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1 m k k (u∞, ψ∞) is a solution of (1.2), m sequences of points xn, ,xn M such that limn→∞ xn = x M, 1 · · · m ∈ ∈ for k = 1,...,m and m sequences of real numbers Rn, ,Rn converging to zero, such that: m · · · k 1 i) un = u∞ + vn + o(1) in H (M), k=1 Xm 1 k 2 ii) ψn = ψ∞ + φn + o(1) in H (ΣM), k=1 X m k k iii) E(un, ψn)= E(u∞, ψ∞)+ ER3 (U∞, Ψ∞)+ o(1), Xk=1 Thanks to a symmetry trick as in [27, 29], it is possible to show the existence of sign-changing solutions. This relies mainly on the fact that every non-trivial solution (U, Ψ) on R3 (or on the sphere by stereographic projection) satisfies the following lower bound, proved in [28]: 3 3 + 3 ER3 (U, Ψ) > Y(S , g0) := Y(S , g0)λ (S , g0) , (1.3) where Y(S3, g ) is the Yamabe invariant of the round 3-sphere, and 0 e + 3 3 3 1/3 λ (S , g0) := inf λ1(S , g)vol(S , g) g∈[g0] 3 is the Hijazi-B¨ar-Lott invariant [11, §8.5]. Here λ1(S , g) denotes the first positive Dirac eigenvalue on 3 (S , g) and [g0] is the conformal class of the round metric. In order for one to have a general existence theorem, one needs to understand these bubbles and classify them as is the case of the Yamabe problem in [9].

1.1. Main results. We are then interested in the following coupled system 2 c3∆u = ψ u , − | | on R3. (1.4) ( Dψ/ = u 2ψ | | with c = 6, that describes blow-up profiles stated above since c ∆= L . The first main result of 3 − 3 gR3 the paper deals with regularity and asymptotic behavior of distributional solution to (1.4). Theorem 1.2. Let (u, ψ) L6(R3) L3(ΣR3) be a distributional solution to (1.4), then ∈ × (u, ψ) C∞(R3) C∞(ΣR3) . ∈ × Moreover, the following complete asymptotic expansions hold. There exist z R, α N3, where at least one z = 0, such that for any M N there is a C < α ∈ ∈ 0 α 6 ∈ M ∞ such that α −1 x −2−M u(x) x zα 6 CM x for all x > 1 . (1.5) − | | x |α|1 | | | | |α|16M X | | There are ζ ΣR3, α N3, where at least one ζ = 0, such that for any M N there is a α ∈ ∈ 0 α 6 ∈ C < such that M ∞ α −2 x −3−M ψ(x) x (x) ζα 6 CM x for all x > 1 , (1.6) − | | U x |α|1 | | | | |α|16M X | | α α1 α2 α3 where x = x x x and α 1 = α1 + α2 + α3 , and the matrix (x) is defined as 1 2 3 | | | | | | | | U 0 i x σ (x)= − |x| · , x R3 0 , U i x σ 0 ∈ \ { } |x| · ! DIRAC-EINSTEIN BUBBLES 3

By the above theorem one immediately gets the following decay estimates.

Corollary 1.3. Under the assumptions of Theorem 1.2 there holds C C u(x) 6 , ψ(x) 6 , x R3 , | | 1+ x | | 1+ x 2 ∈ | | | | for some C > 0. Moreover, if u > 0, then there exists α > 0 such that

lim (1 + x )u(x)= α, |x|→∞ | | and if α = 0 then u = 0 and ψ = 0.

In fact the second assertion of the corollary follows from the strong maximum principle applied to the uK, the solution obtained after inversion (see Section 3.2). Define the spaces D1(R3) := u L6(R3) : u L2(R3) { ∈ ∇ ∈ } and D1/2(ΣR3) := ψ L3(ΣR3) : ξ 1/2 ψ(ξ) L2(R3) , { ∈ | | | |∈ } ψ denoting the Fourier transform of ψ. b We now consider solutions to (1.4), (u, ψ) D1(R3) D1/2(ΣR3), corresponding to critical points ∈ × ofb the functional at infinity

1 2 2 2 ER3 (u, ψ)= c3 u + Dψ,ψ/ u ψ dx . (1.7) 2 R3 |∇ | h i − | | | | Z  As recalled in the next section, the functional (1.7) and the equations (1.4) are conformally covariant, 3 so that we can equivalently consider them on the round sphere (S , g0). Then the functional reads

