Existence results for the conformal Dirac-Einstein system
aa r X i v : . [ m a t h . A P ] J a n Existence results for the conformal Dirac-Einstein system
Chiara Guidi (1) & Ali Maalaoui (2) & Vittorio Martino (3)
Abstract
In this paper we consider the coupled system given by the first variation of the con-formal Dirac-Einstein functional. We will show existence of solutions by means of perturbationmethods.Keywords: Conformally invariant operators, perturbation methods.2010 MSC. Primary: 58J05, 58E15. Secondary: 53A30, 58Z05
Let (
M, g, Σ M ) be a closed (compact, without boundary) three dimensional Rieman-nian Spin manifold where Σ M is its spin bundle. We denote by L g the conformalLaplacian of g and by D g the Dirac operator. We consider the energy functional E M ( v, ψ ) = 12 (cid:18)Z M vL g v + h D g ψ, ψ i − | v | | ψ | d vol g (cid:19) (1.1)and we take its first variation on the related Sobolev space H ( M ) × H (Σ M ); thereforeits critical points satisfy the coupled system L g v = | ψ | uD g ψ = | v | ψ on M. (1.2)This functional arises as the conformal version in the description of a super-symmetricmodel consisting of coupling gravity with fermionic interaction and it generalizes theclassical Hilbert-Einstein energy functional, see for instance [4, 9, 13].Indeed, the total energy functional consists of the Hilbert-Einstein energy which is the Dipartimento di Matematica, Universit`a di Bologna, piazza di Porta S.Donato 5, 40126 Bologna,Italy. E-mail address: [email protected] Department of mathematics and natural sciences, American University of Ras Al Khaimah, POBox 10021, Ras Al Khaimah, UAE. E-mail address: [email protected] Dipartimento di Matematica, Universit`a di Bologna, piazza di Porta S.Donato 5, 40126 Bologna,Italy. E-mail address: [email protected] M , when one restricts it to a fixedconformal class of a given Riemannian metric g , the functional E M shows up.In particular, due to the conformal invariance, the Palais-Smale compactness conditionis violated by this functional and in addition, due to the presence of the Dirac operator,it is strongly indefinite.Regarding the first issue, in [19] the authors studied the lack of compactness and gavea precise description of the bubbling phenomena, characterizing the behaviour of thePalais-Smale sequences, in the spirit of classical works [22, 21, 23, 14, 15, 5, 3]. Forthe strongly indefinite difficulty, in [16, 17, 18] general functionals with these featuresare studied by using methods based on a homological approach. Notice that so far,one cannot apply these homological approaches because of the violation of compactnessstated above.In this paper, we are concerned with the existence of solutions to the coupled system, byusing a perturbation approach, starting from the sphere S equipped with its standardmetric g S .Therefore, let K be a function of the form K = 1 + εk , where k is a function withsuitable assumptions to be determined later; we consider the functional E ( v, ψ ) = 12 (cid:18)Z S vL g S v + h D g S ψ, ψ i − K | v | | ψ | d vol g S (cid:19) (1.3)and we will focus on the existence of solutions to the following coupled system: L g S v = K | ψ | vD g S ψ = Kv ψ on S (1.4)Notice that these solutions converge to the standard bubbles when the parameter ε tends to zero. This is expected from the description of the Palais-Smale sequences ofthe functional E M , but it remains open whether all the solutions on the sphere withpositive scalar component are in fact standard ones.Let us denote by π : S \ { sp } → R the stereographic projection, where sp is the southpole. Our main result is the following Theorem 1.1.
