Existence Results for the Conformal Dirac–Einstein System

Chiara Guidi; Ali Maalaoui; Vittorio Martino

doi:10.1515/ans-2020-2115

Open Access Published by De Gruyter December 15, 2020

Existence Results for the Conformal Dirac–Einstein System

Chiara Guidi , Ali Maalaoui and Vittorio Martino

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2020-2115

Abstract

We consider the coupled system given by the first variation of the conformal Dirac–Einstein functional. We will show existence of solutions by means of perturbation methods.

Keywords: Conformally Invariant Operators; Perturbation Methods

MSC 2010: 58J05; 58E15; 53A30; 58Z05

1 Introduction

Let (M,g,Σ⁢M) be a closed (compact, without boundary) three-dimensional Riemannian Spin manifold where Σ⁢M is its spin bundle (we refer the reader to [10] for a good introduction to spin geometry and to [14] for an introduction and basic tools used in conformal geometry that will be used later in this paper). We denote by Lg the conformal Laplacian of g and by Dg the Dirac operator. We consider the energy functional

E M ⁢ ( v , ψ ) = 1 2 ⁢ ∫ M ( v ⁢ L g ⁢ v + 〈 D g ⁢ ψ , ψ 〉 - | v | 2 ⁢ | ψ | 2 ) ⁢ dvol g

and we take its first variation on the related Sobolev space H1⁢(M)×H12⁢(Σ⁢M); therefore its critical points satisfy the coupled system

{ L g ⁢ v = | ψ | 2 ⁢ v , D g ⁢ ψ = | v | 2 ⁢ ψ , on M .

This functional arises as the conformal version in the description of a super-symmetric model consisting of coupling gravity with fermionic interaction and it generalizes the classical Hilbert–Einstein energy functional, see for instance [4, 9, 13].

Indeed, the total energy functional consists of the Hilbert–Einstein energy which is the total curvature, coupled with a fermionic action. Now, since the energy of the system is invariant under the group of diffeomorphisms of M, when one restricts it to a fixed conformal class of a given Riemannian metric g, the functional EM appears.

In particular, due to the conformal invariance, the Palais–Smale compactness condition is violated by this functional and in addition, due to the presence of the Dirac operator, it is strongly indefinite.

Regarding the first issue, in [20] the authors studied the lack of compactness and gave a precise description of the bubbling phenomena, characterizing the behavior of the Palais–Smale sequences, in the spirit of classical works [23, 22, 24, 15, 16, 5, 3]. For the strong indefiniteness difficulty, in [17, 18, 19] general functionals with these features are studied by using methods based on a homological approach. Notice that in our situation one cannot apply these homological approaches because of the violation of compactness stated above.

In this paper, we are concerned with the existence of solutions to the coupled system, by using a perturbation approach, starting from the sphere 𝕊3 equipped with its standard metric g𝕊3. Therefore, let K be a function of the form K=1+ε⁢k, where k is a function that satisfies suitable assumptions to be determined later; we consider the functional

ℰ ⁢ ( v , ψ ) = 1 2 ⁢ ∫ 𝕊 3 ( v ⁢ L g 𝕊 3 ⁢ v + 〈 D g 𝕊 3 ⁢ ψ , ψ 〉 - K ⁢ | v | 2 ⁢ | ψ | 2 ) ⁢ d ⁢ vol g 𝕊 3

and we will focus on the existence of solutions to the following coupled system:

(1.1) { L g 𝕊 3 ⁢ v = K ⁢ | ψ | 2 ⁢ v , D g 𝕊 3 ⁢ ψ = K ⁢ v 2 ⁢ ψ , on 𝕊 3 .

Notice that these solutions converge to the standard bubbles when the parameter ε tends to zero; here we called standard bubbles the solutions of our equation in 𝕊3 (or ℝ3 via stereographic projection), see [20]. This is expected from the description of the Palais–Smale sequences of the functional EM, but it remains open whether all the solutions on the sphere with positive scalar component are in fact standard ones.

Let us denote by π:𝕊3∖{sp}→ℝ3 the stereographic projection, where sp is the south pole. Our main result is the following:

Theorem 1.1.

Let k∈C2⁢(S3) be a Morse function on S3 such that the south pole is not a critical point. Let us set h=k∘π-1 and suppose that

Δ ⁢ h ⁢ ( ξ ) ≠ 0 for all ξ ∈ crit ⁡ [ h ] ,
∑ ξ ∈ crit ⁡ [ h ] , Δ ⁢ h ⁢ ( ξ ) < 0 ( - 1 ) m ⁢ ( h , ξ ) ≠ - 1 ,

where Δ is the standard Laplacian operator on R3, crit⁡[h] denotes the set of critical points of h and m⁢(h,ξ) is the morse index of h at a critical point ξ. Then there exists ε0>0 such that for K=1+ε⁢k and |ε|<ε0, system (1.1) has a solution.

The condition on the critical point at the south pole of the sphere is needed since we are going to use the standard stereographic projection π, however this condition can be always satisfied by making a unitary transformation which does not affect the generality of the result. In fact, assumptions (i) and (ii) can be formulated using directly the function k on the sphere and its Laplacian ΔS3⁢k, but since we are shifting the analysis on ℝ3 we stated the assumptions using h.

