Index of compact minimal submanifolds of the Berger spheres

INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES FRANCISCO TORRALBO AND FRANCISCO URBANO Abstract. The stability and the index of compact minimal submanifolds of the Berger spheres S2n+1 τ, 0 <τ≤1, are studied. Unlike the case of the standard sphere (τ=1), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if τ2≤1/2. Moreover, there are no stable compact minimal d-dimensional submanifolds of S2n+1 τwhen 1/(d+1)< τ2≤1 and the stable ones are classified for τ2=1/(d+1)when the submanifold is embedded. Finally, the compact orientable minimal surfaces of S3 τwith index one are classified for 1/3 ≤τ2≤1. Acknowledgements. The first author is supported by project PID2019.111531GA.I00 funded by MCIN/AEI/10.13039/501100011033 and the Programa Operativo FEDER Andalucía 2014-2020, grant no. E-FQM-309-UGR18. The second author is supported by Regional J. Andalucía grant no. P18-FR-4049. The authors would like to thank the referees for thorough revisions of the manuscript and inestimable suggestions to improve it. Data sharing not applicable to this article as no datasets were generated or analysed during the current study. 1. Introduction The second variation operator of minimal submanifolds of Riemannian manifolds (the Jacobi operator) carries information about stability properties of the submanifold when it is thought of as a critical point of the volume functional. Perhaps the starting point is the paper of J. Simons [19], where he characterized the compact minimal submanifolds of the sphere with the lowest index (number of independent infinitesimal deformations which do decrease the volume), proving that there are no stable ones. Later, H. B. Lawson and J. Simons [10], and Y. Onhita [15], studied similar problems when the ambient Riemannian manifold is a compact rank-one symmetric space. An important particular case in this setting is when the submanifold is a twosided hypersurface. In this case, the normal bundle is trivial and of rank one. Then the Jacobi operator, which acts on the sections of the normal bundle, becomes a Schrödinger operator acting on functions. In this case the study of the 2010 Mathematics Subject Classification. Primary 53C42; Secondary 53C40. Key words and phrases. minimal submanifolds, compact surfaces, stability, Berger spheres. The first author is supported by project pid2019.111531ga.i00 funded by mcin/ aei/10.13039/501100011033 and the Programa Operativo feder Andalucía 2014-2020, grant no. efqm-309-ugr18. The second author is supported by Regional J. Andalucía grant no. p18-fr-4049. 1 arXiv:2110.08027v2 [math.DG] 2 Mar 2022 2FRANCISCO TORRALBO AND FRANCISCO URBANO index of the Jacobi operator has been made for complete (not necessarily compact) minimal hypersurfaces, and the results of D. Fischer-Colbrie and R. Schoen, [7,8], have been fundamental in the growth of this theory. During the last forty years many papers have been devoted to study stability and index of minimal submanifolds in different ambient Riemannian manifolds. Among them, we only mention two of the last ones. First, the recent paper of O. Chodosh and D. Maximo [5], where they get a lower bound of the index of complete minimal surfaces in R3in terms of the genus, the number of ends and the multiplicity of the surfaces. Second, the paper of F. C. Marques and A. Neves, [12] and references therein, where given a compact manifold Mn, 3 ≤n≤7 with a generic metric, they use min-max theory to construct for each positive integer number ka two-sided embedded minimal hypersurface of Mwith index k. In the present work, we are interested in the index of compact minimal submanifolds of the Berger spheres. If S2n+1is the unit sphere of dimension 2n+1, the Berger spheres are a 1-parameter family {(S2n+1,h.iτ): 0 <τ≤1}, where h,i1is the standard metric gon S2n+1and the metric h,iτis given in (2.1). These Berger spheres are often called elliptic Berger spheres. This deformation of the standard metric gis a classical example of a collapse [4], in which the injectivity radius iτof h,iτis τπ and has uniformly bounded positive curvature. This deformation of the standard metric gcan be defined even for τ∈(1, ∞), but the geometry of these Berger spheres, called hyperbolic, is quite different from the elliptic ones. In the paper, although some minor results are also true for τ∈(1, ∞), we only consider Berger spheres with τ∈(0, 1]. The standard sphere S2n+1is a geodesic sphere of Cn+1of radius 1. In a similar way, the Berger sphere S2n+1 τ,τ∈(0, 1), is isometric to a geodesic sphere of the complex projective space CPn+1(4(1−τ2)) of radius arccos(τ) √1−τ2and the restrictions to S2n+1 τof the complex structures of Cn+1and CPn+1(4(1−τ2)) are the same (see Proposition 1). Along the paper, any minimal submanifold of S2n+1 τwill be considered as a submanifold of CPn+1(4(1−τ2)) and its behaviour with respect to the complex structure will play an important role in the proofs of the results. On the other hand, the Hopf fibration π:S2n+1→CPn(4)defines a Riemannian submersion π:S2n+1 τ→CPn(4)for any τ∈(0, 1]. The induced S1-bundles by the Hopf fibration over compact complex submanifolds of CPn(4)(see Example 1) define examples of compact minimal submanifolds of S2n+1 τwhich have a very good behaviour with respect to the second variation of the volume. These kind of minimal submanifolds will provide the more regular examples, some of which will be characterized by their index. In Section 3, we introduce some important examples of compact minimal submanifolds of S2n+1 τ, classifying in Proposition 5the totally geodesic ones. We compute the index and nullity of the above examples in Propositions 6,7,8and 9, and, among them, stable examples appear. The computation of the index and the nullity of these examples is not easy and, to do that, it has been crucial the paper INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 3 of S. Tanno, [23], in which he gives important insight into the spectrum of the Laplacian of S2n+1 τ. The main results about stability of compact minimal submanifolds of S2n+1 τare obtained in Theorems 1and 2and they can be summarized as follows: There are no stable immersed compact minimal d-dimensional submanifolds of S2n+1 τwhen 1 d+1<τ2≤1. If τ2=1 d+1for some positive integer d, then an embedded compact minimal submanifold Mdof S2n+1 τis stable if and only if d =2m+1and M2m+1is the induced S1-bundle by the Hopf fibration over an embedded compact complex submanifold N2mof CPn(4). If 0<τ2≤1 2m+2for some integer m ≥0, then any immersed submanifold M2m+1which is the induced S1-bundle by the Hopf fibration π:S2n+1 τ→CPn(4), over a compact complex submanifold N2mof CPn(4)is stable. In the proof of the two first above results, we use test normal sections on the second variation coming from parallel vector fields of the complex Euclidean space Cn+1and from holomorphic vector fields of the complex projective space CPn+1(4(1−τ2)). When M2nis a compact orientable minimal hypersurface of S2n+1 τ, then Mis unstable (Proposition 2). Hence, in this case, is natural to study the hypersurfaces of index 1. From Proposition 9, we know that the Clifford hypersurfaces have index 1 when 0 <τ2≤1 2n+1. In Theorem 5, we give an answer to this problem when Mis a surface, proving: If 1 3≤τ2≤1, then a compact orientable minimal surface of S3 τhas index one if and only if M is either the unique minimal sphere of S3 τor the Clifford surface in S3 1/√3. For the proof of this result, we embed isometrically CP2(4(1−τ2)) in the Euclidean space (see appendix), so we can see our surface isometrically immersed in a sphere S7(r), for certain radius r. In this case, we use as test functions on the second variation the restriction to the surface of the components of certain conformal transformations of S7(r). 2. The Berger spheres The Berger spheres S2n+1 τ, 0 <τ≤1, are the Riemannian manifolds (S2n+1,h·,·iτ) where h·,·iτis the Riemannian metric on the unit sphere S2n+1={z∈Cn+1:|z|2= 1}given by hv,wiτ=g(v,w)−(1−τ2)g(v,iz)g(w,iz),v,w∈TzS2n−1, (2.1) where gis the Euclidean metric in Cn+1and iis the imaginary unit. Notice that h·,·i1=g. In the sequel we will denote the Berger metric simply by h·,·i omitting the subindex τ. The isometry group of S2n+1 τ, 0 <τ<1, is the subgroup of the orthogonal group O(2n+2)given by {A∈O(2n+2):AJ=±JA}where J ∈ O(2n+2) 4FRANCISCO TORRALBO AND FRANCISCO URBANO is the matrix associated with the multiplication by iin Cn+1. The subgroup {A∈ O(2n+2):AJ=JA}is the real representation of the unitary group U(n+1) and hence the dimension of the isometry group of S2n+1 τis (n+1)2. In this context, it is clear that the group U(n+1)acts transitively on S2n+1 τand the isotropy subgroup at a point of S2n+1 τis isomorphic to U(n). Hence the Berger sphere is a homogeneous Riemannian manifold diffeomorphic to U(n+1)/U(n). On the other hand, the Hopf fibration π:S2n+1→CPn(4), where CPn(4) denotes the complex projective space endowed with the Fubini-Study metric of constant holomorphic sectional curvature 4, is a Riemannian submersion from S2n+1 τonto CPn(4), whose fibers are circles of length 2πτ. The unit vector field ξ on S2n+1 τ, defined by ξz=1 τiz,∀z∈S2n+1 τ, (2.2) spans the vertical line of the Hopf fibration and its uniparametric group is given by ζ(t,z) = cos(t τ)z+sin(t τ)iz,t∈R,z∈S2n+1 τ. (2.3) Hence the action of S1over S2n+1can be written as eit ·z=ζ(τt,z), 0 ≤t≤ 2π. Moreover, for any z∈S2n+1 τ, the tangent space TzS2n+1 τcan be orthogonally decomposed as hξzi⊕Hz, where dπz:Hz→Tπ(z)CPn(4)is a linear isometry. We note that the metrics h·,·i and gcoincide on H, and iv ∈ Hzfor any v∈ Hz. Along the paper we will see the Berger spheres S2n+1 τas geodesic spheres of the complex projective space CPn+1(4(1−τ2)). The following result describes it explicitly. Proposition 1.The map F :S2n+1 τ→CPn+1(4(1−τ2)) given by F(z1, . . . , zn+1) = hτ √1−τ2,z1, . . . , zn+1i, where [w] = π(w)for any w ∈S2n+3(1 √1−τ2)and π:S2n+3(1 √1−τ2)→CPn+1(4(1− τ2)is the canonical projection, is an isometric embedding of the Berger sphere S2n+1 τ into the complex projective space of constant holomorphic sectional curvature 4(1−τ2). Hence, the Berger sphere S2n+1 τcan be identified with the set S2n+1 τ≡n[(w0,w1, . . . , wn+1)] ∈CPn+1(4(1−τ2)):|w0|2=τ2 1−τ2o, which is a geodesic sphere of CPn+1(4(1−τ2)) with radius 1 √1−τ2arccos(τ)and center [( 1 √1−τ2, 0, . . . , 0)]. The Fubini-Study metric of CPn+1(4(1−τ2)) will be also denoted by h·,·i. Moreover, if J denotes the complex structure of CPn+1(4(1−τ2)), then JdFz(ξz)is a unit normal vector to S2n+1 τin CPn+1(4(1−τ2)), and dFz(iu) = JdFz(u)for any horizontal vector u ∈TzS2n+1 τ. Finally, the second fundamental form of the embedding F is given by ˆ σ(X,Y) = τhX,Yi− 1−τ2 τhX,ξihY,ξiJξ,∀X,Y∈X(S2n+1 τ). (2.4) Proof. Firstly, we can write F=π◦ˆ F, where ˆ F:S2n+1 τ→S2n+3(1 √1−τ2)is given by ˆ F(z) = τ √(1−τ2),z, for any z∈S2n+1 τ. INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 5 Then, for any u∈TzS2n+1 τwe have d ˆ Fz(u)=(0, u)and hence its horizontal component with respect to πis (0, u)H= (0, u)−(1−τ2)g(u,iz)iˆ F(z). From here, and taking into account (2.1), it follows that hdFz(u), dFz(v)i=g(dˆ Fz(u)H, d ˆ Fz(v)H) = hu,vi,∀u,v∈TzS2n+1 τ, (2.5) which means that Fis an isometric immersion. As clearly Fis injective, we obtain that Fis an isometric embedding. Also, if u∈TzS2n+1 τis a horizontal vector, that is, g(u,iz) = 0, we have, from the definition of the complex structure Jof CPn+1(4(1−τ2)), that JdFz(u) = dπˆ F(z)(i(0, u)H) = dπˆ F(z)((0, iu)) = dFz(iu). Moreover, using (2.2) we obtain that JdFz(ξz) = dπˆ F(z)(i(0, ξz)H) = dπˆ F(z)(p1−τ2,−τz). (2.6) So, for every u∈TzS2n+1 τ, we have hdFz(u),JdFz(ξz)i=g(dˆ Fz(u)H,(p1−τ2,−τz)) = 0, and hence z∈S2n+1 τ7→ JdFz(ξz)defines a unit normal vector field to F. Finally, it is clear that F(S2n+1 τ)is a geodesic sphere of CPn+1(4(1−τ2)) of radius 1 √1−τ2arccos(τ)and hence it is a pseudoumbilical hypersurface with two constant principal curvatures: τwith multiplicity 2nand 2τ2−1 τwith multiplicity 1, see [21]. As ξ≡F∗ξis an eigenvector for the eigenvalue 2τ2−1 τ, the second fundamental form ˆ σof the embedding Fis given by (2.4).  