An example

i¹₂

|Ω|_g =−2(tr`^ψ)²+ 2|`^ψ|², (2.26) which implies equality in (2.11).

Proposition 2.4.2. On a compact Riemannian Spin^c manifold (Nⁿ, g) of dimension n > 3, assume that the first eigenvalue λ1 of the Dirac operator to which is attached an eigenspinor ψ satisfies the equality case in (2.13). Then, |`^ψ| is constant and if h > 0 denotes an eigenfunction of the Yamabe operator corresponding to µ₁, then for any vector field X

g(X, `^ψ(dh)−λ₁dh) = g(λ₁X−`^ψ(X), dh) = 0. (2.27) Proof. Ifn >3 and equality holds in (2.13), we consider the positive functionv>0 defined bye^2v =hⁿ⁻²⁴ wherehis an eigenfunction of the Yamabe operator corresponding to µ₁. Inequality (2.20) with u = v gives |`^ψ| is constant, ∇_Xϕ = −`^ϕ(X) · ϕ and Ω · ϕ = i^c₂ⁿ|Ω|_gϕ. By Proposition 2.4.1, Equality (2.26) and (2.25) can be considered for the conformal metric g =e^2vg =hⁿ⁻²⁴ g to get

(tr`^ϕ)² :=f² = 1

4Sv− c_n

4 |Ω|g+|`^ϕ|², gradf =−div`^ϕ.

It is straightforward to see that these two equalities give (2.27).

2.5 An example

If the lower bound (33) is achieved, automatically equality holds in (2.11). Here, we will give an example where equality holds in (2.11) but not in (33).

Let (N³, g) = (S³, can) be endowed with its unique Spin structure and consider a real Killing spinorψ with Killing constant ¹₂. As the norm ofψ is constant, we may suppose that |ψ|= 1. Let ξ be the Killing vector field on N defined by

ig(ξ, X) =hX·ψ, ψi.

2.5. AN EXAMPLE 73 First, we have idξ(X, Y) = − hX∧Y ·ψ, ψi for any X, Y ∈Γ(T N). In fact,

2idξ(X, Y) = X(hY ·ψ, ψi)−Y(hX·ψ, ψi)− h[X, Y]·ψ, ψi

= hY · ∇_Xψ, ψi+hY ·ψ,∇_Xψi − hX· ∇_Yψ, ψi − hX·ψ,∇_Yψi

= 1

2hY ·X·ψ, ψi −1

2hX·Y ·ψ, ψi −1

2hX·Y ·ψ, ψi+1

2hY ·X·ψ, ψi

= hY ·X·ψ, ψi − hX·Y ·ψ, ψi

= −2hX∧Y ·ψ, ψi.

Using the Hodge operator ∗ and the volume form which acts as the identity on the spinor bundle, we get

idξ(X, Y) = ig(dξ, X∧Y)

= − hX∧Y ·ψ, ψi

X∧Y · ∗(X∧Y)· ∗(X∧Y)

|X∧Y |² ·ψ, ψ

= 1

|X∧Y |²

|X∧Y |² v_g · ∗(X∧Y)·ψ, ψ

= h∗(X∧Y)·ψ, ψi

= ig

ξ,∗(X∧Y)

= ig(∗ξ, X∧Y), hence dξ=∗ξ. We have

2dξ(X, Y) = X(g(ξ, Y))−Y(g(ξ, X))−g(ξ,[X, Y])

= g(∇_Xξ, Y) +g(ξ,∇_XY)−g(∇_Yξ, X)−g(ξ,∇_YX)

−g(ξ,∇_XY) +g(ξ,∇_YX)

= g(∇_Xξ, Y)−g(∇_Yξ, X)

= 2g(∇Xξ, Y).

Moreover, d|ξ|² =−2dξ(ξ, .) =−2∇_ξξ. Indeed, d|ξ|²(X) = X(g(ξ, ξ))

= g(∇_Xξ, ξ) +g(ξ,∇_Xξ)

= 2g(∇_Xξ, ξ)

= −2g(∇_ξξ, X)

= −2dξ(ξ, X).