The previous result is the analogous of several ones obtained with this kind of hypothesis of Bahri–Coron type on the function k: for instance, for the standard Riemannian case of prescribing the scalar curvature and its generalization to the Qγ curvature see [2, 6, 7]; in the case of prescribing the Webster curvature in the CR setting and its fractional generalization see [21] and [8]; for the spinorial Yamabe type equations involving the Dirac operator on the sphere see [12].

The idea of the proof follows the abstract perturbation method introduced in [1]. The difficulties in dealing with such system of equations come from the strongly indefiniteness of one of the operator involved and the characterization of the critical manifold of the unperturbed problem. Fortunately, the first difficulty does not affect our analysis in this situation (even though it was a crucial difficulty to circumvent in the general problem as in [20]). The second difficulty is actually central in our situation. In fact, even after characterizing the critical manifold and showing its non-degeneracy, when going through the finite-dimensional reduction of the functional, another kind of degeneracy appears which is due to the invariance with respect to one of the parameters of the problem (see Remark 3.6). This constitute a major bifurcation from the problems found in the literature.

2 Notations and Definitions

Let (M,g) be a closed (compact, without boundary) three-dimensional Riemannian manifold.

We start to describe briefly the first operator appearing in the system. We denote by Lg the conformal Laplacian acting on functions

L g = - Δ g + 1 8 ⁢ R g .

Here Δg is the standard Laplace–Beltrami operator and Rg is the scalar curvature. Lg is a conformally covariant operator. More precisely, given a metric g~=f2⁢g in the conformal class of g, we have

L g ~ ⁢ u = f - 5 2 ⁢ L g ⁢ ( f 1 2 ⁢ u ) .

We recall that the usual Sobolev space on M, denoted by H1⁢(M), continuously embeds in Lp⁢(M) for 1≤p≤6. Moreover, for 1≤p<6, the embedding is compact. In particular, if we assume M to be the sphere

𝕊 3 = { ( x ′ , x 4 ) ∈ ℝ 3 × ℝ : | x ′ | 2 + x 4 2 = 1 }

equipped with its standard metric g𝕊3, then it is possible to identify 𝕊3∖{sp}, being sp=(0,-1) the south pole, with ℝ3, by means of the stereographic projection

π : 𝕊 3 ∖ { sp } → ℝ 3 ,

( x , x 4 ) ↦ y = x 1 + x 4 .

The standard metric gℝ3 on ℝ3 and the metric g~=(π-1)*⁢g𝕊3 are conformal, more precisely g~=f2⁢gℝ3, with f=21+|y|2. Thus the standard conformal Laplacian on the sphere Lg𝕊3 and the one on ℝ3, which we denote as usual Lgℝ3=-Δ, are related by the following identity:

L g 𝕊 3 ⁢ v = [ f - 5 2 ⁢ ( - Δ ) ⁢ ( f 1 2 ⁢ v ∘ π - 1 ) ] ∘ π , v ∈ H 1 ⁢ ( 𝕊 3 ) .

Now, let us describe the second operator involved. Let Σ⁢M be the canonical spinor bundle associated to M, whose sections are simply called spinors on M. This bundle is endowed with a natural Clifford multiplication

Cliff : C ∞ ⁢ ( T ⁢ M ⊗ Σ ⁢ M ) → C ∞ ⁢ ( Σ ⁢ M ) ,

a hermitian metric and a natural metric connection

∇ Σ : C ∞ ⁢ ( Σ ⁢ M ) → C ∞ ⁢ ( T * ⁢ M ⊗ Σ ⁢ M ) .

We denote by Dg the Dirac operator acting on spinors

D g : C ∞ ⁢ ( Σ ⁢ M ) → C ∞ ⁢ ( Σ ⁢ M ) , D g = Cliff ∘ ∇ Σ ,

where the composition Cliff∘∇Σ is meaningful provided that we identify T*⁢M≃T⁢M by means of the metric g. We also have a conformal invariance that in our situation, g~=f2⁢g, reads as follows: there exists an isomorphism of vector bundles F:Σ⁢(M,g)→Σ⁢(M,g~) such that

(2.1) D g ~ ⁢ ψ = F ⁢ [ f - 2 ⁢ D g ⁢ ( f ⁢ F - 1 ⁢ ψ ) ] .

The functional space that we are going to define is the Sobolev space H12⁢(Σ⁢M). First we recall that the Dirac operator Dg on a compact manifold is essentially self-adjoint in L2⁢(Σ⁢M), has compact resolvent and there exists a complete L2-orthonormal basis of eigenspinors {ψi}i∈ℤ of the operator

D g ⁢ ψ i = λ i ⁢ ψ i ,

and the eigenvalues {λi}i∈ℤ are unbounded, that is, |λi|→∞ as |i|→∞. In this way every function in L2⁢(Σ⁢M) has a representation in this basis, namely:

ψ = ∑ i ∈ ℤ a i ⁢ ψ i , ψ ∈ L 2 ⁢ ( Σ ⁢ M ) .