Let k ∈ C ( S ) be a Morse function on S such that the south pole isnot a critical point. Let us set h = k ◦ π − and suppose that ( i ) ∆ h ( ξ ) = 0 , ∀ ξ ∈ crit[ h ] , ( ii ) X ξ ∈ crit[ h ]∆ h ( ξ ) < ( − m ( h,ξ ) = − , where ∆ is the standard Laplacian operator on R , crit[ h ] denotes the set of criticalpoints of h and m ( h, ξ ) is the morse index of h at a critical point ξ .Then, there exists ε > such that for K = 1 + εk and | ε | < ε , the system (1.4) hasa solution. π , however this conditioncan be always satisfied by making a unitary transformation which does not affect thegenerality of the result.The previous result is the analogous of several ones obtained with this kind of hypothesisof Bahri-Coron type on the function k : for instance, for the standard Riemannian caseof prescribing the scalar curvature and its generalization to the Q γ curvature see [2, 6, 8];in the case of prescribing the Webster curvature in the CR setting and its fractionalgeneralization see [20] and [7]; for the spinorial Yamabe type equations involving theDirac operator on the sphere see [12].The idea of the proof follows the abstract perturbation method introduced in [1].The difficulties in our situation come from the fact of having a system, from the stronglyindefiniteness of one of the operator involved and finally from the degeneracy of thecritical points of the finite dimensional reduction of the functional, which is due to theinvariance with respect to one of the parameters of the problem (see Remark 3.6). Let (
M, g ) be a closed (compact, without boundary) three dimensional Riemannianmanifold.We start to describe shortly the first operator appearing in the system. We denote by L g the conformal Laplacian acting on functions L g = − ∆ g + 18 R g . Here ∆ g is the standard Laplace-Beltrami operator and R g is the scalar curvature. L g is a conformally invariant operator. More precisely, given a metric ˜ g = f g in theconformal class of g , we have L ˜ g u = f − L g ( f u ) . We recall that the usual Sobolev space on M , denoted by H ( M ), continuously embedsin L p ( 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 = { ( x ′ , x ) ∈ R × R : | x ′ | + x = 1 } equipped with its standard metric g S , then it is possible to identify S \ { sp } , being sp = (0 , −
1) the south pole, with R , by means of the stereographic projection π : S \ { sp } → R ( x, x ) y = x x . The standard metric g R on R and the metric ˜ g = ( π − ) ∗ g S are conformal, moreprecisely ˜ g = f g R , with f = | y | . Thus the standard conformal Laplacian on the3phere L g S and the one on R , which we denote as usual L g R = − ∆, are related bythe following identity L g S v = h f − ( − ∆) (cid:16) f v ◦ π − (cid:17)i ◦ π, v ∈ H ( S ) . (2.1)Now, let us describe the second operator involved. Let Σ M be the canonical spinorbundle associated to M , whose sections are simply called spinors on M . This bundleis endowed with a natural Clifford multiplicationCliff : 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 D g 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 = f g , reads as follows: there exists an isomorphism of vector bundles F : Σ( M, g ) → Σ( M, ˜ g ) such that D ˜ g ψ = F (cid:2) f − D g (cid:0) f F − ψ (cid:1)(cid:3) . (2.2)The functional space that we are going to define is the Sobolev space H (Σ M ). Firstwe recall that the Dirac operator D g on a compact manifold is essentially self-adjointin L (Σ M ), has compact resolvent and there exists a complete L -orthonormal basisof eigenspinors { ψ i } i ∈ Z of the operator D g ψ i = λ i ψ i , and the eigenvalues { λ i } i ∈ Z are unbounded, that is | λ i | → ∞ , as | i | → ∞ . In this wayevery function in L (Σ M ), it has a representation in this basis, namely: ψ = X i ∈ Z a i ψ i , ψ ∈ L (Σ M ) . We define the unbounded operator | D g | s : L (Σ M ) → L (Σ M ) by | D g | s ( ψ ) = X i ∈ Z a i | λ i | s ψ i and we denote by H s (Σ M ) the domain of | D g | s , namely ψ ∈ H s (Σ M ) if and only if X i ∈ Z a i | λ i | s < + ∞ . s (Σ M ) coincides with the usual Sobolev space W s, (Σ M ) and for s < H s (Σ M ) isdefined as the dual of H − s (Σ M ).For s >