1 2 2 3 ES (u, ψ)= uLg0 u + D/ g0 ψ, ψ u ψ dvolg0 , 2 S3 h i − | | | | Z  while the equations are given by 2 Lg0 u = ψ u , | | on S3. (1.8) 2 (D/ g0 ψ = u ψ | | Definition 1.4. We say that a non-trivial solution (u, ψ) D1(R3) D1/2(ΣR3) is a ground state ∈ × solution if equality in (1.3) holds, that is

3 ES3 (u, ψ)= Y(S , g0) . (1.9)

Theorem 1.5. Let (u, ψ) H1(S3) H1/2(Σ S3) be a ground state solution to (1.4), and assume that ∈ × g0 e u > 0 . Then, up to a conformal diffeomorphism, u 1 and ψ is a ( 1 )-. More precisely, ≡ − 2 there exists a ( 1 )-Killing spinor Ψ on (S3, g ) and a conformal diffeomorphism f Conf(S3, g ) such − 2 0 ∈ 0 that 1 u = (det(df)) 6 and 1 3 ∗ ∗ ψ = (det(df)) βf g0,g0 (f Ψ),

∗ where βf g0,g0 is the spinor identification for conformal metrics. 4 W. BORRELLI AND A. MAALAOUI

Corollary 1.6. Let (u, ψ) H1(R3) H1/2(Σ R3) be a ground state solution to (1.4), with u > 0. ∈ × g0 Then there exists Φ ΣR3 and x R3, λ> 0 such that 0 ∈ 0 ∈

e 1 2λ 2 u(x)= , x R3 (1.10) λ2 + x x 2 ∈  | − 0|  and 3 2λ 2 x x ψ(x)= 1 γ − 0 Φ , x R3 , (1.11) λ2 + x x 2 − λ 0 ∈  | − 0|     where γ( ) denotes the Clifford multiplication. e · These ground state solutions are of extreme importance, since one can show that they are non- degenerate, and as a consequence obtain existence of solutions to the perturbed problem on the sphere (see [14] for more details). We remark that the classification of all solutions of (1.4) having u > 0 remains an open problem though. Ground state bubbles for conformally invariant Dirac equations have been recently classified in [8]. Such equations appear in the blowup analysis of critical Dirac equations on manifolds [19] and in the study of the spinorial Yamabe problem and related question, see e.g. [2, 13] and references therein. Recently, two-dimensional critical Dirac equations also attracted a considerable interest as effective models in mathematical physics, see e.g. [5, 6].

2. Some preliminaries

In this section we recall some notions useful in the sequel, for the convenience of the reader. We refer, in particular, to [25] for more details on spin structures and the Dirac operator.

2.1. and the Dirac operator. Let (M, g) be an oriented Riemannian manifold, and let PSO(M, g) be its frame bundle.

Definition 2.1. A spin structure on (M, g) is a pair (PSpin(M, g),σ), where PSpin(M, g) is a Spin(n)- and σ : P (M, g) P (M, g) is a 2-fold covering map, which is the non-trivial Spin → SO covering λ : Spin(n) SO(n) on each fiber. → In other words, the quotient of each fiber by 1, 1 Z is isomorphic to the frame bundle of M, {− }≃ 2 so that the following diagram commutes σ PSpin(M, g) PSO(M, g)

M A Riemannian manifold (M, g) endowed with a spin structure is called a spin manifold. n n In particular, the euclidean space (R , gRn ) and the round sphere (S , g0), with n > 2, admit a unique spin structure.

Definition 2.2. The complex ΣM M is the vector bundle associated to the Spin(n)- → principal bundle PSpin(M, g) via the complex spinor representation of Spin(n).

n The complex spinor bundle ΣM has rank N = 2[ 2 ]. It is endowed with a canonical spin connection ∇ (which is a lift of the Levi-Civita connection, denoted by the same symbol) and a Hermitian metric g which will be abbreviated as , if there is no confusion. h· ·i DIRAC-EINSTEIN BUBBLES 5