We define the unbounded operator |Dg|s:L2⁢(Σ⁢M)→L2⁢(Σ⁢M) by

| D g | s ⁢ ( ψ ) = ∑ i ∈ ℤ a i ⁢ | λ i | s ⁢ ψ i

and we denote by Hs⁢(Σ⁢M) the domain of |Dg|s, namely ψ∈Hs⁢(Σ⁢M) if and only if

∑ i ∈ ℤ a i 2 ⁢ | λ i | 2 ⁢ s < + ∞ .

Note that Hs⁢(Σ⁢M) coincides with the usual Sobolev space Ws,2⁢(Σ⁢M) and for s<0, Hs⁢(Σ⁢M) is defined as the dual of H-s⁢(Σ⁢M).

For s>0, we define the following inner product: for ψ,ϕ∈Hs⁢(Σ⁢M),

〈 ψ , ϕ 〉 s = 〈 | D g | s ⁢ ψ , | D g | s ⁢ ϕ 〉 L 2 ,

which induces an equivalent norm in Hs⁢(Σ⁢M); we will take

〈 ψ , ψ 〉 := 〈 ψ , ψ 〉 1 2 = ∥ ψ ∥ 2

as our standard norm for the space H12⁢(Σ⁢M). In this case as well, the embedding Hs⁢(Σ⁢M)↪Lp⁢(Σ⁢M) is continuous for 1≤p≤3 and it is compact if 1≤p<3.

Then we decompose H12⁢(Σ⁢M) in a natural way. Let us consider the L2-orthonormal basis of eigenspinors {ψi}i∈ℤ: we denote by ψi- the eigenspinors with negative eigenvalue, ψi+ the eigenspinors with positive eigenvalue and ψi0 the eigenspinors with zero eigenvalue; we also recall that the kernel of Dg is finite dimensional. Now we set

H 1 2 , - := span { ψ i - } i ∈ ℤ ¯ , H 1 2 , 0 := span { ψ i 0 } i ∈ ℤ , H 1 2 , + := span { ψ i + } i ∈ ℤ ¯ ,

where the closure is taken with respect to the H12-topology. Therefore we have the orthogonal decomposition of H12⁢(Σ⁢M), which reads as

H 1 2 ⁢ ( Σ ⁢ M ) = H 1 2 , - ⊕ H 1 2 , 0 ⊕ H 1 2 , + .

Also, we let P+ and P- be the projectors on H12,+ and H12,-, respectively.

Again, if we assume M to be the sphere 𝕊3 and we identify 𝕊3 minus the south pole with ℝ3 via stereographic projection, the conformal invariance of the Dirac operator reads as

D g 𝕊 3 ⁢ ψ = F ⁢ { [ f - 2 ⁢ D ⁢ ( f ⁢ F - 1 ⁢ ( ψ ∘ π - 1 ) ) ] ∘ π } , ψ ∈ H 1 2 ⁢ ( Σ ⁢ 𝕊 3 ) ,

where Dg𝕊3 and Dgℝ3=D denote the Dirac operators on the standard sphere and ℝ3, respectively; moreover f=21+|y|2 and F:Σ⁢(ℝ3,gℝ3)→Σ⁢(𝕊3,g𝕊3) the isomorphism of vector bundles in (2.1).

In the sequel we will need the following function spaces on ℝ3:

D 1 2 ⁢ ( Σ ⁢ ℝ 3 ) = { ψ ∈ L 3 ⁢ ( Σ ⁢ ℝ 3 ) : | ξ | 1 2 ⁢ | ψ ^ | ∈ L 2 ⁢ ( ℝ 3 ) } ,

D 1 ⁢ ( ℝ 3 ) = { u ∈ L 6 ⁢ ( ℝ 3 ) : | ∇ ⁡ u | ∈ L 2 ⁢ ( ℝ 3 ) } .

Here ψ^ is the Fourier transform of ψ. Moreover, for a more detailed exposition on conformal and spin geometry as well as on the functional spaces involved we address the reader to [10, 12, 13] and the references therein.

3 Proof of the Main Result

Our existence result will be obtained by means of the abstract perturbation method illustrated in [1]. We recall it in the following theorem and then we will show how it can be applied in our setting.

Theorem 3.1.

(see [1]) Let A be a Hilbert space and assume J0∈C2⁢(A,R) satisfies the following conditions:

J 0 has a finite-dimensional manifold Z of critical points,
J 0 ′′ ⁢ ( z ) is a Fredholm operator of index zero for every z ∈ Z ,
T z ⁢ Z = ker ⁡ J 0 ′′ ⁢ ( z ) for every z ∈ Z .

For G∈C2⁢(A,R), let ε>0 small enough, we denote by Jε=J0-ε⁢G the perturbed functional, by V the orthogonal complement of Tz⁢Z in A and by P:A→V the orthogonal projection. Then, for any z∈Z, there exists v⁢(z)∈V such that P⁢(Jε′⁢(z+v⁢(z)))=0.

Moreover, if there exists a compact set Ω⊂Z such that Jε|Z has a critical point z∈Ω, then z+v⁢(z) is a critical point of the perturbed functional Jε in A.