0, we define the inner product, for ψ, φ ∈ H s (Σ M ) h ψ, φ i s = h| D g | s ψ, | D g | s φ i L , which induces an equivalent norm in H s (Σ M ); we will take h ψ, ψ i := h ψ, ψ i = k ψ k as our standard norm for the space H (Σ M ). In this case as well, the embedding H s (Σ M ) ֒ → L p (Σ M ) is continuous for 1 ≤ p ≤ ≤ p < H (Σ M ) in a natural way. Let us consider the L -orthonormalbasis of eigenspinors { ψ i } i ∈ Z : we denote by ψ − i the eigenspinors with negative eigen-value, ψ + i the eigenspinors with positive eigenvalue and ψ i the eigenspinors with zeroeigenvalue; we also recall that the kernel of D g is finite dimensional. Now we set: H , − := span { ψ − i } i ∈ Z , H , := span { ψ i } i ∈ Z , H , + := span { ψ + i } i ∈ Z , where the closure is taken with respect to the H -topology. Therefore we have theorthogonal decomposition of H (Σ M ), which reads as: H (Σ M ) = H , − ⊕ H , ⊕ H , + . Also, we let P + and P − be the projectors on H , + and H , − respectively.Again, if we assume M to be the sphere S and we identify S minus the south polewith R via stereographic projection, the conformal invariance of the Dirac operatorreads as D g S ψ = F (cid:8)(cid:2) f − D (cid:0) f F − ( ψ ◦ π − ) (cid:1)(cid:3) ◦ π (cid:9) , ψ ∈ H (Σ S ) (2.3)where D g S and D g R = D denote the Dirac operators on the standard sphere and R respectively; moreover f = | y | and F : Σ( R , g R ) → Σ( S , g S ) the isomorphism ofvector bundles in (2.2).In the sequel we will need the following function spaces on R : D (Σ R ) = n ψ ∈ L (Σ R ) : | ξ | | b ψ | ∈ L ( R ) o ; D ( R ) = (cid:8) u ∈ L ( R ) : |∇ u | ∈ L ( R ) (cid:9) . Here b ψ is the Fourier transform of ψ . Our existence result will be obtained by means of the abstract perturbation methodillustrated in [1].We recall it in the following theorem and then we will show how it can be applied inour setting. 5 heorem 3.1. (see [1]) Let A be an Hilbert space and assume J ∈ C ( A, R ) satisfiesthe following conditions1. J has a finite-dimensional manifold Z of critical points,2. J ′′ ( z ) is a Fredholm operator of index zero for every z ∈ Z ,3. T z Z = ker J ′′ ( z ) , for every z ∈ Z .For G ∈ C ( A, R ) , we denote by J ε = J − εG the perturbed functional, by V theorthogonal complement of T z Z in A and by P : A → V the orthogonal projection.Then, for any z ∈ Z there exists v ( z ) ∈ V such that P ( J ′ ε ( z + v ( z ))) = 0 .Moreover, if there exists a compact set Ω ⊂ Z such that J ε | Z has a critical point z ∈ Ω , then z + v ( z ) is a critical point of the perturbed functional J ε in A . In order to apply the previous result to our situation, we introduce the following map H ( S ) × H (Σ S ) ∋ ( v, ψ ) ( u, φ ) = (cid:16) f v ◦ π − , f F − ( ψ ◦ π − ) (cid:17) , which gives a one to one correspondence between solutions to (1.4) on S and solutionsto the equivalent system on R − ∆ u = H | φ | uDψ = Hu φ on R (3.1)where we set H = K ◦ π − . Hence let us consider this last problem and let us denote A = D ( R ) × D (Σ R ) . We take w = ( u, ψ ) ∈ A and we set J ( w ) = 12 Z R − u ∆ u + h Dφ, φ i − | u | | φ | ,G ( w ) = 12 Z R h | u | | φ | , J ε ( w ) = J ( w ) − εG ( w )with h = k ◦ π − . We are going to define the manifold of critical points of J . Let λ ∈ R + , y, ξ ∈ R , a ∈ Σ R with | a | = 1, it is well known that the functions¯ U λ,ξ ( y ) = √ λ / ( λ + | y − ξ | ) / are a family of positive solutions to − ∆ u = u in R and the spinors¯Φ λ,ξ,a ( x ) = 2 λ ( λ + | y − ξ | ) / ( λ − ( y − ξ )) · a Dφ = | φ | φ in Σ R . Using this fact, and the equality | ¯Φ λ,ξ,a | = | y | , one cancheck that the pairs ( U λ,ξ , Φ λ,ξ,a ) = √ U λ,ξ , √
32 ¯Φ λ,ξ,a ! ∈ A are critical points of J . Hence Z = (cid:8) W λ,ξ,a = ( U λ,ξ , Φ λ,ξ,a ) : λ ∈ R + , ξ ∈ R and a ∈ Σ R , | a | = 1 (cid:9) ⊂ A is a 7-dimensional manifold of critical points of J . Let us fix any a ∈ Σ R with | a | =1, in the sequel we will use the notation U = U , , Φ = Φ , ,a and W = ( U , Φ ).Now we will check assumption 2 in Theorem 3.1. We have h J ′′ ( W λ,ξ,a )[ w ] , w i = Z R − u ∆ u − u u | Φ λ,ξ,a | − u U λ,ξ h Φ λ,ξ,a , φ i + Z R h Dφ − | U λ,ξ | φ , φ i − U λ,ξ u h Φ λ,ξ,a , φ i . Therefore J ′′ is a compact perturbation of the identity, hence it is a Fredholm operatorof index zero for all W λ,ξ,a ∈ Z .Now it remains to check that T W λ,ξ,a Z = ker J ′′ ( W λ,ξ,a ) for every λ ∈ R + , ξ ∈ R and a ∈ Σ R with | a | = 1. Since J ′′ is invariant with respect to translations and dilationsit will be enough to prove T W Z = ker J ′′ ( W ) . We will need the following Remark.