Taking into account Proposition 1and the expression for the curvature tensor in the complex projective space, we get that the curvature tensor R of S2n+1 τis given by R(X,Y,Z,W) = hY,ZihX,Wi−hX,ZihY,Wi + (1−τ2)hJY,ZihJX,Wi−hJX,ZihJY,Wi−2hJX,YihJZ,Wi + (1−τ2)hZ,ξihX,ξihY,Wi−hY,ξihX,Wi + (1−τ2)hW,ξihY,ξihX,Zi−hX,ξihY,Zi, (2.7) for every X,Y,Z,W∈X(S2n+1 τ). In particular, the sectional curvature of the plane Π⊂TpS2n+1 τgenerated by an orthonormal frame {v,w}is given by K(Π) = 1+ (1−τ2)3hv,Jwi2−|ξΠ|2, (2.8) where ()Πdenotes the tangent component to Π. Moreover, the Ricci curvature Ric of a unit vector vis Ric(v) = 2n+2(1−τ2)[1−(n+1)hv,ξi2], (2.9) and the scalar curvature is ρ=2n[2(n+1)−τ2]. It is interesting to note that Ric ≥2nτ2>0 . In the next result we relate some geometric objects of the metrics h·,·i and g(·,·)on the sphere S2n+1. 6FRANCISCO TORRALBO AND FRANCISCO URBANO Lemma 1.Let ∇and g∇be the Levi-Civita connections of the Berger metric h·,·i and the standard metric g(·,·)on S2n+1respectively. Then: (1) For any X,Y∈X(S2n+1) g∇XY=∇XY+1−τ2 τhY,ξiJ(X−hX,ξiξ) + hX,ξiJ(Y−hY,ξiξ), (2.10) where ξis the vector field defined in (2.2). (2) For any X ∈X(S2n+1 τ), ∇Xξ=τJ(X−hX,ξiξ), (2.11) So ξis a unit geodesic Killing field on S2n+1 τ. (3) The gradients and the Laplacians of a function f :S2n+1→Rwith respect to both metrics are related by g∇f=∇f−(1−τ2)(Lξf)ξ, g¯ ∆f=¯ ∆f−(1−τ2)(Lξ)2f,(2.12) where Lξis the Lie derivative with respect to the vector field ξ. (4) The spectrum of the Laplacian operator ¯ ∆of S2n+1 τis contained in the set nµk,p=λk+1−τ2 τ2(k−2p)2:k∈Z+,p∈Z, 0 ≤p≤ bk 2co, where {λk=k(2n+k),k∈Z+}is the spectrum of (S2n+1,g)and b·c stands for integer part. Moreover, g¯ ∆◦Lξ=Lξ◦g¯ ∆and the eigenspace V(λk)of the eigenvalue λkdecomposes as V(λk) = V(µk,0)⊕···⊕V(µk,[k 2]), where some of V(µk,p)may be trivial. Moreover, for each ϕ∈V(µk,p)we have (Lξ)2ϕ+1 τ2(k−2p)2ϕ=0. Remark 1.For n=0, the metrics h,iτand gare homothetic on S1. Hence, we have that V(λk) = V(µk,0)and so V(µk,p)are trivial for all p6=0. For n≥1, Tanno [22] shows that V(µk,0),V(µk,k/2)and V(µk,(k−2)/2)(keven), and V(µk,(k−1)/2)(kodd) are always non-trivial. Proof. Using the expression of the Levi-Civita connection given in the Koszul formula and the relation between the metrics h·,·i and g(·,·)given in (2.1), it is straightforward to get (2.10). Now using that g∇Xiz is the tangential component to the sphere of iX and (2.10) we easily get (2.11). If fis a smooth function on S2n+1then X(f) = h∇f,Xi=g(g∇f,X). Now, using (2.1) we get the relations between the gradients given in (2.12). Under this condition, the relations between the Laplacians given in (2.12) is a direct consequence of (2.10). Finally, item (4) was proved by S. Tanno [23].  3. Minimal submanifolds in the Berger spheres Let Φ:Md→S2n+1 τbe an immersion of a compact d-dimensional manifold Min the Berger sphere S2n+1 τ. Then, Φ∗TS2n+1 τ=TM ⊕T⊥M, where T⊥Mis the normal bundle of Φ. Let ∇be the Levi-Civita connection of S2n+1 τ,∇be INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 7 the Levi-Civita connection of the induced metric on M, and ∇⊥be the normal connection. Suppose now that Φis a minimal immersion, that is Φis a critical point for the volume functional. The second variation operator, known as the Jacobi operator of Φand which will be denote by L, is a strongly elliptic operator acting on the sections of the normal bundle, L:X⊥(M)→X⊥(M)given by L=∆⊥+A+R, where ∆⊥is the normal Laplacian ∆⊥= d ∑ i=1∇⊥ ei∇⊥ ei−∇⊥ ∇eiei, and A,Rare the endomorphisms of the normal bundle defined as follows A(η) = d ∑ i=1 σ(ei,Aηei),R(η) = d ∑ i=1 (R(η,ei)ei)⊥, where {e1, . . . , ed}is an orthonormal reference tangent to M,σis the second fundamental form of Φ,Aηis the Weingarten endomorphism associated to η∈ X⊥(M)and ⊥stands for normal component. Let Q:X⊥(M)→Rbe the quadratic form associated to the Jacobi operator L, defined by Q(η) = −ZMhLη,ηidv. We will denote by Ind(Φ)and Nul(Φ)(they will be also denoted by Ind(M)and Nul(M)) the index and the nullity of the quadratic form Q, which are respectively the number of negative eigenvalues of Land the multiplicity of zero as an eigenvalue of L. The immersion Φwill be called stable if Ind(Φ) = 0. Using (2.7) we obtain that Lη=∆⊥η+Aη+d−(1−τ2)|ξ>|2η−d(1−τ2)hη,ξiξ⊥−3(1−τ2)[J(Jη)>]⊥, (3.1) where >and ⊥denote respectively tangential and normal components to M. In the particular case Mis an orientable hypersurface, i.e. d=2n, and Nis a unit normal vector field to Φ, any normal section ηcan be written η=f N and so X⊥(M)≡ C∞(M)(η≡f). The operator Lbecomes in the Schrödinger operator L:C∞(M)→ C∞(M)given by L=∆+|σ|2+2n−2(1−τ2)(n+1)ν2−1, (3.2) where ∆is the Laplacian operator on Mand ν=hξ,Niis the so-called angle function. Considering the constant function 1 as a test function, we obtain that Q(1) = −ZM|σ|2dv−ZM2(n+1−τ2)dv+ZM2(n+1)(1−τ2)ν2dv ≤ −ZM2(n+1−τ2)dv+ZM2(n+1)(1−τ2)dv=−ZM2nτ2dv, where we have used that ν2≤1. Hence we get the following result: Proposition 2.Every orientable compact minimal hypersurface of S2n+1 τis unstable and the first eigenvalue of Lis λ1≤ −2nτ2. 8FRANCISCO TORRALBO AND FRANCISCO URBANO Now we define an important family of minimal submanifolds of S2n+1 τ. Example 1.