In order to apply the previous result to our situation, we introduce the map

H 1 ⁢ ( 𝕊 3 ) × H 1 2 ⁢ ( Σ ⁢ 𝕊 3 ) ∋ ( v , ψ ) ↦ ( u , ϕ ) = ( f 1 2 ⁢ v ∘ π - 1 , f ⁢ F - 1 ⁢ ( ψ ∘ π - 1 ) ) ,

which gives a one to one correspondence between solutions to (1.1) on 𝕊3 and solutions to the equivalent system on ℝ3

{ - Δ ⁢ u = H ⁢ | ϕ | 2 ⁢ u , D ⁢ ψ = H ⁢ u 2 ⁢ ϕ , on ℝ 3 ,

where we set H=K∘π-1. Hence let us consider this last problem and let us denote

A = D 1 ⁢ ( ℝ 3 ) × D 1 2 ⁢ ( Σ ⁢ ℝ 3 ) .

We take w=(u,ψ)∈A and we set

J 0 ⁢ ( w ) = 1 2 ⁢ ∫ ℝ 3 ( - u ⁢ Δ ⁢ u + 〈 D ⁢ ϕ , ϕ 〉 - | u | 2 ⁢ | ϕ | 2 ) ,

G ⁢ ( w ) = 1 2 ⁢ ∫ ℝ 3 h ⁢ | u | 2 ⁢ | ϕ | 2 ,

J ε ⁢ ( w ) = J 0 ⁢ ( w ) - ε ⁢ G ⁢ ( w )

with h=k∘π-1. We are going to define the manifold of critical points of J0. Let λ∈ℝ+, y,ξ∈ℝ3, a∈Σ⁢ℝ3 with |a|=1, it is well known that the functions

U ¯ λ , ξ ⁢ ( y ) = 3 4 ⁢ λ 1 2 ( λ 2 + | y - ξ | 2 ) 1 2

are a family of positive solutions to -Δ⁢u=u5 in ℝ3 and the spinors

Φ ¯ λ , ξ , a ⁢ ( x ) = 2 ⁢ λ ( λ 2 + | y - ξ | 2 ) 3 2 ⁢ ( λ - ( y - ξ ) ) ⋅ a

solve D⁢ϕ=32⁢|ϕ|⁢ϕ in Σ⁢ℝ3. Using this fact, and the equality |Φ¯λ,ξ,a|=21+|y|2, one can check that the pairs

( U λ , ξ , Φ λ , ξ , a ) = ( 3 4 ⁢ U ¯ λ , ξ , 3 2 ⁢ Φ ¯ λ , ξ , a ) ∈ A

are critical points of J0. Hence

Z = { W λ , ξ , a = ( U λ , ξ , Φ λ , ξ , a ) : λ ∈ ℝ + , ξ ∈ ℝ 3 ⁢ and ⁢ a ∈ Σ ⁢ ℝ 3 , | a | = 1 } ⊂ A

is a seven-dimensional manifold of critical points of J0. Let us fix any a0∈Σ⁢ℝ3 with |a0|=1, in the sequel we will use the notation U0=U1,0, Φ0=Φ1,0,a0 and W0=(U0,Φ0).

Now we will check assumption (2) in Theorem 3.1. We have

〈 J 0 ′′ ⁢ ( W λ , ξ , a ) ⁢ [ w 1 ] , w 2 〉 = ∫ ℝ 3 ( - u 2 ⁢ Δ ⁢ u 1 - u 2 ⁢ u 1 ⁢ | Φ λ , ξ , a | 2 - 2 ⁢ u 2 ⁢ U λ , ξ ⁢ 〈 Φ λ , ξ , a , ϕ 1 〉 )

+ ∫ ℝ 3 ( 〈 D ⁢ ϕ 1 - | U λ , ξ | 2 ⁢ ϕ 1 , ϕ 2 〉 - 2 ⁢ U λ , ξ ⁢ u 1 ⁢ 〈 Φ λ , ξ , a , ϕ 2 〉 ) .

Therefore J0′′ is a compact perturbation of the identity, hence it is a Fredholm operator of index zero for all Wλ,ξ,a∈Z.

Now it remains to check that TWλ,ξ,a⁢Z=ker⁡J0′′⁢(Wλ,ξ,a) for every λ∈ℝ+, ξ∈ℝ3 and a∈Σ⁢ℝ3 with |a|=1. Since J0′′ is invariant with respect to translations and dilations, it will be enough to prove TW0⁢Z=ker⁡J0′′⁢(W0). We will need the following remark.

Remark 3.2.

Let λ1=34 and μ1=32. The map (v,ψ)↦(ν,η)=(μ1-12⁢v,λ1-12⁢ψ) is a one-to-one correspondence between solution to (1.1) on 𝕊3 and the equivalent rescaled system

{ L g 𝕊 3 ⁢ ν = λ 1 ⁢ | η | 2 ⁢ ν , D g 𝕊 3 ⁢ η = μ 1 ⁢ ν 2 ⁢ η , on 𝕊 3 ,

which in turn it is equivalent to

(3.1) { - Δ ⁢ u = λ 1 ⁢ | ϕ | 2 ⁢ u , D ⁢ ϕ = μ 1 ⁢ u 2 ⁢ ϕ , on ℝ 3

by means of the stereographic projection. Notice that (3.1) arises as the first variation of the functional

J ~ 0 ⁢ ( w ) = 1 2 ⁢ ∫ ℝ 3 ( - λ 1 - 1 ⁢ u ⁢ Δ ⁢ u + μ 1 - 1 ⁢ 〈 D ⁢ ϕ , ϕ 〉 - | ϕ | 2 ⁢ | u | 2 )

and since (Uλ,ξ,Ψλ,ξ,a) are critical points of J0, it follows that

W ~ λ , ξ , a = ( μ 1 - 1 2 ⁢ U λ , ξ , λ 1 - 1 2 ⁢ Ψ λ , ξ , a )

are critical points of J~0.

Lemma 3.3.

We have TW0⁢Z=ker⁡J0′′⁢(W0).

Proof.

It is standard to check that TW0⁢Z⊆ker⁡J0′′⁢(W0), so it suffices to prove the inclusion ker⁡J0′′⁢(W0)⊆TW0⁢Z. Moreover, since dim⁡(TW0⁢Z)=7, it is enough to show that

dim ⁡ ( ker ⁡ J 0 ′′ ⁢ ( W 0 ) ) ≤ 7

and, by means of Remark 3.2, this is equivalent to

dim ⁡ ( ker ⁡ J ~ 0 ′′ ⁢ ( W ~ 0 ) ) ≤ 7 .

On the sphere 𝕊3, the linearization of (3.1) at W~0 reads as

(3.2) { L g 𝕊 3 ⁢ ν = λ 1 ⁢ ν ⁢ | Ψ 1 | 2 + 2 ⁢ λ 1 ⁢ V 1 ⁢ 〈 Ψ 1 , η 〉 , D g 𝕊 3 ⁢ η = μ 1 ⁢ | V 1 | 2 ⁢ η + 2 ⁢ μ 1 ⁢ ν ⁢ V 1 ⁢ Ψ 1 ,

where

( V 1 , Ψ 1 ) = ( μ 1 - 1 2 ⁢ ( f - 1 2 ⁢ U λ , ξ ) ∘ π , λ 1 - 1 2 ⁢ ( f ∘ π ) - 1 ⁢ F ⁢ ( Φ λ , ξ , a ∘ π ) ) = ( 1 , Ψ 1 ) .

Notice that Ψ1 satisfies

(3.3) D g 𝕊 3 ⁢ Ψ 1 = 3 2 ⁢ | Ψ 1 | ⁢ Ψ 1 and | Ψ 1 | = 1 ,

so it is an eigenspinor of Dg𝕊3 with eigenvalue 32. We set η=∑k∈ℤfk⁢Ψk, where Ψk is a trivialization with Killing spinors of the spinor bundle of 𝕊3 and we write f1=g1+i⁢h1, where g1 and h1 are real-valued functions; here by trivialization we mean a global basis of the spinor bundle of 𝕊3 formed by Killing spinors. For more details regarding such a basis, we refer the reader to [10] or [12, formula (5.14)]. We will first find f1. Since f1=〈η,Ψ1〉, we have (see [12, Lemma 5.2 and formula (5.16)])

Δ g 𝕊 3 ⁢ f 1 = 〈 Δ g 𝕊 3 ⁢ η , Ψ 1 〉 + 〈 η , Δ g 𝕊 3 ⁢ Ψ 1 〉 + 〈 D g 𝕊 3 ⁢ η , Ψ 1 〉 .

Notice now that, by (3.3) and the Lichnerowicz’s formula on the sphere

D g 𝕊 3 2 = - Δ g 𝕊 3 + 3 2 ,

we have -Δg𝕊3⁢Ψ1=34⁢Ψ1 and

- Δ g 𝕊 3 ⁢ η = D g 𝕊 3 2 ⁢ η - 3 2 ⁢ η

= D g 𝕊 3 ⁢ ( 3 2 ⁢ η + 3 ⁢ ν ⁢ Ψ 1 ) - 3 2 ⁢ η = 3 2 ⁢ ( 3 2 ⁢ η + 3 ⁢ ν ⁢ Φ 1 ) + 3 ⁢ ∇ ⁡ ν ⋅ Ψ 1 + 9 2 ⁢ ν ⁢ Ψ 1 - 3 2 ⁢ η

= 3 4 ⁢ η + 9 ⁢ ν ⁢ Ψ 1 + 3 ⁢ ∇ ⁡ ν ⋅ Ψ 1 .

Therefore

- Δ g 𝕊 3 ⁢ f 1 = 3 4 ⁢ f 1 + 9 ⁢ ν + 3 ⁢ 〈 ∇ ⁡ ν ⋅ Ψ 1 , Ψ 1 〉 + 3 4 ⁢ f 1 - 3 2 ⁢ f 1 - 3 ⁢ ν = 6 ⁢ ν + 3 ⁢ 〈 ∇ ⁡ ν ⋅ Ψ 1 , Ψ 1 〉 .

Since the last addend in the previous equality is purely imaginary, we take the real and imaginary part to have

- Δ g 𝕊 3 ⁢ g 1 = 6 ⁢ ν

and

- Δ g 𝕊 3 ⁢ h 1 = - 3 ⁢ i ⁢ 〈 ∇ ⁡ ν ⋅ Ψ 1 , Ψ 1 〉 .