But dξ(ξ, .) =g(∇_ξξ, .)' ∇_ξξ, so

d|ξ|² =−2dξ(ξ, .)' −2∇_ξξ.

Since dξ(ξ, .) =∗ξ(ξ, .) = 0, we conclude that d|ξ|² = 0 and then |ξ|= cte. Moreover, dξ·ϕ=∗ξ·ϕ=−ξ·ϕfor any spinor field ϕ.

74 CHAPTER 2. LOWER BOUNDS The Killing vector ξ is not zero because if ξ= 0 we get,

hξ·ψ, ψi=ig(ξ, ξ) = i|ξ|² = 0,

hence ξ·ψ ⊥ψ and the vector space TxN ·ψ of dimension 3 is orthogonal to ψ. This is a contradiction because ψ^⊥ has complex dimension 1. Let{ ξ/|ξ|, e₁, e₂} be a local orthonormal frame of N and φ a spinor field generated by ψ^⊥. We have he_i·ψ, ψi = ig(ξ, ei) = 0 hence ei ·ψ ⊥ ψ, i.e, ei ·ψ = aiφ where a1 and a2 are no zero complex functions. Hence,

ξ·ψ =−|ξ|e₁·e₂·ψ =−|ξ|e₁ ·(a₂φ) =|ξ|a₂ a1

ψ,

so there exists a complex function a such that ξ ·ψ = aψ. By definition, we have ig(ξ, ξ) =hξ·ψ, ψi=a, hencei|ξ|² =a. On the one hand,

ξ·ξ·ξ·ξ·ψ =−|ξ|²(−|ξ|²)ψ =|ξ|⁴ψ, and on the other hand, we have

ξ·ξ·ξ·ξ·ψ = |ξ|²|ξ|²ψ =|ξ|²(−ia)ψ

= (−ia)²ψ =−a²ψ

= −a(ξ·ψ) =−ξ(a·ψ)

= −ξ(ξ·ψ) =|ξ|²ψ.

Finally, |ξ|⁴ =|ξ|² and since |ξ| is constant, we get|ξ|= 1. As a conclusion, a=i and ξ·ψ =iψ.

Let h be a real constant such that h >1. We define the metric g^h onN, by:

g^h(ξ, X) = g(ξ, X) for any X ∈Γ(T N), g^h(X, Y) = h⁻²g(X, Y) forX, Y ⊥ξ.

By the following isomorphism,

(T N, g) −→ (T N, g^h) Z 7−→ Z^h =

Z if Z =ξ, hZ if Z ⊥ξ,

if {ξ, e₁, e₂} is a local orthonormal frame of (N, g), {ξ^h = ξ, e^h₁, e^h₂} is a local g^h -orthonormal frame of (N, g^h).

Lemma 2.5.1. The Levi-Civita connection ∇^h associated with the metric g^h is given by

∇^h_ξξ = ∇_ξξ = 0,

g^h(∇^h_ξe^h_i, e^h_j) = g(∇_ξe_i, e_j) + (h² −1)dξ(e_i, e_j), g^h(∇^h_eh

iξ, e^h_j) = dξ(e^h_i, e^h_j) = h²g(∇_e_iξ, e_j).

2.5. AN EXAMPLE 75 Proof. By the Koszul formula, we get

2g^h(∇^h_ξξ, e^h_i) = ξ(g^h(ξ, e^h_i)) +ξ(g^h(ξ, e^h_i))−e^h_i(g^h(ξ, ξ)) +g^h(e^h_i,[ξ, ξ])−g^h(ξ,[ξ, e^h_i])−g^h(ξ,[ξ, e^h_i])

= ξ(hg(ξ, e_i)) +ξ(hg(ξ, e_i))−he_i(g(ξ, ξ)) +hg^h(e_i,0)−2hg^h(ξ,[ξ, e_i])

= −2hg^h(ξ,[ξ, e_i]).