In particular, the spinor bundle of the euclidean space Rn is trivial, so that we can identify spinors with vector-valued functions ψ : Rn CN . → The Clifford multiplication γ : T M EndC(ΣM) verifies the Clifford relation → γ(X)γ(Y )+ γ(Y )γ(X)= 2g(X,Y )Id , − ΣM for any tangent vector fields X,Y Γ(T M), and is compatible with the bundle metric g. ∈ Locally, taking an oriented orthonormal tangent frame (ej) the Dirac operator is defined as D/ gψ := γ(e ) g ψ , ψ Γ(ΣM) . j ∇ej ∈ Given α C, a non-zero spinor field ψ Γ(ΣM) is called α-Killing if ∈ ∈ g ψ = αγ(X)ψ, X Γ(T M). ∇X ∀ ∈ For more information on Killing spinors we refer the reader to [11, Appendix A]. In the paper n [26] Killing spinors on the round sphere (S , g0) are explicitly computed in spherical coordinates. On (Sn, g ) α-Killing spinors only exist for α = 1/2, and via the stereographic projection the pull-back 0 ± of the 1 -Killing spinors have the form ± 2 n 2 2 Ψ(x)= (1 γ n (x)) Φ (2.1) 1+ x 2 ± R 0  | |  where 1 denotes the identity endomorphism of the spinor bundle Σ Rn and Φ CN , see e.g. [2]. gRn 0 ∈ 2.2. Conformal covariance. In this section we recall known formulas that relate the Dirac and conformal Laplace operators for conformally equivalent metrics, see e.g. [11, 18]. To this aim we explicitly label the various geometric objects with the metric g, e.g. Σ M, g, D/ g, g ∇ etc. Let f C∞(M) and consider the conformal metric g = e2f g. This induces an isometric isomor- ∈ f phism of spinor bundles β β : (Σ M, g) (Σ M, g ) . (2.2) ≡ g,gf g → gu f There holds g n−1 n+1 g D/ f β(e− 2 f ψ)= e− 2 f β(D/ ψ),

n−2 n+2 − 2 f − 2 f Lgf (e u)= e Lgu . The following quantities are conformally invariant. Setting

n−1 n−2 ϕ := β(e− 2 f ψ) , v := e− 2 f u (2.3) there holds

uL u dvol = vL v dvol , D/ gψ, ψ dvol = D/ gf ϕ, ϕ dvol g g gf gf h i g h i gf ZM ZM ZM ZM u 6 dvol = v 6 dvol , ψ 3 dvol = ϕ 3 dvol (2.4) | | g | | gf | | g | | gf ZM ZM ZM ZM u 2 ψ 2 dvol = v 2 ϕ 2 dvol . | | | | g | | | | gf ZM ZM Consequently the action (1.1) is conformally invariant, and hence also equation (1.2). 6 W. BORRELLI AND A. MAALAOUI

In particular, by a conformal change of metric (1.4) can be reinterpreted as an equation on the 3 3 3 3 round sphere (S , g0). Indeed, considering the stereographic projection π : S N R , where N S −1 ∗ 4 3 \ →1 3 1/2 3 ∈ 4 is the north pole, we have (π ) g = 2 2 gR3 with x R . Then if (u, ψ) D (R ) D (R , C ) 0 (1+|x| ) ∈ ∈ × 2f 4 is a weak solution to (1.2), consider (u, ϕ) defined in by (2.3) with e = (1+|x|2)2 . Such pair is in H1(S3) H1/2(Σ S3) by (2.4), and it is a weak solutions to (1.8). Notice that a priori (v, ϕ) is a weak × g0 solution only on S3 N . However, the removability of the singularity at the north pole N can be \ { } proved by a cut-off argument similar to the one contained in Section 3.2, see also [1, Theorem 5.1]. Moreover, by (2.4) the functional (1.1) is also conformally invariant, i.e.

ER3 (u, ψ)= ES3 (v, ϕ) .

3. Regularity and asymptotics

3.1. Regularity. We first need the following (see e.g. [7, 12] ) Liouville-type result.

Lemma 3.1. Let p,q > 1 and assume that (u, ψ) Lp(R3) Lq(ΣR3) satisfies ∈ × c ∆u = 0 , − 3 ( Dψ/ = 0 on R3 in the sense of distributions. Then u 0 and Dψ/ 0. ≡ ≡ We now use the above lemma to rewrite the first equation in (1.4) in integral form. The Green’s functions of c ∆ and D/ are given, respectively, by − 3 1 ı x y G(x y)= , Γ(x y)= α − − −4π x y − 4π · x y 3 | − | | − | where α = (α , α , α ) and the α are 4 4 Hermitian matrices satisfying the anticommutation 1 2 3 j × relations

αjαk + αkαj = 2δj,k , 1 6 j, k 6 3 . (3.1) We choose the α of a particular block-antidiagonal form, namely, let σ ,σ ,σ be 2 2 Hermitian j 1 2 3 × matrices satisfying analogous anticommutation relations as in (3.1), that is,

σjσk + σkσj = 2δj,k, 1 6 j, k 6 3.