In particular, recalling that Lg𝕊3=-Δg𝕊3+34 and the first equation in (3.2), we have the system

{ - Δ g 𝕊 3 ⁢ ν = 3 2 ⁢ g 1 , - Δ g 𝕊 3 ⁢ g 1 = 6 ⁢ ν .

Hence,

Δ g 𝕊 3 2 ⁢ g 1 = 9 ⁢ g 1

from which we deduce that g1 is the first eigenfunction of the Laplacian on the sphere and ν=g12. So, the first equation in (3.2) becomes

L g 𝕊 3 ⁢ g 1 2 = 3 4 ⁢ g 1 2 + 3 ⁢ f 1

and recalling the definition of Lg𝕊3, from the quality above we get

f 1 = g 1 .

Using this fact, system (3.2) becomes

{ ν = g 1 2 , D g 𝕊 3 ⁢ η = 3 2 ⁢ η + 3 2 ⁢ 〈 η , Ψ 1 〉 ⁢ Ψ 1 .

Hence we need to compute the dimension of

Λ = { η ∈ H 1 2 ⁢ ( Σ ⁢ 𝕊 3 ) : D g 𝕊 3 ⁢ η = 3 2 ⁢ η + 3 2 ⁢ 〈 η , Ψ 1 〉 ⁢ Ψ 1 } .

This computation has been carried out by Isobe in [12] for general dimensions of the sphere 𝕊m, so in our situation it suffices to take m=3 in [12, Lemma 5.1] to get dim⁡(Λ)=7 as desired. ∎

Now we will focus on the reduced functional. For a fixed a∈Σ⁢ℝ3, with |a|=1, we set Vλ,ξ=|Uλ,ξ|2⁢|Φλ,ξ,a|2, so that Vλ,ξ⁢(x)=1λ3⁢V1,0⁢(1λ⁢(x-ξ)) and let

Γ ⁢ ( λ , ξ ) = 1 2 ⁢ ∫ ℝ 3 h ⁢ ( x ) ⁢ V λ , ξ ⁢ ( x ) ⁢ d x

for (λ,ξ)∈(0,+∞)×ℝ3. Then we have the following:

Proposition 3.4.

The function Γ is of class C2 on (0,+∞)×R3 and it can be extended to a C1 function at λ=0 by

Γ ⁢ ( 0 , ξ ) = c 0 ⁢ h ⁢ ( ξ ) , c 0 = 1 2 ⁢ ∫ ℝ 3 V 1 , 0 ⁢ ( x ) ⁢ d x .

Also,

lim λ → 0 ⁡ ∇ ξ 2 ⁡ Γ ⁢ ( λ , ξ ) = c 0 ⁢ ∇ 2 ⁡ h ⁢ ( ξ ) ,

uniformly on every compact of R3. Moreover, for any compact set Σ⊂R3, there exists a constant C=CΣ such that

| ∂ λ ⁡ Γ ⁢ ( λ , ξ ) - c 1 ⁢ λ ⁢ Δ ⁢ h ⁢ ( ξ ) | ≤ C Σ ⁢ λ 2

for all λ>0 and all ξ∈Σ, being

c 1 = ∫ ℝ 3 | y | 2 ⁢ V 1 , 0 ⁢ ( y ) ⁢ d y .

Proof.

We have by a change of variable that

Γ ⁢ ( λ , ξ ) = ∫ ℝ 3 h ⁢ ( λ ⁢ x + ξ ) ⁢ V 1 , 0 ⁢ ( x ) ⁢ d x .

Using the smoothness of h and the dominated convergence, we have that

lim λ → 0 ⁡ Γ ⁢ ( λ , ξ ) = c 0 ⁢ h ⁢ ( ξ ) .

The same reasoning applies to show that one has

∇ ξ ⁡ Γ ⁢ ( 0 , ξ ) = c 0 ⁢ ∇ ⁡ h ⁢ ( ξ ) , ∇ ξ 2 ⁡ Γ ⁢ ( 0 , ξ ) = c 0 ⁢ ∇ 2 ⁡ h ⁢ ( ξ ) and ∇ λ ⁡ Γ ⁢ ( 0 , ξ ) = 0 .

The last equality follows from the oddness of the integral, that is,

∫ ℝ 3 x i ⁢ V 1 , 0 ⁢ ( x ) ⁢ d x = 0 , i = 1 , 2 , 3 .

We fix now a compact set Σ; then by Taylor expansion of y↦h⁢(y+ξ), we have

| ∂ ξ i ⁡ h ⁢ ( y + ξ ) - ∂ ξ i ⁡ h ⁢ ( ξ ) - ∑ j = 1 3 ∂ ξ i ⁢ ξ j 2 ⁡ h ⁢ ( ξ ) ⁢ y j | ≤ C Σ ⁢ | y | 2 .

Also, notice that since

∫ ℝ 3 y i ⁢ y j ⁢ V 1 , 0 ⁢ ( y ) ⁢ d y = 0 if ⁢ i ≠ j ,

we have for our choice of c1,

c 1 ⁢ λ ⁢ Δ ⁢ h ⁢ ( ξ ) = ∫ ℝ 3 ∑ i = 1 3 ( ∂ ξ i ⁡ h ⁢ ( ξ ) + ∑ j = 1 3 ∂ ξ i ⁢ ξ j 2 ⁡ h ⁢ ( ξ ) ⁢ λ ⁢ y j ) ⁢ y i ⁢ V 1 , 0 ⁢ ( y ) ⁢ d ⁢ y .