Similary, 2g^h(∇_ξξ, e^h_i) =−2hg^h(ξ,[ξ, e_i]).Again, by the Koszul formula, we haveg^h(∇_ξξ, ξ) = 0 and g^h(∇^h_ξξ, ξ) = 0. Hence, ∇^h_ξξ=∇_ξξ= 0 since ∇_ξξ = 0. It follows

2g^h(∇^h_eh

iξ, e^h_j) = e^h_i(g^h(ξ, e^h_j)) +ξ(g^h(e^h_i, e^h_j))−e^h_j(g^h(e^h_i, ξ)) +g^h(e^h_j,[e^h_i, ξ])−g^h(ξ,[e^h_i, e^h_j])−g^h(e^h_i,[ξ, e^h_j])

= he_i(hg^h(ξ, e_j)) +ξ(h²g^h(e_i, e_j))−he_j(hg^h(e_i, ξ)) +h²g^h(e_j,[e_i, ξ])−h²g^h(ξ,[e_i, e_j])−h²g^h(e_i,[ξ, e_j])

= h²h

g^h(e_j,[e_i, ξ])−g^h(ξ,[e_i, e_j])−g^h(e_i,[ξ, e_j])i

= h²h

h⁻²g(e_j,[e_i, ξ]) + (1−h⁻²)g(ξ, e_j)g(ξ,[e_i, ξ])

−h⁻²g(ξ,[e_i, e_j])−(1−h⁻²)g(ξ, ξ)g(ξ,[e_i, e_j])

−h⁻²g(e_i,[ξ, e_j])−(1−h⁻²)g(ξ, e_i)g(ξ,[ξ, e_j])i

= h²h

h⁻²g(e_j,∇_e_iξ)−h⁻²g(e_j,∇_ξe_i)−g(ξ,[e_i, e_j])

−h⁻²g(ei,∇ξej) +h⁻²g(ei,∇ejξ) i

= h²h

−g(ξ,[e_i, e_j])−h⁻²ξ(g(e_i, e_j))i

= −h²g(ξ,[ei, ej]).

Moreover, we have

2g(∇_e_iξ, e_j) = e_i(g(ξ, e_j)) +ξ(g(e_i, e_j))−e_j(g(e_i, ξ)) +g(e_j,[e_i, ξ])−g(ξ,[e_i, e_j])−g(e_i,[ξ, e_j])

= −g(ξ,[e_i, e_j]) +g(e_j,∇_e_iξ)−g(e_j,∇_ξe_i)

−g(e_i,∇_ξe_j) +g(e_i,∇_e_jξ) = −g(ξ,[e_i, e_j]).

Thus, 2g^h(∇^h_e

ihξ, e^h_j) = −h²g(ξ,[ei, ej]) = 2h²g(∇eiξ, ej) and finally g^h(∇^h_e

ihξ, e^h_j) =h²g(∇_e_iξ, e_j) =h²dξ(e_i, e_j) =dξ(e^h_i, e^h_j).

76 CHAPTER 2. LOWER BOUNDS Again, by the Koszul formula,

2g^h(∇^h_ξe^h_i, e^h_j) = ξ(g^h(e^h_i, e^h_j)) +e^h_i(g^h(ξ, e^h_j))−e^h_j(g^h(ξ, e^h_i)) +g^h(e^h_j,[ξ, e^h_i])−g^h(e^h_i,[ξ, e^h_j])−g^h(ξ,[e^h_i, e^h_j])

= h²g^h(e_j,[ξ, e_i])−h²g^h(e_i,[ξ, e_j])−h²g^h(ξ,[e_i, e_j])

= h²h

h⁻²g(e_j,[ξ, e_i]) + (1−h⁻²)g(ξ, e_j)g(ξ,[ξ, e_i])i

−h²h

h⁻²g(ei,[ξ, ej]) + (1−h⁻²)g(ξ, ei)g(ξ,[ξ, ej]) i

−h²g(ξ,[e_i, e_j])

= g(e_j,[ξ, e_i])−g(e_i,[ξ, e_j])−h²g(ξ,[e_i, e_j])

= 2g(∇ξei, ej)−2g(∇eiξ, ej)−h²g(ξ,[ei, ej])

= 2g(∇_ξe_i, e_j) +g(ξ,[e_i, e_j])−h²g(ξ,[e_i, e_j])

= 2g(∇_ξe_i, e_j) + (1−h²)g(ξ,[e_i, e_j]), since we have

2g(∇_e_iξ, e_j) = g(e_j,[e_i, ξ])−g(ξ,[e_i, e_j])−g(e_i,[ξ, e_j])

= g(e_j,∇_e_iξ)−g(e_j,∇_ξe_i)−g(ξ,[e_i, e_j])

−g(ei,∇ξei) +g(ei,∇ejξ)

= −g(ξ,[e_i, e_j]).