Then the matrices

0 σj αj = , 1 6 j 6 3, σj 0 ! satisfy (3.1) and we shall work in the following with this choice. Such matrices exist and form a representation of the Clifford algebra of the euclidean space R3, so that

3 D/ := ıα = ı α ∂ − ·∇ − j xj Xj=1 Lemma 3.2. If (u, ψ) L6(R3) L3(ΣR3) solves (1.4) in the sense of distributions, then ∈ × u = G ( ψ 2u) , ψ =Γ ( u 2ψ) . (3.2) ∗ | | ∗ | | DIRAC-EINSTEIN BUBBLES 7

Proof. We note the Green functions have the following weak-Lebesgue integrability, namely, G L3,∞ ∈ and Γ L3/2,∞. Since ψ L3 and u L6, we have ψ 2u L6/5 and therefore by the weak Young ∈ ∈ ∈ | | ∈ inequality, the function u˜ := G ( ψ 2u) ∗ | | satisfies u˜ L6(R3) . ∈ Moreover, it is easy to see that c ∆˜u = ψ 2u in R3 − 3 | | in the sense of distributions. This implies that

∆(u u˜)=0 in R3 − − in the sense of distributions and therefore, by Lemma 3.1, u =u ˜, as claimed. With obvious modifica- tions the argument applies to ψ, thus concluding the proof. 

We remark that regularity of weak solutions to (1.4) does not follow by standard bootstrap argu- ments, as we are dealing with critical equations. In the following proposition we follow the strategy of [7], where the authors deal with critical Dirac equations.

Proposition 3.3. Any distributional solution (u, ψ) L6(R3) L3(ΣR3) to (1.4) is smooth. ∈ × Proof. Since the nonlinearity in (1.4) is smooth, it suffices to prove that (u, ψ) L∞(R3) L∞(ΣR3), ∈ × as standard elliptic regularity then applies to give smoothness. We only deal with u as the argument for the spinorial part ψ follows along the same lines, with straightforward modifications. Fix r> 6. We claim that u Lr(R3) for 6 6 r< . To this aim, we prove that there exists C > 0 ∈ ∞ such that for all M > 0 there holds

SM := sup vu dx : v r′ 6 1 , v 6/5 6 M 6 C , (3.3) R3 k k k k  Z  so that

6/5 sup vu dx : v r′ 6 1 , v L 6 C , R3 k k ∈  Z  r and then by density and duality, u L . ∈ Now fix M > 0 and let ε> 0 be a constant to be chosen later. The function

2

f := ψ ½ ε | | {δ6|ψ|6µ} is bounded and supported on a set of finite measure. Moreover, we have

ψ 2 f 3/2 = ψ 3 dx < ε k| | − εk3/2 | | Z{|ψ|<δ}∪{|ψ|>µ} ′ for suitable δ,µ > 0, as ψ L3. Define g := ψ 2 f and take v Lr L6/5, with v ′ 6 1 and ∈ ε | | − ε ∈ ∩ k kr v 6 M. k k6/5 By (3.2), there holds

vu dx = v(G (fεu)) dx + v(G (gεu)) dx . R3 R3 ∗ R3 ∗ Z Z Z 8 W. BORRELLI AND A. MAALAOUI

Using Fubini’s theorem we can rewrite the second integral on the right-hand side, as follows:

v(G (gεu)) dx = dx v(x) G(x y)(gε(y)u(y)) dy R3 ∗ R3 R3 − Z Z Z = dy dx(G(x y)v(x)) gε(y)u(y) R3 R3 − Z Z = (G v)gεu dy . R3 ∗ Z Applying again the same argument, and since u = G ( ψ 2)u, we can further rewrite the last integral ∗ | | to get

vu dx = v(G (fεu)) dx + hεu dx , (3.4) R3 R3 ∗ R3 Z Z Z where h := ψ 2G (g (G v)) . (3.5) ε | | ∗ ε ∗ 3r We now estimate the two terms in (3.4). Choosing s := 3+2r , the first integral in (3.4) can be bounded using the H¨older and Young inequalities

v(G (fεu)) dx 6 v r′ G (fεu) r 6 v r′ G 3,∞ fεu s R3 ∗ k k k ∗ k k k k k k k Z (3.6)