(1)Induced S1-bundle by the Hopf fibration over a complex submanifold of CPn(4). Let Ψ:N2m→CPn(4)be a complex immersion of a complex manifold Nand M2m+1 0={(p,x)∈N×S2n+1 τ:Ψ(p) = π(x)}, where π:S2n+1 τ→CPn(4)is the Hopf fibration. We define π0:M0→N by π0(p,x) = pand Φ0:M0→S2n+1 τby Φ0(p,x) = x. Then, the following diagram is commutative: M2m+1 0 Φ0 −−−−→ S2n+1 τ π0  y  yπ N2m−−−−→ Ψ CPn(4) π◦Φ0=Ψ◦π0, (3.3) and M2m+1 0is a S1-bundle over N2n, where the action S1×M0→M0is given by z·(p,x)=(p,zx)for any z∈S1and (p,x)∈M0. Now, as ξis tangent to M0, i.e. ξ⊥=0, and Ψis a minimal immersion, it is easy to check that Φ0 is also a minimal immersion, which will be called the induced bundle by the Hopf fibration over the complex immersion Ψ:N2m→CPn(4). We observe that Φ0is an embedding if and only if Ψis an embedding. (2)S1-bundle compatible with the Hopf fibration over a complex submanifold of CPn(4). An immersion Φ:M2m+1→S2n+1 τis called a S1-bundle compatible with the Hopf fibration if there exists a complex immersion Ψ:N2n→CPn(4)such that Mis a S1-bundle over Nwhose associated projection ˆ π:M→Nsatisfies Ψ◦ˆ π=π◦Φ. Since in this case ξis also tangent to Mand Ψis minimal we get that Φis a minimal immersion. Moreover, if Φ0:M2m+1 0→S2n+1 τis the induced S1-bundle over N, then there exists an integer s≥1 and a s-sheeted covering map e π:M→M0such that π0◦e π=ˆ πand Φ=Φ0◦e π, where π0:M0→Nis the projection. In such case, the following diagrams are commutative M2m+1˜ π// Φ && ˆ π$$ M2m+1 0 Φ0// π0  S2n+1 τ π  N2m Ψ //CPn(4) (3.4) In fact, let ϕ:R×M→Mbe the uniparametric group of ˆ ξ, where ˆ ξis the restriction of ξto M. Then (Φ◦ϕ)(t,p) = ζ(t,Φ(p)) (see (2.3)). If λ∈Ris the minimum period of all the curves ϕp(see [3]), then ζ(t+λ,Φ(p)) = ζ(t,Φ(p)) for any p∈Mand so there exists an integer s≥1 such that λ=2πτs. INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 9 The action S1×M→Mis given by eit ·p=ϕ(sτt,p), 0 ≤t≤2π, so the subgroup Gs⊂S1given by Gs={1, ei2π s, . . . , ei2π(s−1) s}also acts on M and ˆ Φ:M/Gs→S2n+1 τgiven by ˆ Φ([p]) = Φ(p)is well-defined and it is also aS1-bundle over Ncompatible with the Hopf fibration. Let Φ0:M2m+1 0→S2n+1 τbe the induced S1-bundle over the complex submanifold Ψ:N→CPn(4). Then it is easy to see that F:M/Gs→M0 F([p]) = ( ˆ π(p),Φ(p)) is a S1-bundle isomorphism and so the assertion follows. The integer s≥1 will be called the order of the immersion Φ. In the next result we characterize two important families of minimal submanifolds of S2n+1 τwhich will play an important role along the paper. Proposition 3.Let Φ:Md→S2n+1 τbe a minimal immersion of a d-dimensional manifold M. (i) The normal component of the Killing vector field ξvanishes identically and the normal bundle of Φis invariant under the complex structure J of CPn+1(4(1−τ2)) if and only if, d =2m+1is odd and M is a S1-bundle ˆ π:M2m+1→N2mover a complex submanifold Ψ:N2m→CPn(4)compatible with the Hopf fibration. (ii) The tangent component of the Killing vector field ξvanishes identically if and only if, J(TM)⊂T⊥M. In particular Φ:Md→S2n+1 τ⊂CPn+1(4(1−τ2)) is a totally real immersion and d ≤n. In this case the metrics induced on M by the Berger metric h·,·i and the standard metric g are the same. In both cases, the immersion Φ:Md→S2n+1 τ0is also minimal for all τ0∈(0, 1]. Proof. (i) Suppose that ξ⊥=0 and the normal bundle of Φis invariant under the complex structure J. Then, as the normal bundle is of even dimension, we get that d=2m+1 for some integer m≥0. Also, as ξ⊥=0, the restriction ˆ ξof ξto Mis tangent to Mand so TM orthogonally decomposes as TM =D⊕hˆ ξi, where the subbundle Dis invariant under J. If ηis the 1-form on Mgiven by η(v) = hˆ ξ,vi, then using (2.10) and that ξ⊥= 0, it follows that the differential of ηis given by dη(X,Y) = 2τhJX,Yifor any vector fields X,Ytangent to M. So, if {e1, . . . , em,Je1, . . . , Jem}is an orthonormal basis of Dp,p∈M, we have that (η∧(dη)m)p(ˆ ξp,e1, . . . , em,Je1, . . . , Jem)6=0, and hence ηdefines a contact structure on M. We remark that, in particular, Mis orientable. Now, from a result of W. Boothby and H. Wang [3] and A. Morimoto [13], we have that Mis a S1-principal bundle over a complex manifold N2m. If ˆ π:M→Nis the projection, then we define Ψ:N→CPn(4)by Ψ(q) = π(Φ(p)), where p∈ˆ π−1(q). It is clear that Ψis well defined, i.e., it is independent of the point pin ˆ π−1(q), and so Ψdefines a complex immersion of 16 FRANCISCO TORRALBO AND FRANCISCO URBANO with associated eigensection f a +τs s−1(Lξf)ia if s6=1 or aand ia if s=1, where fis any eigenfunction of ∆associated to the (s−1)-th eigenvalue. Hence, Φsis unstable if τ2>1 2sand the proof finishes.  Proposition 7.Let Φ0:RP3→S5 τbe the minimal embedding given in Example 2.(2). Then: Ind(RP3) =        8if 1 2<τ2≤1, 6if 1 4<τ2≤1 2 0if τ2≤1 4. , Nul(RP3) =        16 if τ2=1or τ2=1 4, 12 if τ2=1 2, 10 otherwise. Moreover, if Φ:(S3, 2h·,·i√2τ)→S5 τis the minimal isometric immersion given in Example 2.(2), then Ind(S3) =        ≥14 if 1 4<τ2≤1, 8if 1 8<τ2≤1 4 0if τ2≤1 8. , Nul(S3) =        16 if τ2=1or τ2=1 4, 18 if τ2=1 8, ≥10 otherwise. Proof. Let consider the global orthonormal reference of the normal bundle of Φ given by {N,iN}where N(z,w)= ( ¯ w2,−√2¯ z¯ w,¯ z2). Then, given an arbitrary normal section η=f N +giN,f,g∈ C∞(S3), straightforward computations shows that the Jacobi operator L(see (3.5)) satisfies Lη=h∆f+42−1 τ2f+2(2−τ2) τLξgiN+h∆g+42−1 τ2g−2(2−τ2) τLξfiiN. Decomposing f=∑fk,pand g=∑gk,p, where fk,pand gk,pare eigenfunction of the Laplacian associated to the eigenvalue µk,p(see Lemma 1.(4)), and following a simular argument as in the proof of Proposition 6we can deduce that the eigenvalues of Ltake the form ρ±(k,p) = 1 2(1+k(2+k)) + 1 4τ2(k−2p±4)2−8−1 2(k−2p±1)2. for certain integers k≥0 and 0 ≤p≤ bk 2c. Moreover, the associated eigensections to ρ±(k,p)are fk,pN±2τ k−2pLξfk,piN if k−2p6=0, or fk,pNand fk,piN if k=2p. Hence, the multiplicity associated to ρ±(k,p)is dim V(µk,p)if k6=2p, or 2 ·dim V(µk,k/2)if k=2p. The result about the index and the nullity of Φfollows by a careful analysis of the sign of ρ±(k,p). In the case of the embedding Φ0, it is clear that Nand iN project on RP3and they give also a global orthonormal reference of the normal bundle of Φ0. Hence, the eigenvalues of its Jacobi operator are those of the Jacobi operator of Φwith k even. From here the result for the index and nullity of Φ0follows easily.  