Hence, g^h(∇^h_ξe^h_i, e^h_j) =g(∇_ξe_i, e_j) + (h²−1)dξ(e_i, e_j).Similary, we get g^h(∇^h_eh

ie_i^h, e_j^h) = hg(∇_e_ie_i, e_j).

The following isomorphism

P_SO₃N_g −→ P_SO₃N_g^h

u={ξ, e₁, e₂} 7−→ u^h ={ξ, e^h₁, e^h₂}, can be extended in a natural way to

P_Spin₃N_g −→ P_Spin₃N_gh

u 7−→ u˜^h.

This isomorphism defines the following isomorphism of vector spaces:

Σ_gN −→ Σ_g^hN

ψ = [˜u, σ] 7−→ ψ^h = [˜u^h, σ].

The Clifford multiplication with respect togandg^h will be denoted by the same symbol.

Hence, we have

hψ₁, ψ₂i_Σ

gN =

ψ₁^h, ψ₂^h

Σ_ghN, (X·ψ)^h =X^h·ψ^h for every X ∈Γ(T N).

2.5. AN EXAMPLE 77 Lemma 2.5.2. The spinorial Levi-Civita connection ∇^h of the Spin manifold (N, g^h) is given by

78 CHAPTER 2. LOWER BOUNDS We consider the trivial S¹-principal bundle (N ×S¹, π, N) and let L be the trivial line bundle associated with this S¹-principal fiber bundle via the standard representation.

We know that if∇^Ldenotes the covariant derivative onL, then there exists a real 1-form α and a global section l of L such that

∇^L_Xl =iα(X)l,

for every X ∈ Γ(T M). We choose α = (1 −h²)ξ and we consider the connection

∇^h,L =∇^h⊗Id + Id⊗ ∇^L on the bundle Σ_g^hN ⊗L. We have

∇^h,L_Xh(ψ^h⊗l) = h²

2X^h·(ψ^h⊗l) + 2i

(1−h²)ξ

(X^h)(ψ^h⊗l).

Hence,

∇^h,L_eh 1

(ψ^h⊗l) = h²

2 e^h₁ ·(ψ^h⊗l),

∇^h,L

e^h₂ (ψ^h⊗l) = h²

2 e^h₂ ·(ψ^h⊗l),

∇^h,L_ξ (ψ^h⊗l) = (−3h²

2 + 2)ξ·(ψ^h⊗l).

The spinor field ψ^h⊗l is a section of Σ_ghN ⊗L, which is the Spin^c bundle associated with the Spin^c structure whose auxiliary line bundle is given by L². This spinor field is an eigenspinor associated with the eigenvalue ^h₂² −2. In fact,

D(ψ^h⊗l) = ξ· ∇^h,L_ξ (ψ^h⊗l) +e^h₁ · ∇^h,L_eh 1

(ψ^h⊗l) +e^h₂ · ∇^h,L_eh 2

(ψ^h⊗l)

− h² 2 − h²

2 −(−3h²

2 + 2)

(ψ^h⊗l)

= (h²

2 −2)(ψ^h⊗l).

It is clear that ψ^h⊗l is not a Spin^c Killing spinor field since h > 1, and hence (N, g^h) is not a limiting manifold for the Friedrich type Spin^c inequality. But, it is a limiting manifold for (2.11) since we will prove that

∇^h,L_Xh(ψ^h ⊗l) = −`^ψ^h^⊗l(X^h)·(ψ^h⊗l), dα·(ψ^h⊗l) = i^c₂ⁿ|dα|_g^h(ψ^h⊗l),

where idα is the curvature form associated with the connection ∇^L of L. We have idξ(e^h₁, e^h₂) = ih²dξ(e₁, e₂) =−h²he₁ ·e₂·ψ, ψi=h²hξ·ψ, ψi=ih²,

idξ(ξ, e^h₁) =ihdξ(ξ, e₁) = −ihhξ·e₁·ψ, ψi=−hhe₁·ψ, ψi= 0, idξ(ξ, e^h₂) =ihdξ(ξ, e₂) = −ihhξ·e₂·ψ, ψi=−hhe₂·ψ, ψi= 0.