Therefore,

| ∂ λ ⁡ Γ ⁢ ( λ , ξ ) - c 1 ⁢ λ ⁢ Δ ⁢ h ⁢ ( ξ ) | ≤ C Σ ⁢ λ 2 . ∎

Proposition 3.5.

Let k and h be functions as in Theorem 1.1. Then there exists an open set Ω⊂(0,+∞)×R3 such that ∇⁡Γ≠0 on ∂⁡Ω and

deg ⁡ ( ∇ ⁡ Γ , Ω , 0 ) = ∑ ξ ∈ crit ⁡ [ h ] Δ ⁢ h ⁢ ( ξ ) < 0 ( - 1 ) m ⁢ ( h , ξ ) + 1 .

Proof.

Let s>0; we consider the set

ℬ s = { ( λ , ξ ) ∈ ( 0 , + ∞ ) × ℝ 3 : | ( λ , ξ ) - ( s , 0 ) | ≤ s - 1 s } .

We will show that for s large enough, we can choose Ω=ℬs. First, we set

crit ⁡ [ h ] = { ξ 1 , ξ 2 , … , ξ l }

for some l∈ℕ. Since the south pole is not a critical point for k, we have that for r large enough,

crit ⁡ [ h ] ⊂ A r = { ξ ∈ ℝ 3 : | ξ | ≤ r } .

Since h is a Morse function (as well as k), it follows from the non-degeneracy condition (i) that there exist constants μ∈(0,r) and δ>0 such that

| Δ ⁢ h ⁢ ( ξ ) | > δ for all ⁢ ξ ∈ ⋃ i = 1 l B μ ⁢ ( ξ i ) ,

where Bμ⁢(ξi) denote as usual the balls of centers ξi and radius μ. By using Proposition 3.4, we have that for s sufficiently large and μ even smaller if necessary,

∂ λ ⁡ Γ ⁢ ( λ , ξ ) ≠ 0 in ⁢ ∂ ⁡ ℬ s ∩ ( ( 0 , μ ) × ⋃ i = 1 l B μ ⁢ ( ξ i ) ) .

Hence,

∇ ⁡ Γ ≠ 0 ⁢ in ⁢ ∂ ⁡ ℬ s ∩ ( ( 0 , μ ) × ⋃ i = 1 l B μ ⁢ ( ξ i ) ) .

Again, by Proposition 3.4, since Γ extends to a C1 function at λ=0 and ∇ξ⁡Γ⁢(0,ξ)=c0⁢∇⁡h⁢(ξ), we have that

∇ ⁡ Γ ≠ 0 in ⁢ ∂ ⁡ ℬ s ∩ ( ( 0 , μ ) × A 2 ⁢ r ∖ ⋃ i = 1 l B μ ⁢ ( ξ i ) ) .

Hence,

∇ ⁡ Γ ≠ 0 in ⁢ ∂ ⁡ ℬ s ∩ ( ( 0 , μ ) × A 2 ⁢ r ) .

So it remains to study Γ on the component of ∂⁡ℬs outside (0,μ)×A2⁢r. So we consider the Kelvin reflection

τ : ℝ 3 ∖ { 0 } → ℝ 3 ∖ { 0 } , τ ⁢ ( x ) = x | x | 2 .

We notice that

τ * ⁢ ( g ℝ 3 ) = 1 | x | 4 ⁢ g ℝ 3 .

Hence, for all F∈L6⁢(ℝ3), by putting y=τ⁢(x), we have

∫ ℝ 3 h ⁢ ( y ) ⁢ | F ⁢ ( y ) | 6 ⁢ d y = ∫ ℝ 3 h ⁢ ( τ ⁢ ( x ) ) ⁢ | F ⁢ ( τ ⁢ ( x ) ) | 6 ⁢ f 3 ⁢ ( x ) ⁢ d x = ∫ ℝ 3 h ⁢ ( τ ⁢ ( x ) ) ⁢ | F * ⁢ ( x ) | 6 ⁢ d x ,

where

F * ⁢ ( x ) = 1 | x | 2 ⁢ F ⁢ ( x | x | 2 ) .

In particular, if we set

λ ~ = λ λ 2 + | ξ | 2 , ξ ~ = ξ λ 2 + | ξ | 2 ,

we have that

| V λ , ξ * ⁢ ( x ) | = | V λ ~ , ξ ~ ⁢ ( x ) | .

So we define

Γ ~ = 1 2 ⁢ ∫ ℝ 3 h ⁢ ( τ ⁢ ( x ) ) ⁢ | V λ , ξ ⁢ ( x ) | ⁢ d x ,

and we have that

Γ ⁢ ( λ , ξ ) = Γ ~ ⁢ ( λ ~ , ξ ~ ) .