Proposition 8.Let Φ:Sd→S2n+1 τbe the totally geodesic embedding given in Proposition 5.(ii) . Then: INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 17 (i) Ind(Sd) =    2n+1+d(d−1) 2if τ2<1, 2n+1−d if τ2=1. (ii) Nul(Sd) =    (d+1)(2n+1−3d 2)if τ2<1, (d+1)(2n+1−d)if τ2=1. Proof. The result for τ2=1 is well-known, see [19], so we will assume that τ2<1. Let Sd,d≤n, be the totally geodesic sphere given in Proposition 5.(ii). We recall that ξ>=0 and Sdis totally real in CPn+1(4(1−τ2)). Then, its normal bundle can be orthogonally decomposed as T⊥Sd=J(TSd)⊕hξi⊕D. Firstly, it is clear that {aj: 1 ≤j≤2(n−d)}defined by aj= (0, . . . , 0, 2(d+1)+j 1, 0, . . . , 0)∈R2n+2 is a global orthonormal reference of the bundle D. Hence any section of the bundle Dcan be written as η=∑2(n−d) j=1fjajwith fj∈ C∞(Sd). Now, since g∇uaj=0 and ajis orthogonal to ξ, we get from (2.10) that ∇⊥ uaj= 0. As a consequence ∆⊥aj=0 and so, using (3.1), Lη= 2(n−d) ∑ j=1 (∆fj+d fj)aj. If Γ(D)is the space of sections of the subbundle D, then L(Γ(D)) ⊂Γ(D). Moreover, since the first eigenvalues of ∆are 0 and dwith multiplicities 1 and d+1 (see Lemma 1.(4)) we get that Ind(L|Γ(D)) = 2(n−d)and Nul(L|Γ(D)) = 2(n−d)(d+1). Secondly, given X∈X(Sd)then JX is normal and, if Dis the Levi-Civita connection of CPn+1(4(1−τ2)), we have that DuJX =JDuXfor any tangent vector u∈TSd. Taking normal components in this equation, using Gauss and Weingarten formulae and (2.4), we get that ∇⊥ uJX =J(∇uX) + Jˆ σ(u,X) = J(∇uX)−τhu,Xiξ. (3.13) Now, thanks to (3.13) we easily get that ∆⊥JX =J(∆X−τ2X)−2τ(div X)ξ. Using this equation and (3.1), we get that, for any f∈ C∞(Sd), L(JX +fξ) = J[∆X+ (d+3−4τ2)X+2τ∇f]+[∆f−2τdiv X]ξ, (3.14) where we have used that ξis a Jacobi field, i.e., Lξ=0. Now, we consider the identification Γ(JTSd⊕hξi)≡Ω1(Sd)⊕C∞(Sd)given by JX +fξ≡(α,f), where αis the 1-form on Sdgiven by α(Y) = hX,Yi, for any Y∈X(Sd)and Ωp(Sd)denotes the space of p-forms on Sd. If ˆ ∆represents the Hodge Laplacian acting on forms of Sd, then the vector field ∆X−Ric(X) = ∆X−(d−1)X 18 FRANCISCO TORRALBO AND FRANCISCO URBANO corresponds with the 1-form ˆ ∆α, and so the Jacobi operator Lacting on the sections of the bundle JTSd⊕ hξicomputed in (3.14) becomes in an operator L:Ω1(Sd)⊕C∞(Sd)→Ω1(Sd)⊕C∞(Sd)given by L(α,f) = ( ˆ ∆α+2(d+1−2τ2)α+2τdf,∆f−2τδα), where d is the differential and δis the codifferential operator on Sdgiven by δα =∑d i=1(∇eiα)(ei)being {e1, . . . , ed}a local orthonormal reference on Sd. It is clear that Ind(L)and Nul(L)are the index and the nullity of the Jacobi operator Lacting on Γ(JTSd⊕hξi). Now, as the first Betti number of Sdis 0, the Hodge decomposition theorem says that Ω1(Sd) = dC∞(Sd)⊕δΩ2(Sd), which allows to write in a unique way any 1-form αas α=dg+δω, with g∈ C∞(Sd)and ω∈Ω2(Sd). Now we can split the operator Las L=L1⊕L2where L1: dC∞(Sd)⊕C∞(Sd)→dC∞(Sd)⊕C∞(Sd) L1(dg,f) = d(∆g+2(d+1−2τ2)g+2τf)),∆(f−2τg),(3.15) and L2:δΩ2(Sd)→δΩ2(Sd) L2(δω, 0) = δ(ˆ ∆ω+2(d+1−2τ2)ω)), 0.(3.16) It is clear that Ind(L) = Ind(L1) + Ind(L2)and Nul(L) = Nul(L1) + Nul(L2). We now compute the index and nullity of both operators. Let ρ≤0 be an eigenvalue of L1with associated eigenfunction (dg,f). Then, from the equality L1(dg,f) + ρ(dg,f) = 0 and the expression of L1in (3.15) we deduce that 0=d[∆g+ρ+2(d+1−2τ2)g+2τf], 0=∆f−2τ∆g+ρf.(3.17) Notice that if dg=0 then from the first equation fis constant and so, from the second one, ρ=0. Hence, (0, a),a∈R, is in the nullity of L1. If dg6=0, we decompose f=∑k≥0fkand g=∑k≥0gkin eigenfunctions of the Laplacian ∆of Sd, so ∆fk+λkfk=0 and ∆gk+λkgk=0 for all k≥0 where λk=k(d+k−1)are the eigenvalues of the Laplacian on Sd. Notice that f0and g0are constant functions. Hence, (3.17) now reads 0= [ρ+2(d+1−2τ2)−λk]dgk+2τdfk, 0=2τλkgk+ (ρ−λk)fk,k≥0. (3.18) We now substitute fkfrom the second equation in the first one to obtain hρ+2(d+1−2τ2)−λk+4τ2λk λk−ρidgk=0, k≥1. (3.19) And so, since we have assumed that dg6=0, the eigenvalue ρ≤0 takes the form ρ=λk−(d+1−2τ2)−q(d+1−2τ2)2+4τ2λk. for some k≥1. Moreover, its associated eigenfunction is dg,2τλk λk−ρkg, where gis an eigenfunction of the Laplacian associated to λk. INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 19 Now, ρ<0 if and only if λk<2(d+1)which only occurs if k=1. As a consequence Ind(L1) = d+1. If ρ=0 then we get from (3.18) fk=2τgk,[2(d+1)−λk]dgk=0, k≥1. so, since dg6=0, we get that g=g2is an eigenfunction of the Laplacian associated to λ2=2(d+1)and f=2τg. Hence, we obtain Nul(L1) = 1 2(d+2)(d+1)which is exactly the multiplicity of the eigenvalue λ2plus 1 since we have previously analysed that (0, a),a∈R, are also in the nullity of L1and corresponds to the case dg=0. Finally, we compute the index and nullity of L2. From [9], it follows that only the eigenvalue 2(d−1)of ˆ ∆acting on Ω2(Sd)provides a non-positive eigenvalue of L2, which is −4(1−τ2). As its multiplicity is 1 2d(d+1), we obtain that Ind(L2) = 1 2d(d+1)and Nul(L2) = 0. From the previous analysis of the index and nullity of L|Γ(D),L1and L2we get the result.  Proposition 9.Let Φ:S2m1+1(r1)×S2m2+1(r2)→S2n+1 τ, with m1+m2+1=n, r2 1=2m1+1 2n, r2 2=2m2+1 2n, be a Clifford hypersurface described in Example 3. Then (i) Ind(Φ) =    1if τ2≤1 2n+1, 2n+3if 1 2n+1<τ2≤1. (ii) Nul(Φ) =    2(m1+1)(m2+1)if τ26=1 2n+1, 2(m1+1)(m2+1) + 2(n+1)if τ2=1 2n+1. Proof. Our first goal is to show a similar relation to (2.12) between the Laplacian ∆ of the induced metric by Φand the Laplacian g∆of the standard product metric of S2m1+1(r1)×S2m2+1(r2)so we can write down an expression for the eigenvalues of ∆following a similar argument as in Lemma 1.