Hence,

dξ =dξ(ξ, e^h₁)ξ∧e^h₁ +dξ(ξ, e^h₂)ξ∧e^h₂ +dξ(e^h₁, e^h₂)e^h₁ ∧e^h₂ =h² e^h₁ ∧e^h₂.

2.5. AN EXAMPLE 79 The Clifford multiplication of dα byψ^h⊗l is given by

dα·(ψ^h⊗l) = (1−h²)dξ·(ψ^h⊗l)

= (1−h²)h²e^h₁ ·e^h₂ ·(ψ^h⊗l)

= −i(1−h²)h² ψ^h⊗l

= i(h²−1)h² ψ^h⊗l.

Hence, it follows that

|dα|²_gh = (1−h²)²|dξ|²_gh = (1−h²)²(dξ(e^h₁, e^h₂))² =h⁴(1−h²)². Since, h >1, |dα|_g^h =h²(h²−1). So, we have

dα·(ψ^h⊗l) = ih²(h²−1)(ψ^h⊗l) =ic_n

2|dα|_g^h(ψ^h⊗l).

Moreover, it is easy to check that `^ψ^h^⊗l(e^h₁, e^h₁) =`^ψ^h^⊗l(e^h₂, e^h₂) = −^h₂².In fact,

`^ψ^h^⊗l(e^h₁, e^h₁) = 1 2Re

h²

2 e^h₁ ·e^h₁ ·(ψ^h⊗l) + h²

2 e^h₁ ·e^h₁ ·(ψ^h⊗l), ψ^h⊗l

|ψ^h⊗l|²

= −h² 2. Similary, we have

`^ψ^h^⊗l(e^h₁, ξ) =`^ψ^h^⊗l(e^h₂, ξ) =`^ψ^h^⊗l(e^h₁, e^h₂) = 0,

`^ψ^h^⊗l(ξ, ξ) = 3h² 2 −2.

Now, we compute

−`^ψ^h^⊗l(e^h₁)·(ψ^h⊗l) = −`^ψ^h^⊗l(e^h₁, e^h₁)e^h₁ ·(ψ^h⊗l)

= h²

2 e^h₁ ·(ψ^h⊗l)

= ∇^h,L_eh 1

(ψ^h ⊗l).

Similary, we have

−`^ψ^h^⊗l(e^h₂)·(ψ^h⊗l) = h²

2 e^h₂ ·(ψ^h⊗l) =∇^h,L_eh 2

(ψ^h⊗l),

−`^ψ^h^⊗l(ξ)·(ψ^h⊗l) = (−3h²

2 + 2)ξ·(ψ^h⊗l) =∇^h,L_ξ (ψ⊗l).

Finally, the manifold (N, g^h) is a limiting manifold for (2.11).

Chapter 3 The Hijazi Inequalities on Complete Riemannian Spin ^c Manifolds ¹

3.1 Introduction

We recall that on a compact Riemannian Spin^cmanifold (Nⁿ, g) of dimensionn>2, any eigenvalue λ of the Dirac operator satisfies a Friedrich type inequality [HM99, Fri80]:

λ² > n

4(n−1)inf

N (S−c_n|Ω|), (3.1)

Equality holds if and only if the eigenspinorψ associated with the first eigenvalueλ₁ is a Spin^cKilling spinor satisfying Ω·ψ =i^c₂ⁿ|Ω|ψ, i.e. for everyX ∈Γ(T N) the eigenspinor ψ satisfies

∇_Xψ =−^λ_n¹X·ψ,

Ω·ψ =i^c₂ⁿ|Ω|ψ. (3.2)