6 v r′ G 3,∞ fε 6s u 6 6 Cε , k k k k k k 6−s k k where the constant Cε depends on ε, r, ψ, u but not on M. We now turn to the second integral on the righ-hand side of (3.4). By (3.5), H¨older and Young inequalities give 2 2 h ′ 6 ψ G (g (G v)) ′ 6 ψ G g (G v) ′ k εkr k| | k3/2k ∗ ε ∗ ks k| | k3/2k k3,∞k ε ∗ kr 2 2 2 6 ψ G g G v ′ 6 ψ G g v ′ (3.7) k| | k3/2k k3,∞k εk3/2k ∗ ks k| | k3/2k k3,∞k εk3/2k kr 6 C′ g v ′ , k εk3/2k kr where the constant C′ > 0 depends on ψ. Similarly, we get h 6 ψ 2 G (g (G v)) 6 ψ 2 G g (G v) k εk6/5 k| | k3/2k ∗ ε ∗ k6 k| | k3/2k k3,∞k ε ∗ k6/5 6 ψ 2 G g (G v) 6 ψ 2 G 2 g v (3.8) k| | k3/2k k3,∞k εk3/2k ∗ k6 k| | k3/2k k3,∞k εk3/2k k6/5 6 C′ g v . k εk3/2k k6/5 The estimates (3.7) and (3.8) imply that

′ ′ hεu dx 6 C gε 3/2SM 6 C εSM , R3 k k Z by (3.3). Then taking ε = (2C′) −1 and combining the above estimate with (3.6) we get

′′ 1 vu dx 6 C + SM , R3 2 Z where C′′ is the constant C for ε = (2 C′)−1. Taking the supremum over all v we have ε 1 S 6 C′′ + S = S 6 2C′′ , M 2 M ⇒ M and the claim is proved. A similar argument works for ψ, so that we conclude that

(u, ψ) Lr(R3) Ls(ΣR3) , for all r > 6,s > 3 , ∈ × DIRAC-EINSTEIN BUBBLES 9 so that ψ 2u Lt(R3) for all t> 3/2. By (3.2), we have | | ∈ u(x)= G(x y) ψ(y) 2u(y)dy , R3 − | | Z and then writing G as the sum of a function in Lp and one in Lq, for some 1

3.2. Existence of asymptotics. The existence of an asymptotic expansion at infinity for solutions to (1.4) follows by the regularity results of the previous section, via the Kelvin transform (see [7, Section 4] for the spinorial case). Given a function u and a spinor ψ on R3, it is defined as

−1 2 uK(x) := x u(x/ x ) , | | | | x R3 0 , (3.9) −2 2 (ψK(x) := x (x)ψ(x/ x ) ∈ \ { } | | U | | where the unitary matrix (x) is defined as U 0 i x σ (x)= − |x| · , x R3 0 , U i x σ 0 ∈ \ { } |x| · ! where σ = (σ1,σ2,σ3) (see previous Section). Let (u, ψ) L6(R3) L3(ΣR3) be a distributional solution to (1.4). Then, as proved in the previous ∈ × section, (u, ψ) C∞(R3) C∞(ΣR3) and it solves the equations in the classical sense. The transformed ∈ × couple (uK, ψK) is smooth as well, and there holds

∆u = x −4( ∆u) , Dψ/ = x −2(Dψ/ ) . (3.10) K | | − K K | | K The computation for the scalar part is a classical one from potential theory [15, Section 1.8], while the spinorial one can be found in [7, Section 4]. Using (3.9) and (3.10) it is not hardy to see that,

uK( ∆uK)dx = u( ∆u)dx , Dψ/ K, ψK dx = Dψ,ψ/ dx R3 − R3 − R3 h i R3 h i Z Z Z Z 6 6 3 3 uK dx = u dx , ψK dx = ψ dx R3 | | R3 | | R3 | | R3 | | Z Z Z Z 2 2 2 2 uK ψK dx = u ψ dx . R3 | | | | R3 | | | | Z Z Moreover, by (3.10), one can easily verify that (u , ψ ) is smooth and solves (1.4) on R3 0 , and K K \ { } thus is a distributional solution on that set. We now prove that the equation is actually verified in distributional sense on the whole R3, that is, the origin is a removable singularity.