(4). We will denote by ∇,g∇ the Levi-Civita connections of the Clifford hypersurfaces induced by the Berger h·,·i and the standard product metric grespectively. Firstly, notice that, for any tangent vector field Xand any smooth function f, g(g∇f,X) = X(f) = h∇f,Xi. Hence, using (2.1), we easily get ∇f=g∇f+1−τ2 τ2(Lξf)ξ. Secondly, by Example 3, we can take orthonormal references on the Clifford hypersurface {e1, . . . , en−1,ξ}(respectively {e1, . . . , en−1,τξ}) with respect to the metric h·,·i (respectively with respect to the metric g). Now, using (2.10) and the above relation between the gradients we easily deduce ∆f=g∆f+ (1−τ2)(Lξ)2f. (3.20) Now, it is well-known that each eigenvalue λof g∆is of the form λ=λk1+λk2, where λkj=1 r2 j kj(2mj+kj)is the kj-th eigenvalue of the Laplacian of (S2mj+1(rj),g). The associated eigenspace V(λk1+λk2)to λk1+λk2is generated by f1(p)·f2(q) for eigenfunctions fjassociated to the eigenvalue λkj. More precisely, fjis the restriction to S2mj+1(rj)of a homogeneous harmonic polynomial of degree kj. 20 FRANCISCO TORRALBO AND FRANCISCO URBANO Hence, f1·f2is the restriction to S2m1+1(r2)×S2m2+1(r2)of a homogeneous harmonic polynomial of degree k1+k2. As a consequence, so it is any eigenfunction h∈V(λk1+λk2). Since Lξand g∆commutes and Lξis skew-symmetric we get that (Lξ)2is a self-adjoint linear transformation of V(λk1+λk2)with positive real eigenvalues. Then, we can decompose h=∑php, with hp∈V(λk1+λk2)eigenfunctions of (Lξ)2. But, thanks to the previous paragraph and following [23, Lemma 3.1], (Lξ)2hp+1 τ2(k1+k2−2p)2hp=0, for certain 0 ≤p≤ b1 2(k1+k2)c. (3.21) Therefore, using (3.20), each eigenvalue µk1,k2,pof ∆takes the form µk1,k2,p=2(m1+m2+1)k12m1+k1 2m1+1+k22m2+k2 2m2+1+1−τ2 τ2(k1+k2−2p)2, (3.22) for some integers kj≥0 and 0 ≤p≤ b1 2(k1+k2)c, where we have taken into account the relation between the radii r1,r2and m1,m2. Finally, from (3.2) and Example 3, the Jacobi operator of the Clifford hypersurface is given by L=∆+4(m1+m2+1) = ∆+4n. Then, thanks to (3.22) the non-positive eigenvalues of Lare: (i) −4n<0 with multiplicity 1 (corresponds to µ0,0,0). (ii) 1 τ2−(2n+1)<0 with multiplicity 2(n+1)when τ2>1 2n+1(corresponds to µ1,0,0 and µ0,1,0). (iii) 0 with multiplicity 2(n+1)when τ2=1 2n+1(corresponds to µ1,0,0 and µ0,1,0) (iv) 0 with multiplicity 2(m1+1)(m2+1)(corresponds to µ1,1,1). where the multiplicities are obtained from the fact that the multiplicities of the first two eigenvalues of (S2m+1(r),g)are given by 1 and 2m+2 respectively and a careful analysis of the condition (3.21) in the case of µ1,1,1. So the proof follows.  4. Stable compact minimal submanifolds of S2n+1 τ Propositions 6and 7compute the index of the S1-bundles compatible with the Hopf fibration over the totally geodesic CPm(4)⊂CPn(4)and the Veronese surface CP1(2)⊂CP2(4), showing that they are stable when τ2≤1 s(d+1), where dand sare respectively the dimension and the order of the submanifold. In the next result we generalize this property to any minimal submanifold of S2n+1 τwhich is a S1-bundle compatible with the Hopf fibration over a complex submanifold of CPn(4). Theorem 1.Let Φ:M2m+1→S2n+1 τbe a S1-bundle over a complex immersion Ψ: N2m→CPn(4)compatible with the Hopf fibration and order s. If 0<τ2≤1 s(2m+2), then Φis stable. Remark 3.In the case Φis the induced S1-bundle then s=1 and so it is stable if τ2≤1 2m+2. Proof. Following the proof of Proposition 4,TM =D ⊕ hξi,Dand T⊥Mare invariant under the complex structure Jof CPn+1(4(1−τ2)) and the quadratic INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 21 form associated to the Jacobi operator is Q(η) = ZM|∇⊥η|2−hAη,ηi−(2m+τ2)|η|2dv. (4.1) We consider, for any X∈Γ(D), the operator DX:X⊥(M)→X⊥(M)given by DXη=∇⊥ JXη−J∇⊥ Xη. Let {e1, . . . , em,Je1, . . . , Jem,ξ}be a local tangent orthonormal reference to M. Then m ∑ i=1|Deiη|2= m ∑ i=1{|∇⊥ Jeiη|2+|J∇⊥ eiη|2−2h∇⊥ Jeiη,J∇⊥ eiηi} =|∇⊥η|2−|∇⊥ ξη|2−2 m ∑ i=1h∇⊥ Jeiη,J∇⊥ eiηi. Now, let consider, for any η∈X⊥(M), the 1-form on Mdefined by α(u) = h∇⊥ J(u−hu,ξiξ)η,Jηi. Then, taking into account (3.6) and (3.7), its codifferential is δα = m ∑ i=1{ei(α(ei)) −α(∇eiei) + (Jei)(α(Jei)) −α(∇JeiJei) + ξ(α(ξ)) −α(∇ξξ)} = m ∑ i=1{h∇⊥ ei∇⊥ Jeiη,Jηi−h∇⊥ Jei∇⊥ eiη,Jηi+2h∇⊥ Jeiη,J∇⊥ eiηi−α(∇eiei)−α(∇JeiJei)} = m ∑ i=1{R⊥(ei,Jei,η,Jη) + h∇⊥ [ei,Jei]η,Jηi+2h∇⊥ Jeiη,J∇⊥ eiηi−α(∇eiei)−α(∇JeiJei)}, where R⊥is the normal curvature of the immersion Φ. Now, again from (3.7) we easily get that m ∑ i=1h∇⊥ [ei,Jei]η,Jηi−α(∇eiei)−α(∇JeiJei) = −2τmh∇⊥ ξη,Jηi. Moreover, from Ricci equation and using (2.7) and (3.7) m ∑ i=1 R⊥(ei,Jei,η,Jη) = m ∑ i=1 R(ei,Jei,η,Jη) + h[Aη,AJη]ei,Jeii =−2m(1−τ2)|η|2−hAη,ηi. As a consequence of all the above computations we obtain that m ∑ i=1|Deiη|2=|∇⊥η|2−hAη,ηi−|∇⊥ ξη|2−2m(1−τ2)|η|2−2τmh∇⊥ ξη,Jηi−δα. Using the last expression in (4.1), the quadratic form Qover any normal vector field ηcan be written as Q(η) = ZM m ∑ i=1|Deiη|2+|∇⊥ ξη|2−τ2(2m+1)|η|2+2τmh∇⊥ ξη,Jηi!dv ≥ZM|∇⊥ ξη|2−τ2(2m+1)|η|2+2τmh∇⊥ ξη,Jηidv, (4.2) where we have used that |Deiη|2≥0. 