Once again, by using Proposition 3.4, we have that Γ~ can be extended to a C1 function up to the origin (0,0)∈[0,∞)×ℝ3. Since (λ,ξ)↦(λ~,ξ~) is a diffeomorphism, we have ∇⁡Γ⁢(λ,ξ)=0 if and only if Γ~⁢(λ~,ξ~)=0. But by assumption, the south pole is not a critical point of h, hence 0 is not a critical point of h⁢(τ⁢(x)). Therefore, ∇⁡Γ~≠0 in a neighborhood of the origin and so ∇⁡Γ≠0 in a neighborhood of infinity. Finally, we have that for r and s large enough,

∇ ⁡ Γ ≠ 0 on ⁢ ∂ ⁡ ℬ s ∖ ( ( 0 , μ ) × A ¯ 2 ⁢ r ) .

The degree computation is by now standard and it follows for instance as in [11]. ∎

Remark 3.6.

We want explicitly to notice that at this point we cannot directly conclude as in the classical cases (see for instance [2, 21]), since the critical points of Γ on Z are degenerate: this is due to the invariance of the functional with respect to the parameters a and this degeneracy causes the degree to vanish.

We recall that Z is a non-degenerate manifold of critical points of J0 and J0′′ is Fredholm of index zero, therefore we have that there exists ε>0 such that for all z∈Zc⊂Z with Zc compact, there exists a unique w⁢(z)∈Tz⁢Z⊥ such that

P ⁢ J ε ′ ⁢ ( z + w ⁢ ( z ) ) = 0 ,

where P:A→Tz⁢Z⊥ is the orthogonal projection. Now, to find a solution to our problem, it is enough to find a critical point for the function Φε:Z→ℝ defined by

Φ ε ⁢ ( z ) = J ε ⁢ ( z + w ⁢ ( z ) ) .

In order to do this, we will consider the set of the parameters a

{ a ∈ Σ ⁢ ℝ 3 : | a | = 1 } ≃ 𝕊 3

as a Lie group. Hence, we will consider the natural action of 𝕊3 on Z≃(0,+∞)×ℝ3×𝕊3, being Z parameterized by (λ,ξ,a). Also, we notice that (J0)|Z and G|Z are invariant under this action: then we need to extend the action to the whole space D12⁢(Σ⁢ℝ3). In order to do this, we recall that the spinor bundle of ℝ3 can be trivialized by Killing spinors that are either constant (parallel spinors) or spinors of the form x⋅ϕ with ϕ constant. So we fix an orthonormal basis of Σ⁢ℝ3 of the form {a1,a2,x⋅a1,x⋅a2}, where a1,a2 are (distinct) constant spinors with |a1|=|a2|=1. Hence, if ϕ∈D12⁢(Σ⁢ℝ3), there exist f1,f2,g1,g2 such that

ϕ ⁢ ( x ) = ( f 1 ⁢ ( x ) + g 1 ⁢ ( x ) ⁢ x ) ⋅ a 1 + ( f 2 ⁢ ( x ) + g 2 ⁢ ( x ) ⁢ x ) ⋅ a 2 .

Since a1 and a2 can be seen as elements in 𝕊3, we can define the action for a general w∈𝕊3 and ϕ∈D12⁢(Σ⁢ℝ3) by

w ⁢ ϕ = ( f 1 ⁢ ( x ) + g 1 ⁢ ( x ) ⁢ x ) ⋅ w ⁢ a 1 + ( f 2 ⁢ ( x ) + g 2 ⁢ ( x ) ⁢ x ) ⋅ w ⁢ a 2 .

In this way, this last action extends the one previously defined on Z and in addition both J0 and G are invariant under this action. Therefore, Φε descends to a C1 function Φ~ε defined on the quotient

Z / 𝕊 3 ≃ ( 0 , ∞ ) × ℝ 3 .

The same argument works for Γ; therefore for ε small enough, we have that

Φ ~ ε ′ = ε ⁢ Γ + o ⁢ ( ε ) .

At this point, from the invariance of the degree by homotopy, we have that

deg ⁡ ( Φ ~ ε ′ , ℬ s , 0 ) = ∑ ξ ∈ crit ⁡ [ h ] Δ ⁢ h ⁢ ( ξ ) < 0 ( - 1 ) m ⁢ ( h , ξ ) + 1 .

Finally, by assumption (ii), by contradiction (for the argument, see for instance [1, 2]) if

∑ ξ ∈ crit ⁡ [ h ] Δ ⁢ h ⁢ ( ξ ) < 0 ( - 1 ) m ⁢ ( h , ξ ) ≠ - 1 ,

then Φ~ε has a critical point that can be lifted as a critical orbit of Φε, which in turn ends the proof of the Main Theorem 1.1.

Communicated by Enrico Valdinoci

Acknowledgements

We thank the anonymous referee for her/his helpful comments and suggestions.

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Received: 2020-01-18

Revised: 2020-07-12

Accepted: 2020-07-13

Published Online: 2020-12-15

Published in Print: 2021-02-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Existence Results for the Conformal Dirac–Einstein System

Abstract

1 Introduction

Theorem 1.1.

2 Notations and Definitions

3 Proof of the Main Result

Theorem 3.1.

Remark 3.2.

Lemma 3.3.

Proof.

Proposition 3.4.

Proof.

Proposition 3.5.

Proof.

Remark 3.6.

Acknowledgements

References

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