We only work out the argument for the scalar part uK, as a similar argument applies to ψK. We aim at showing that

2 ∞ R3 ( ∆f)uK dx = fuK ψK dx , f Cc ( ) . (3.11) R3 − R3 | | ∀ ∈ Z Z Let η C∞(R3) be (real-valued) a cutoff such that η 1 outside the unit ball and η 0 near the ∈ ≡ ≡ origin, and define η (x) := η (x/ε). Since η f C∞(R3 0 ), there holds ε ε ε ∈ c \ { } 2 ∞ R3 ∆(ηεf)uK dx = ηεfuK ψK dx , f Cc ( ) . (3.12) R3 − R3 | | ∀ ∈ Z Z 10 W. BORRELLI AND A. MAALAOUI

We have that η fu ψ 2 L1, η 6 η and η 1 almost everywhere, as ε 0. Then, ε K| K| ∈ | ε| k k∞ ε → → by dominated convergence, the right-hand side of (3.12) converges to the second integral in (3.11). Observe that ∆(η f)= η ( ∆f) 2 η f ( ∆η )f . (3.13) − ε ε − − ∇ ε ·∇ − − ε Arguing as above, one sees that the contribution to (3.12) of the first term on the right-hand side of (3.13) converges to the first integral in (3.11). The other to terms go to zero, as ε 0. Indeed, there holds →

ηε uKf dx = ηε uKf dx R3 ∇ ·∇ {|x|<ε} ∇ ·∇ Z Z 1/2 6 f ∞ ηε ∞ uK 2 x < ε . √ε , k k k∇ k k∇ k |{| | }| and

( ∆ηε)uKf dx = ( ∆ηε)uKf dx R3 − {|x|<ε} − Z Z 5/6 6 f ∞ ∆ηε ∞ uK 6 x < ε . √ε , k k k k k k |{| | }| Then by the regularity result of the previous section we conclude that (uK, ψK) is a smooth solution of 3 (1.4) on R . Formulas (1.5) and (1.6) follow by Taylor expanding uK and ψK at the origin and taking the inverse Kelvin transform. Observe that at least one of the coefficients zα in (1.5) must be non-zero, and the same holds for the ζα in (1.6), as by unique continuation principle a non-trivial solution to (1.4) cannot have derivative of arbitrary order vanishing at a point ( see [3] for the Laplacian and [24] concerning the Dirac operator).

4. Classification of ground states

Proof of Theorem 1.5. Let (u, ψ) H1(S3) H1/2(Σ S3) be a ground state solution to (1.8), with ∈ × g0 u > 0. By the strong maximum principle one infers that actually u> 0. Then consider the conformal change of metric 4 g := u4g , on S3, 9 0 and the corresponding isometry of spinor bundles β :Σ S3 Σ S3 g0 → g By formulas (2.2) 3 3 v := , ϕ := u−2β(ψ) , r2 2 solves equations (1.8) in the metric g, that is 2 Lgv = ϕ v , | | S3 3 on .  D/ ϕ = ϕ .  g 2 By the definition of the Yamabe invariant, we get 1/3 S3 6 2 2 Y( , [g0]) u dvolg0 6 uLg0 u dvolg0 = u ϕ dvolg0 , S3 S3 S3 | | Z  Z Z and since (u, ψ) is a ground state, using (1.9) we find 1/3 6 + S3 u dvolg0 6 λ ( , g0) . S3 Z  DIRAC-EINSTEIN BUBBLES 11

Observe that the volume of the metric g is

S3 8 6 vol( , g)= u dvolg0 27 S3 Z The Hijazi’s inequality [16, 17] gives

2/3 3 S3 9 S3 2/3 6 + S3 2 Y( , g0) 6 vol( , g) = u dvolg0 6 λ ( , g0) . 8 4 S3 Z  But since the last and the first term of these inequalities coincide, we have equality in the Hijazi inequality, and thus the spinor ϕ is Killing, with constant 1 , by the second equation in (1.8). − 2 Combining this with the second equation in (1.8), we conclude that ϕ is a twistor spinor, so that by [11, Prop. A.2.1] 9 3 ϕ = (D/ )2ϕ = R ϕ , 4 g 8 g and the scalar curvature of g is then Rg = Rg0 = 6. By a result of Obata [30] there exists an isometry

f : (S3, g) (S3, g ) , → 0 ∗ 4 4 such that f g0 = g = 9 u g0. Then we obtain 8 3 dvol ∗ = det(df)= u6 dvol = u = det(df)1/6 . f g0 27 g0 ⇒ 2   r Recall that the isometry f induces an isometry of spinor bundles F , so that the following diagram is commutative