22 FRANCISCO TORRALBO AND FRANCISCO URBANO Now, to get a lower bound of (4.2) we change the Berger metric h,iin S2n+1 τby the standard one g. It is clear that the volumen forms dvof (M,Φ∗h·,·i)and dvg of (M,Φ∗g)are related by dv=τdvg. Also, thanks to (2.10) and (3.6), we get ∇⊥ ξη=1 τg∇⊥ Vη−(1−τ2)Jη,Ag ηV=0, (4.3) where Agis the shape operator of Φ:M→S2n+1, and Vis the tangent vector field on Mdefined by Vp=ip,∀p∈M. Hence, by the previous formula and (2.2) the inequality in (4.2) becomes Q(η)≥1 τZMh|g∇⊥ Vη|2 g+ (τ2(2m+2)−2)g(g∇⊥ Vη,Jη) + (1−τ2(2m+2))|η|2 gidvg. Claim:The operator G :X⊥(M)→X⊥(M)given by Gη=g∇⊥ Vg∇⊥ Vη−g∇⊥ g∇VVη=g∇⊥ Vg∇⊥ Vη, is a self-adjoint operator with spectrum {k2 s2:k∈Z,k≥0}. Now, we can write η=∑k≥0ηk, with ηkan eigensection of Gassociated to the eigenvalue k2 s2, i.e., Gηk+k2 s2ηk=0. In particular η0satisfies g∇⊥ Vη0=0. Hence from the last expression for the quadratic form Qand as G(g∇⊥ Vηk) + k2 s2g∇⊥ Vηk= 0 we get Q(η)≥1 τZM(1−τ2(2m+2))|η0|2 gdvg +1 τ∑ k≥1ZMh|g∇⊥ Vηk|2 g+τ2(2m+2)−2g(g∇⊥ Vηk,Jηk) + 1−τ2(2m+2)|ηk|2 gidvg ≥1 τ∑ k≥1ZMk2 s2+1−τ2(2m+2)|ηk|2 g+τ2(2m+2)−2g(g∇⊥ Vηk,Jηk)dvg, (4.4) where we have used that 1 −τ2(2m+2)≥0. Now, for any k≥1 we have that 0≤1 2ZM|g∇⊥ Vηk−k sJηk|2 gdvg=k2 s2ZM|ηk|2 gdvg−k sZMg(g∇⊥ Vηk,Jηk)dvg. As τ2(2m+2)−2<0, using the above inequality in (4.4) we obtain Q(η)≥1 τ∑ k≥1k s−1k s−1+2τ2(m+1)ZM|ηk|2 g ≥1 τ s−1 ∑ k=1k s−1k s−1+2τ2(m+1)ZM|ηk|2 g≥0, because for 1 ≤k≤s−1 we have that (k s−1)2τ2(m+1)≥(k s−1)1 s. Hence Φis stable. Proof of the claim: Firstly, since divgV=0, 0=ZMdivg(g(g∇⊥ Vη,ζ)V)dvg=ZMhg(g∇⊥ Vg∇⊥ Vη,ζ) + g(g∇⊥ Vη,g∇⊥ Vζ)idvg, =ZMg(Gη,ζ) + g(g∇⊥ Vη,g∇⊥ Vζ)dvg, and so RMg(Gη,ζ)dvg=RMg(η,Gζ)dvg, which proves that Gis self-adjoint. INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 23 In fact, Gis the vertical normal Laplacian with respect to the fibration ˆ π: M2m+1→N2m, because if p∈Mand Fpis the fiber through p, then (Gη)|Fp=∆⊥ p(ηx|Fp), where ∆⊥ pis the normal Laplacian of the totally geodesic immersion Fp⊂M2m+1→ S2n+1. As the order of Mis s, all these immersions are congruent to the immersion : ν:S1→S2n+1given by z7→ (zs, 0 . . . , 0), and so the eigenvalues of the normal Laplacian ∆⊥ pwill be those of the normal Laplacian of ν. If ∆⊥ νis the normal Laplacian of the immersion ν, and {e1, . . . , en,Je1, . . . , Jen}is a global orthonormal reference of the normal bundle, then if ν=∑n i=1{fiei+giJei}is a normal section, then ∆⊥ νν= n ∑ i=1{(∆fi)ei+ (∆gi)Jei}. Hence the eigenvalues of ∆⊥ νwill be those of the Laplacian of S1endowed with the induced metric by ν, i.e., the set {k2 s2:k∈Z,k≥0}. Since any compact minimal submanifold of the sphere (S2n+1,g)is unstable, any compact minimal submanifold of S2n+1 τis also unstable for τnext to 1. In the following result we obtain the first value of τfor which the instability disappears. This value is τ2=1 d+1, where dis the dimension of the submanifold and for this τ we classify the stable compact minimal embedded submanifolds. This value of τ can be also interpreted as the first value of τfor which the minimal submanifold Mdof S2n+1 τis also minimal in CPn+1(4(1−τ2)). In fact, from (2.4), the mean curvature Hof Mdin CPn+1(4(1−τ2)) is given by H=1 dτd−1−τ2 τ|ξ>|2Jξ. Hence H=0 if and only if τ2d= (1−τ2)|ξ>|2. As |ξ>|2≤1, if H=0 then τ2≤1 d+1. It is clear that if τ2=1 d+1and ξ⊥=0 then H=0. Theorem 2.Let Φ:Md→S2n+1 τbe a minimal immersion of a compact d-manifold M in the Berger sphere S2n+1 τ. If 1 d+1≤τ2≤1and Φis stable then τ2=1 d+1, d=2m+1and M is a S1-bundle ˆ π:M2m+1→N2mover a complex submanifold Ψ:N2m→CPn(4)compatible with the Hopf fibration. As a consequence of Theorems 1,2, and Proposition 3.(i) we get the following result: Corollary 3.Let Φ:Md→S2n+1 τbe a minimal embedding of a compact d-manifold M in the Berger sphere S2n+1 τ. If 1 d+1≤τ2≤1, then Φis stable if and only if τ2=1 d+1, d=2m+1and M is the induced S1-bundle by the Hopf fibration over a complex embedded submanifold Ψ:N2m→CPn(4). Remark 4.The authors believe that the minimal examples described in Proposition 3.(i) are unstable when τ2=1 2m+2and the order s≥2, as Example 2.(1) for m=0 and Example 2.(2) corroborate (see Propositions 6and 7). 24 FRANCISCO TORRALBO AND FRANCISCO URBANO Proof. Claim 1:If 1 d+1≤τ2≤1and Φis stable then τ2=1 d+1and ξ⊥=0. Firstly, we are going to define certain vector fields on S2n+1 τ, whose normal components will be test sections for the quadratic form Q. To do that, for each a∈Cn+1let Xa∈X(S2n+1) = X(S2n+1 τ)be the vector field given by (Xa)p=a−g(a,p)p, for all p∈S2n+1. If gDis the Levi-Civita connection of the Euclidean metric gin Cn+1, then we get for any u∈TpS2n+1, 0=gDua=gDu(Xa+g(a,p)p) = g∇uXa+g(a,p)u, where we recall that g∇is the Levi-Civita connection of (S2n+1,g)(see Lemma 1). So g∇uXa=−fau, where fa:S2n+1→Ris given by fa(p) = g(a,p). Now, using the relation between the Levi-Civita connections ∇of S2n+1 τand g∇given in (2.10), we get the following behaviour of the vector field Xawith respect to ∇ h∇uXa,vi=−fahu,vi− 1−τ2 τ(hXa,ξihJu,vi+hu,ξihJXa,vi). (4.5) for any u,v∈TpS2n+1 τ. In this situation, given the immersion Φ:Md→S2n+1 τ, we decompose Xa∈ X(S2n+1 τ)in its tangent a normal component to Φ, i.e. Xa=X> a+X⊥ a. Our goal is to compute the Jacobi operator (3.1) acting on X⊥ a. From (4.5) we deduce that, for any u∈TpM, ∇uX> a=−fau+AX⊥ au−1−τ2 τhXa,ξi(Ju)>+hu,ξi(JXa)>, (4.6) ∇⊥ uX⊥ a=−σ(u,X> a)−1−τ2 τhXa,ξi(Ju)⊥+hu,ξi(JXa)⊥. (4.7) We will make the computation of LX⊥ aat a point p∈Mand we will use a orthonormal tangent reference {e1, . . . , ed}to Msatisfying (∇ekej)p=0. Hence, taking normal derivatives with respect to eiin (4.7) and using Codazzi equation we deduce ∇⊥ ei∇⊥ eiX⊥ a= (R(X> a,ei)ei)⊥−σ(ei,∇eiX> a)−1−τ2 τ∇⊥ eihXa,ξi(Jei)⊥+hei,ξi(JXa)⊥. But, using (4.6), the minimality assumption and that ∑iσ(ei,(Jei)>) = 0 because the skew-symmetry of J, we get d ∑ i=1 σ(ei,∇eiX> a) = AX⊥ a−1−τ2 τσ(ξ>,(JXa)>). As a consequence ∆⊥X⊥ a= d ∑ i=1 (R(X> a,ei)ei)⊥−AX⊥ a+1−τ2 τσ(ξ>,(JXa)>) −1−τ2 τ d ∑ i=1∇⊥ eihXa,ξi(Jei)⊥+hei,ξi(JXa)⊥. (4.8) INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 25 We now compute the last sum. Firstly, using the minimality assumption, (2.11) and (4.5), we get ei(hXa,ξi) = −fahei,ξi−τhJXa,eiiand d ∑ i=1 ei(hei,ξi) = 0. (4.9) As a direct consequence of the previous equation d ∑ i=1∇⊥ eihXa,ξi(Jei)⊥+hei,ξi(JXa)⊥=−fa(Jξ>)⊥−τ[J(JXa)>]⊥ + d ∑ i=1hXa,ξi∇⊥ ei(Jei)⊥+hei,ξi∇⊥ ei(JXa)⊥. Now, in order to compute the last term of the above formula we are going to obtain a general expression for ∇⊥ u(JY)⊥where Yis any vector field in S2n+1 τ and u∈TpM. To do so, we are going to consider S2n+1 τisometrically embedded in CPn+1(4(1−τ2)) (see Proposition 1). On the one hand, decomposing JY ∈ X(CPn+1(4(1−τ2))) as