Remark 3.2.
Let λ = and µ = . The map ( v, ψ ) ( ν, η ) = ( µ − v, λ − ψ ) is aone to one correspondence between solution to (1.4) on S and the equivalent rescaledsystem L g S ν = λ | η | νD g S η = µ ν η on S (3.2) which in turn it is equivalent to − ∆ u = λ | φ | uDφ = µ u φ on R (3.3) by means of the stereographic projection. Notice that (3.3) arises as the first variationof the functional ˜ J ( w ) = 12 Z R − λ − u ∆ u + µ − h Dφ, φ i − | φ | | u | and since ( U λ,ξ , Ψ λ,ξ,a ) are critical points of J , then ˜ W λ,ξ,a = (cid:18) µ − U λ,ξ , λ − Ψ λ,ξ,a (cid:19) are critical points of ˜ J . emma 3.3. We have T W Z = ker J ′′ ( W ) . Proof.
It is standard to check that T W Z ⊆ ker J ′′ ( W ), so it suffices to prove theinclusion ker J ′′ ( W ) ⊆ T W Z. Moreover, since dim( T W Z ) = 7 it is enough to showthat dim(ker J ′′ ( W )) ≤ J ′′ ( ˜ W )) ≤ . On the sphere S , the linearization of (3.3) at ˜ W reads as L g S ν = λ ν | Ψ | + 2 λ V h Ψ , η i D g S η = µ | V | η + 2 µ νV Ψ (3.4)where ( V , Ψ ) = (cid:18) µ − ( f − U λ,ξ ) ◦ π, λ − ( f ◦ π ) − F (Φ λ,ξ,a ◦ π ) (cid:19) = (1 , Ψ ). Noticethat Ψ satisfies D g S Ψ = 32 | Ψ | Ψ and | Ψ | = 1 , (3.5)so it is an eigenspinor of D g S with eigenvalue . We set η = P k ∈ Z f k Ψ k where Ψ k isa trivialization with Killing spinors and we write f = g + ih , where g and h arereal valued functions. We will first find f . Since f = h η, Ψ i , we have (see Lemma5.2 and Formula 5.16 in [12])∆ g S f = h ∆ g S η, Ψ i + h η, ∆ g S Ψ i + h D g S η, Ψ i . Notice now that, by (3.5) and the Lichnerowicz’s formula on the sphere D g S = − ∆ g S + 32 , we have − ∆ g S Ψ = Ψ and − ∆ g S η = D g S η − η = D g S (cid:18) η + 3 ν Ψ (cid:19) − η = 32 (cid:18) η + 3 ν Φ (cid:19) + 3 ∇ ν · Ψ + 92 ν Ψ − η = 34 η + 9 ν Ψ + 3 ∇ ν · Ψ . Therefore − ∆ g S f = 34 f + 9 ν + 3 h∇ ν · Ψ , Ψ i + 34 f − f − ν = 6 ν + 3 h∇ ν · Ψ , Ψ i . (3.6)8ince the last addend in the previous equality is purely imaginary, we take the real andimaginary part to have − ∆ g S g = 6 ν and − ∆ g S h = − i h∇ ν · Ψ , Ψ i . In particular, recalling that L g S = − ∆ g S + and the first equation in (3.4), we havethe system − ∆ g S ν = g − ∆ g S g = 6 ν (3.7)Hence, ∆ g S g = 9 g from which we deduce that g is the first eigenfunction of the Laplacian on the sphereand ν = g . So, the first equation in (3.4) becomes L g S g g f and recalling the definition of L g S , from the quality above we get f = g . Using this fact, the system (3.4) becomes ν = g D g S η = η + h η, Ψ i Ψ . (3.8)Hence we need to compute the dimension ofΛ = (cid:26) η ∈ H (Σ S ) : D g S η = 32 η + 32 h η, Ψ i Ψ (cid:27) . This computation has been carried out by Isobe in [12] for general dimensions of thesphere S m , so in our situation it suffices to take m = 3 in [12, Lemma 5.1] to getdim(Λ) = 7 as desired.Now we will focus on the reduced functional. For a fixed a ∈ Σ R , with | a | = 1, we set V λ,ξ = | U λ,ξ | | Φ λ,ξ,a | , so that V λ,ξ ( x ) = λ V , ( λ ( x − ξ )) and letΓ( λ, ξ ) = 12 Z R h ( x ) V λ,ξ ( x ) dx, for ( λ, ξ ) ∈ (0 , + ∞ ) × R . Then we have the following9 roposition 3.4. Γ is of class C on (0 , + ∞ ) × R and it can be extended to a C function at λ = 0 by Γ(0 , ξ ) = c h ( ξ ) , c = 12 Z R V , ( x ) dx Also, lim λ → ∇ ξ Γ( λ, ξ ) = c ∇ h ( ξ ) , uniformly on every compact of R . Moreover, for any compact set Σ ⊂ R , there existsa constant C = C Σ such that | ∂ λ Γ( λ, ξ ) − c λ ∆ h ( ξ ) | ≤ C Σ λ , for all λ > and all ξ ∈ Σ , being c = Z R | y | V , ( y ) dy . Proof.
We have by a change of variable thatΓ( λ, ξ ) = Z R h ( λx + ξ ) V , ( x ) dx Using the smoothness of h and the dominated convergence, we have thatlim λ → Γ( λ, ξ ) = c h ( ξ ) . The same reasoning applies to show that one has ∇ ξ Γ(0 , ξ ) = c ∇ h ( ξ ); ∇ ξ Γ(0 , ξ ) = c ∇ h ( ξ ) and ∇ λ Γ(0 , ξ ) = 0 . The last equality follows from the oddness of the integral, that is Z R x i V , ( x ) dx = 0 , i = 1 , , . We fix now a compact set Σ, then by Taylor expansion of y h ( y + ξ ), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ i h ( y + ξ ) − ∂ ξ i h ( ξ ) − X j =1 ∂ ξ i ξ j h ( ξ ) y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Σ | y | . Also, notice that since Z R y i y j V , ( y ) dy = 0 , if i = j, we have for our choice of c : c λ ∆ h ( ξ ) = Z R X i =1 ∂ ξ i h ( ξ ) + X j =1 ∂ ξ i ξ j h ( ξ ) λy j y i V , ( y ) dy. Therefore | ∂ λ Γ( λ, ξ ) − c λ ∆ h ( ξ ) | ≤ C Σ λ . roposition 3.5. Let k and h be functions as in the main Theorem 1.1. Then thereexists an open set Ω ⊂ (0 , + ∞ ) × R such that ∇ Γ = 0 on ∂ Ω and deg ( ∇ Γ , Ω ,
0) = X ξ ∈ crit[ h ]∆ h ( ξ ) < ( − m ( h,ξ ) + 1 . Proof.
Let s >
0, we consider the set B s = (cid:26) ( λ, ξ ) ∈ (0 , + ∞ ) × R ; | ( λ, ξ ) − ( s, | ≤ s − s (cid:27) . We will show that for s large enough, we can choose Ω = B s . First, we setcrit[ h ] = { ξ , ξ , · · · , 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 = (cid:8) ξ ∈ R ; | ξ | ≤ r (cid:9) . Since h is a Morse function (as well as k ), then by the non-degeneracy condition ( i ),there exist constants µ ∈ (0 , r ) and δ > | ∆ h ( ξ ) | > δ, ∀ ξ ∈ l [ i =1 B µ (cid:0) ξ i (cid:1) , where B µ (cid:0) ξ i (cid:1) denote as usual the balls of centers ξ i and radius µ . By using Proposition3.4, we have that for s sufficiently large and µ even smaller if necessary, ∂ λ Γ( λ, ξ ) = 0 , in ∂ B s ∩ (0 , µ ) × l [ i =1 B µ (cid:0) ξ i (cid:1)! . Hence, ∇ Γ = 0 in ∂ B s ∩ (0 , µ ) × l [ i =1 B µ (cid:0) ξ i (cid:1)! . Again, by Proposition 3.4, since Γ extends to a C function at λ = 0 and ∇ ξ Γ(0 , ξ ) = c ∇ h ( ξ ), we have that ∇ Γ = 0 in ∂ B s ∩ (0 , µ ) × A r \ l [ i =1 B µ (cid:0) ξ i (cid:1)! . Hence, ∇ Γ = 0 in ∂ B s ∩ ((0 , µ ) × A r ) . So it remains to study Γ on the component of ∂ B s outside (0 , µ ) × A r . So we considerthe Kelvin reflection τ : R \ { } → R \ { } , τ ( x ) = x | x | .