In Chapter 2, it is shown that on a compact Riemannian Spin^c manifold any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies an Hijazi type inequality [Hij95] involving the energy-momentum tensor`^ψ and the scalar curvature:

λ² >inf

N (1

4S− c_n

4|Ω|+|`^ψ|²), (3.3)

Equality holds in (3.3) if and only, for all X ∈Γ(T N), we have ∇Xψ =−`^ψ(X)·ψ,

Ω·ψ =i^c₂ⁿ|Ω|ψ, (3.4)

where ψ is an eigenspinor associated with the first eigenvalue λ₁. By definition, the trace tr(`^ψ) of `^ψ, where ψ is an eigenspinor associated with an eigenvalue λ, is equal to λ. Hence, Inequality (3.3) improves Inequality (3.1) since, by the Cauchy-Schwarz inequality, |`^ψ|² > _n¹(tr(`^ψ))² = _n¹λ².

1This chapter is the subject of a published paper [Nak11b]

82 CHAPTER 3. THE HIJAZI INEQUALITIES ON COMPLETE MANIFOLDS

In the same spirit as in [Hij86], A. Moroianu and M. Herzlich (see [HM99]) general-ized the Hijazi inequality [Hij86], involving the first eigenvalue of the Yamabe operator L, to the case of compact Spin^c manifolds of dimension n >3: any eigenvalue λ of the Dirac operator satisfies

λ² > n

4(n−1)µ₁, (3.5)

where µ₁ is the first eigenvalue of the perturbed Yamabe operator defined by L^Ω = L−cn|Ω|g = 4ⁿ⁻¹_n−24+S−cn|Ω|g.The limiting case of (3.5) is equivalent to the limiting case in (3.1). The Hijazi inequality [Hij95], involving the energy-momentum tensor and the first eigenvalue of the Yamabe operator, is then proved in Chapter 2 for compact Spin^c manifolds. In fact, any eigenvalue of the Dirac operator to which is attached an eigenspinor ψ satisfies

λ² > 1

4µ₁+ inf

N |`^ψ|². (3.6)

Equality in (3.6) holds if and only, for all X ∈Γ(T N), we have ∇_Xϕ=−`^ϕ(X) · ϕ,

Ω·ψ =i^c₂ⁿ|Ω|_gψ, (3.7)

whereϕ=e⁻ⁿ⁻¹² ^uψ, the spinor fieldψ is the image of ψ under the isometry between the spinor bundles of (Nⁿ, g) and (Nⁿ, g =e^2ug) andψ is an eigenspinor associated with the first eigenvalue λ1 of the Dirac operator. Again, Inequality (3.6) improves Inequality (3.5). In this chapter we examine these lower bounds on open manifolds, and especially on complete Riemannian Spin^c manifolds. We prove the following:

Theorem 3.1.1. Let(Nⁿ, g)be a complete RiemannianSpin^c manifold of finite volume.

Then any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies the Hijazi type inequality (3.3). Equality holds if and only if the eigenspinor associated with the first eigenvalue λ₁ satisfies (3.4).

The Friedrich type inequality (3.1) is derived for complete Riemannian Spin^c man-ifolds of finite volume. This was proved by N. Grosse in [Nad08a] and [Nad08b] for complete Spin manifolds of finite volume. Using the conformal covariance of the Dirac operator we prove:

Theorem 3.1.2. Let (Nⁿ, g)be a complete Riemannian Spin^c manifold of finite volume and dimension n > 2. Any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies the Hijazi type inequality (3.6). Equality holds if and only if Equation (3.7) holds.

Now, the Hijazi type inequality (3.5) can be derived for complete Riemannian Spin^c manifolds of finite volume and dimension n > 2 and equality holds if and only if the eigenspinor associated with the first eigenvalue λ₁ satisfies (3.2). This was also proved

3.2. PRELIMINARIES 83

Dans le document The DART-Europe E-theses Portal (Page 73-84)

2.5 An example

Chapter 3

The Hijazi Inequalities on Complete Riemannian Spin c Manifolds 1

3.1 Introduction

The Hijazi Inequalities on Complete Riemannian Spin ^c Manifolds ¹