F ∗ S3 ∗ S3 (Σf g0 ,f g0) (Σg0 , g0) ,

3 ∗ f 3 (S ,f g0) (S , g0) where the vertical arrows are the projections defining the spinor bundles. Then the spinor Ψ := F ϕ f −1 is 1 -Killing with respect to the round metric g , as this property ◦ ◦ − 2 0 is clearly preserved by isometries. Thus we can rewrite

ϕ = F −1 Ψ f f ∗Ψ , ◦ ◦ ≡ and 2 ψ = u2β−1(ϕ) = det(df)1/3β−1(f ∗Ψ) . 3 

Proof of Corollary 1.6. Formula (1.10) is the well-known one for standard bubbles for the classical Yamabe problem [9]. So we only deal with the spinorial part of the solution. As recalled in (2.1), the pullback on R3 of 1 -Killing spinors on the round sphere is given by − 2 3 2 2 Ψ(x)= 1 γ (x) Φ , x R3 , (4.1) 1+ x 2 − R3 0 ∈  | |  4  where Φ0 C . The other solutions to the second equation in (1.2) are obtained from those of the ∈ 3 form (4.1) by applying conformal transformations of the euclidean space (R , gR3 ). 12 W. BORRELLI AND A. MAALAOUI

We deal first with the composition of translation and a scaling. For fixed x R3,λ> 0, consider 0 ∈ the map f : R3 R3 defined by x0,λ → x x f (x) := − 0 , x R3 . x0,λ λ ∈ ∗ −2 3 3 Then f gR3 = λ gR3 . Recall that the frame bundle is trivial, i.e. P (R , gR3 ) = R SO(3), so x0,λ SO × that f induces a map f : R3 SO(3) R3 SO(3), with x0,λ x0,λ × → × f (x, v , v , v ) = (f (x), v , v , v ) , e x0,λ 1 2 3 x0,λ 1 2 3 3 3 acting as the identity on SO(3), which then lifts to a map on P (R , gR3 ) = R Spin(3) which e Spin × also acts as the identity on the second component. Since also the spinor bundle is trivial, that is ΣR3 = R3 C4, we finally get a map F : R3 C4 R3 C4, and the transformation on Ψ is given × x0,λ × → × by

−1 −2 ψ(x)=βλ gR3 ,gR3 Fx0,λΨ(fx0,λ(x)) 3 2 2λ −1 x x0 = β −2 F 1 γ − Φ , λ2 + x x 2 λ gR3 ,gR3 x0,λ − R3 λ 0  | − 0|     −2 where βλ gR3 ,gR3 is the conformal identification of spinors induced by the conformal change of metric. −1 −2 By the above discussion we can take βλ gR3 ,gR3 Fx0,λ to be the identity map, so that (1.11) holds. To conclude the proof we need to prove that the transformations on spinors of the form (1.11), induced by a rotation on R3, give another spinor of the same form suitably choosing new parameters. Let R SO(3) and consider the induced map F :ΣR3 ΣR3. Taking a spinor of the form (1.11) ∈ R → we get 3 2 −1 2λ −1 Rx x0 F (ψ(Rx)) = β −2 F 1 γ − Φ R λ2 + Rx x 2 λ gR3 ,gR3 R − R3 λ 0  0     | − | 3 (4.2) 2λ 2 x R−1x = 1 γ − 0 F −1(Φ ) , λ2 + x R−1x 2 − R3 λ R 0  | − 0|     where we have used the fact that, given v R3, ψ ΣR3, there holds F γ (v)ψ = γ (Rv)F (ψ). ∈ ∈ R R3 R3 R Then the claim follows, as (4.2) gives a spinor of the (1.11), with parameters R−1x R3, λ> 0 and 0 ∈ F −1(Φ ) ΣR3.  R 0 ∈ References

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(W. Borrelli) Scuola Normale Superiore, Centro De Giorgi, Piazza dei Cavalieri 3, I-56100 , Pisa, Italy. Email address: [email protected]

(A. Maalaoui) Department of mathematics and natural sciences, American University of Ras Al Khaimah, PO Box 10021, Ras Al Khaimah, UAE. Email address: [email protected]