 1 n

|λ − inα|

²

Z

Ω

|ϕ|

²

− Z

Ω

h R

^α

ϕ, ϕ, i 0 6

n − 1

n |λ|

²

+ n

²

α

²

− 2nα Im(λ) Z

Ω

|ϕ|

²

− Z

Ω

h R

^α

ϕ, ϕ, i 0 6

n − 1

n |λ|

²

− 2nα Im(λ) Z

Ω

|ϕ|

²

− Z

Ω

h R ϕ, ϕ, i ,

whihispossibleonlyif

|λ|

²

>

ⁿ

(n−1)

R

₀

+ 2nα Im(λ)

^. ^Now^just^take

α = α

₀

=

_2(n−1)¹

H

₀^MIT

sothat

K

^α_MIT⁰ îs ^learly âusal ând ^future ôriented ând ^we ôbtain ^the ^desired înequality^,

namely

|λ|

²

> n (n − 1)

R

₀

+ H

₀^MIT

Im(λ) .

(9)

The equality ase in (3.2 ) is not easy to treat in general, but we an however dedue

someinformation under anatural additional assumption.

3.6. Theorem. Under the assumptions of Theorem 3.3, equality in (3.2) always leads to

the existene of an imaginary

∇

^Killing ^spinor ^on

Ω

^of ^Killing ^numb^er

−iα

0

= −λ/n

^.

If we assume furthermore that

k

^∂Ω

(ν ) = 0

^along ^the ^boundary

∂Ω

^, ^then

∂Ω

^is ^a ^totally

umbilial andonstant mean urvature hypersurfae of

Ω

^.

Proof: Suppose now that equality holds in (3.2). Thereby, there exists a nonzero

λ

eigenspinor eld

ϕ

^suh ^that

(3.7)

P

^α⁰

ϕ ≡ 0 ,

e

₀

· K

^α_MIT⁰

· ϕ, ϕ

≡ 0 .

The seond equation of (3.7 ) simply says that the vetor eld

K

^α_MIT⁰ ^is ^lightlike, ^whih

meansinother words

tr

_ℓ

θ −

k

^∂Ω

(ν)

≡ 2(n − 1)α

₀

.

Therst equationof(3.7 ) an bereformulated as

∇

_X

ϕ + λ

n

X · ϕ = 0 , ∀X ∈ Γ(T Ω),

thatis,

ϕ

îsâ^Killing^spinor. Îtîs^well^known^that^the^Killing^number ^−λ_n

hastobeeither

real or purely imaginary. But remind that

Im(λ) > 0

^, ^and ^so

λ ∈ i R

^∗₊. For later use, we set

λ = iµ

^with

µ

â ^positive ^real ^number. ^Wê ônsider ^now âny ^vetor êld

X ∈ Γ(T ∂Ω)

tangent to the boundary,and we ompute

−i µ n

X · ϕ = ∇

X

ϕ

= ∇

_X

(iν · ϕ)

= i ∇

_X

ν · ϕ + ν · ∇

_X

ϕ

= i

∇

X

ν · −k(X, ν )e

0

· −i µ n

ν · X·

ϕ

= i

−θ(X) · −k(X, ν)e

₀

· + µ n

X·

ϕ .

Thisnallyleads to

(3.8)

∀X ∈ Γ(T ∂Ω) ,

−θ(X) · −k(X, ν)e

₀

· + 2µ

n

X·

ϕ = 0 .

Theondition(3.8)notably impliesthatforanyvetoreld

X ∈ Γ(T ∂Ω)

^,^the^vetor^eld

n

−θ(X) − k(X, ν)e

₀

+

2µ n

X o

islightlike, andonsequently we have

(3.9)

∀X ∈ Γ(T ∂Ω) , k(X, ν)

²

=

θ(X) − 2 µ n

X

²

g

.

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Thestrikingrelation(3.9 )somehowmeasuresthefailureof

∂Ω

^to^be^totally^umbilial. ^W^e

nowprove that

µ = nα

₀^. Îndeed, ûsing^{(3.8 ),} ^we ômpute âlong^the^boundary

2µ

n X

n

j=2

he

_j

· ϕ, e

_j

· ϕi = 2(n − 1)µ n |ϕ|

²

= X

n j=2

he

j

· ϕ, {θ(e

j

) · +k(e

j

, ν)e

0

} · ϕi

= − X

n j=2

hϕ, e

_j

· {θ(e

_j

) · +k(e

_j

, ν)e

₀

} · ϕi

= D ϕ, n

tr

_ℓ

θ − k

^∂Ω

(ν) · e

₀

· o ϕ E

=

ϕ, e

₀

· K

⁰MIT

· ϕ ,

whih anbewritteninshortas

ϕ, e

₀

· K

⁰_MIT

ϕ

=

^2µ(n−1)_n

|ϕ|

²^. ^Besides,^using ^the^seond

equationof (3.7 ),we knowthat

0 = Z

∂Ω

e

₀

· K

^α_MIT⁰

· ϕ, ϕ

= Z

∂Ω

ϕ, e

0

· K

⁰MIT

· ϕ

− 2(n − 1)α

0

Z

∂Ω

|ϕ|

²

= 2(n − 1) µ

n − α

₀

Z

∂Ω

|ϕ|

²

,

whih entails

µ = nα

₀ ^sine ^the eigenspinor

ϕ

ânnot ^vanish îdentially âlong

∂Ω

^. ^Sup-

posefurthermore, that

k

^∂Ω

(ν) = 0

ôn ^the^boundary^, ^then îtîs ^lear ^from^(3.9) ^that

∂Ω

^is

atotally umbilial and onstant mean urvature hypersurfae.

3.2. APS BoundaryCondition. TheAtiyahPatodiSinger(APS)boundaryondition

wasoftenusedinorderto prove positive masstheorem forasymptotially at blakholes

(that is to say for asymptotially at manifolds with boundary [8, 14℄). More preisely,

we denote by

Π

_± ^the

L

²-orthogonal projetions on the spaes of eigenspinors of positive (respetivelynegative)eigenvaluesoftheDiraoperatoroftheboundary

D e := P

j>2

e

_j

·

_∂Ω

∇ e

_e_j^,

where

·

_∂Ω^denotes^the^Cliord^ation^with^the^respet^to^boundary^metri

ℓ

^. ^In^this^setion,

ouraimistondalowerboundfor

|λ|

²^where

λ

îsâny^nonzeroômplex^(a^priori)^number

involvinginthe following elliptirst orderboundaryproblem

(

^APS

)

D ϕ = λϕ

^on

Ω Π

+

(ϕ) = 0

^on

∂Ω ,

where

D

still denotes the DiraWitten operator. We rst prove that the spetrum of (APS)isreal.

3.7. Lemma. The spetrum of (APS) isa disrete set of real numbers.

Proof: Let

λ ∈ C

be anyeigenvalue of

(APS)

^with

ϕ

^a orresponding eigenspinor eld.

Now,

·

N

, ·

Ω

, ·

_∂Ω ^the ^Cliord^ations^ofrespetively

γ, g, ℓ

^satisfy^the^following ^relations

(3.10)

X ·

_∂Ω

ψ = X ·

_Ω

ν ·

_Ω

ψ, X ·

_Ω

ψ = iX ·

_N

e

₀

·

_N

ψ .