11e notice that τ ∗ ( g R ) = 1 | x | g R . Hence, for all F ∈ L ( R ), by putting y = τ ( x ), we have Z R h ( y ) | F ( y ) | dy = Z R h ( τ ( x )) | F ( τ ( x )) | f ( x ) dx = Z R h ( τ ( x )) | F ∗ ( x ) | dx, where F ∗ ( x ) = 1 | x | F (cid:18) x | x | (cid:19) . In particular, if we set ˜ λ = λλ + | ξ | , ˜ ξ = ξλ + | ξ | , we have that | V ∗ λ,ξ ( x ) | = | V ˜ λ, ˜ ξ ( x ) | . So we define ˜Γ = 12 Z R h ( τ ( x )) | V λ,ξ ( x ) | dx, 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 C functionup to the origin (0 , ∈ [0 , ∞ ) × R . Since ( λ, ξ ) (˜ λ, ˜ ξ ) is a diffeomorphism, then ∇ Γ( λ, ξ ) = 0 if and only if ˜Γ(˜ λ, ˜ ξ ) = 0. But by assumption, the south pole is not acritical point of h , hence 0 is not a critical point of h ( τ ( x )). Therefore, ∇ ˜Γ = 0 in aneighborhood of the origin and so ∇ Γ = 0 in a neighborhood of infinity. Finally, wehave that for r and s large enough, ∇ Γ = 0 on ∂ B s \ (cid:0) (0 , µ ) × A r (cid:1) . 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 concludeas in the classical cases (see for instance [2, 20]), since the critical points of Γ on Z aredegenerate: 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 J and J ′′ is Fredholmof index zero, therefore we have that there exists ε > z ∈ Z c ⊂ Z with Z c compact, there exists a unique w ( z ) ∈ T z Z ⊥ such that P J ′ ε ( z + w ( z )) = 012here P : A → T z Z ⊥ is the orthogonal projection. Now, to find a solution to ourproblem, it is enough to find a critical point for the function Φ ε : Z → R defined byΦ ε ( z ) = J ε ( z + w ( z )) . In order to do this, we will consider the set of the parameters a (cid:8) a ∈ Σ R : | a | = 1 (cid:9) ≃ S as a Lie group. Hence, we will consider the natural action of S on Z ≃ (0 , + ∞ ) × R × S ,being Z parameterized by ( λ, ξ, a ). Also, we notice that ( J ) | Z and G | Z are invariantunder this action: then we need to extend the action to the whole space D (Σ R ). Inorder to do this, we recall that the spinor bundle of R can be trivialized by Killingspinors that are either constant (parallel spinors) or spinors of the form x · φ with φ constant. So we fix an orthonormal basis of Σ R of the form { a , a , x · a , x · a } , where a , a are (distinct) constant spinors with | a | = | a | = 1. Hence, if φ ∈ D (Σ R ),there exist f , f , g , g such that φ ( x ) = ( f ( x ) + g ( x ) x ) · a + ( f ( x ) + g ( x ) x ) · a . Since a and a can be seen as elements in S , we can define the action for a general w ∈ S and φ ∈ D (Σ R ) by wφ = ( f ( x ) + g ( x ) x ) · wa + ( f ( x ) + g ( x ) x ) · wa . In this way, this last action extends the one previously defined on Z and in additionboth J and G are invariant under this action. Therefore, Φ ε descends to a C function˜Φ ε defined on the quotient Z/ S ≃ (0 , ∞ ) × R . 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 thatdeg( ˜Φ ′ ε , B s ,
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