JY = (JY)>+ (JY)⊥+hY,ξiJξand using (2.4) and (2.11) we deduce (DuJY)⊥= (Du(JY)>)⊥+ (Du(JY)⊥)⊥+hY,ξi(DuJξ)⊥ =σ(u,(JY)>) + ∇⊥ u(JY)⊥+1−τ2 τhY,ξihu,ξiξ⊥, where Dis the Levi-Civita connection of CPn+1(4(1−τ2)). On the other hand, using (2.4) (DuJY)⊥= (JDuY)⊥= (J∇uY)⊥−τhu,Yi− 1−τ2 τhu,ξihY,ξiξ⊥. Therefore, from both expressions of (DuJY)⊥, we get ∇⊥ u(JY)⊥= (J∇uY)⊥−σ(u,(JY)>)−τhu,Yiξ⊥. (4.10) As a consequence, taking into account the minimality of Φand (4.10), we get d ∑ i=1∇⊥ ei(Jei)⊥=−τdξ⊥, (4.11) and also, by (4.5) and (4.10), d ∑ i=1hei,ξi∇⊥ ei(JXa)⊥=−fa(Jξ>)⊥−21−τ2 τhXa,ξi|ξ>|2ξ⊥+1−τ2 τ|ξ>|2X⊥ a −τhXa,ξ>iξ⊥−σ(ξ>,(JXa)>). Lastly, using (2.7) we get d ∑ i=1 (R(X> a,ei)ei)⊥=−3(1−τ2)[J(JX> a)>]⊥+ (1−τ2)(1−d)hX> a,ξiξ⊥. (4.12) 32 FRANCISCO TORRALBO AND FRANCISCO URBANO τ2=1 3 τ2=1 1 1 2 v Figure 1. Each vector vin the shaded region represents a conformal structure on a torus (generated by the lattice {(1, 0),v}). (i) Ind(M) = 1if and only if Φis an embedding and M is either the minimal sphere S2or τ2=1 3and M is the Clifford surface T2 1/√3. (ii) Ind(M)≥g 4, where g is the genus of M. Proof. (i) From Proposition 9, the Clifford surface T2 1/√3has index one. Also, in [25, §4] it was proved that the index of the minimal sphere S2is also one for any τ∈(0, 1]. Suppose now that Ind(M) = 1. If g=0, then Mis the minimal sphere S2. Hence we can assume that the genus of Mis g≥1. Thanks to Proposition 10, we can consider the isometric embeddings CP2(4(1−τ2)) ⊂S7(c,r)⊂HM1(3)⊂H(3), where S7(c,r)stands for the sphere of center c= ( 1 3√2(1−τ2))Iand radius r= 1 √3(1−τ2), and HM1(3)is the affine hyperplane of the Hermitian matrices of order three H(3)defined in (5.5). We then consider the immersion Ψ=1 r(Φ−c):M→ S7(0, 1). As Mhas index 1, let ϕbe an eigenfunction associated to the first eigenvalue λ1<0 of the Jacobi operator L. Then, by [11] there exists B∈H1(3),|B|<1, and a conformal transformation FB:S7(0, 1)→S7(0, 1), FB(p) = B+1−|B|2 |p+B|2(p+B)∀p∈S7(0, 1), such that ZMϕ·(FB◦Ψ)dv=0. As the second eigenvalue of Lis non-negative, the above expression implies that Q(FB◦Ψ) = ∑8 i=1Q((FB◦Ψ)i)≥0. Now, if Kis the Gauss curvature of Φ, the Gauss equation K=−|σ|2 2+τ2+ 4(1−τ2)ν2of Φjoint with (3.2), allow to write the Jacobi operator of Φas L=∆−2K+4+4(1−τ2)ν2. (5.1) So, using that |FB◦Ψ|2=1, we obtain Q(FB◦Ψ) = ZM|∇(FB◦Ψ)|2+2K−4−4(1−τ2)ν2dv≥0, INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 33 which can be rewritten, using Gauss-Bonnet theorem, as ZM|∇(FB◦Ψ)|2dv≥8π(g−1) + 4ZM[1+ (1−τ2)ν2]dv. (5.2) As FBis a conformal transformation of the sphere and dΨ=1 rdΦwe easily get |∇(FB◦Ψ)|2=2 r2 (1−|B|2)2 |Ψ+B|4, and so, the inequality (5.2) becomes in ZM (1−|B|2)2 r2|Ψ+B|4dv≥4π(g−1) + 2ZM[1+ (1−τ2)ν2]dv. (5.3) To estimate the first term of (5.3), we are going to compute ∆log|Ψ+B|2, which is a well defined function because |B|<1. To do it, we decompose B=B>+hB,NiN+B⊥+hΨ,BiΨ, where B>is the tangent component of Bto Mand B⊥is the normal component of Bto S3 τ⊂S7(c,r). From this equation it is clear that ∇|Ψ+B|2=2 rB>, and so ∆|Ψ+B|2=2 r 2 ∑ i=1heσ(ei,ei),Bi− 4 r2hΨ,Bi, where eσis the second fundamental form of the embedding S3 τ⊂S7(c,r). Hence we obtain that ∆log|Ψ+B|2=2 r|Ψ+B|2 2 ∑ i=1heσ(ei,ei),Bi− 4hΨ,Bi r2|Ψ+B|2−4|B>|2 r2|Ψ+B|4. Now, ZM (1−|B|2)2 r2|Ψ+B|4dv=ZM1 r2−4 r2|Ψ+B|4(|B|2+hΨ,Bi2+hΨ,Bi(1+|B|2))dv ≤ZM1 r2−4 r2|Ψ+B|4(|B>|2+|B⊥|2+2hΨ,Bi2+hΨ,Bi(1+|B|2))dv =ZM"1 r2−4|B⊥|2 r2|Ψ+B|4−2 r|Ψ+B|2 2 ∑ i=1heσ(ei,ei),Bi#dv, (5.4) where the first equality is a direct computation, the second inequality comes from |B|2≥ |B>|2+|B⊥|2+hΨ,Bi2and the third equality comes from RM∆log|Ψ+ B|2dv=0. On the other hand, using the inequality 0≤B⊥ r|Ψ+B|2+1 4 2 ∑ i=1eσ(ei,ei) 2=|B⊥|2 r2|Ψ+B|4+∑2 i=1heσ(ei,ei),Bi 2r|Ψ+B|2+1 16 2 ∑ i=1eσ(ei,ei) 2, in (5.4) we get that ZM (1−|B|2)2 r2|Ψ+B|4dv≤ZM"1 r2+1 4 2 ∑ i=1eσ(ei,ei) 2#dv. 34 FRANCISCO TORRALBO AND FRANCISCO URBANO So, using that 1/r2=3(1−τ2), the previous inequality and (5.3) becomes in 1 4ZM 2 ∑ i=1eσ(ei,ei) 2dv≥4π(g−1) + ZM3τ2−1+2(1−τ2)ν2dv. But the second fundamental form eσof the embedding S3 τ⊂S7(c,r)is given eσ=ˆ σ+σ+1 rh·,·iΨ, where σis the second fundamental form of the embedding CP2(4(1−τ2)) ⊂HM(3). Now, from (2.4) and (5.6), we easily get that 1 4 2 ∑ i=1eσ(ei,ei) 2=1 4τ23τ2−1+ (1−τ2)ν22+ (1−τ2)ν2. Finally, the last inequality above reads 4π(g−1) + 1 4τ2ZM3τ2−1+ (1−τ2)ν2·τ2(1+ν2) + (1−ν2)dv≤0 Since 1 3≤τ2≤1, g≥1 and ν2≤1 we get that g=1, τ2=1 3and ν=0. Therefore, the Killing field ξis tangent to Mand so an orthonormal reference on TM is given by {ξ,JN}, where Nis a unit normal vector field to Φ. Now, from (2.11) it follows that σ(ξ,ξ) = 0, σ(JN,ξ) = −τN, which implies that |σ|2=2τ2. The Gauss equation says us that Mis flat and so Mis congruent to a finite covering of the Clifford torus ([24]). As the Jacobi operator is Lf=∆+4, Mis congruent to the Clifford torus. (ii) The argument we use to prove (ii) is inspired in the papers [17,18]. As Ind(M)≥1, we can assume g≥5. Then, if ˆ ∆is the Hodge Laplacian acting on 1-forms on M, it is well-known that ker ˆ ∆is the space of harmonic 1-forms of M, whose dimension is 2g. Using the metric on M, the 1-forms and the vector fields on Mare identified and we say that a vector field Xis harmonic if the corresponding 1-form is harmonic. It is well-known that this property is equivalent to div X=0, h∇vX,wi=h∇wX,vi, for any tangent vectors v,wto M. Also, if Xis a harmonic vector field on Mthen ∆X=KX, where Kis the Gauss curvature of Mand ∆is the rough Laplacian defined by ∆=∑2 i=1{∇ei∇ei−∇∇eiei}. Given a harmonic vector field Xon Mwe consider the vectorial function X: M→HM1(3)and we are going to compute Q(X) = ∑8 i=1Q(hX,Bii), where Bi: 1 ≤i≤8}is an orthonormal reference of HM1(3). If e ∆demotes the Laplacian of the Euclidean space HM1(3), then he ∆X,Xi=h∆X,Xi−hA2X,Xi− 2 ∑ i=1|ˆ σ(X,ei)|2+|σ(X,ei)|2 =K−1 2|σ|2|X|2− 2 ∑ i=1|ˆ σ(X,ei)|2+|σ(X,ei)|2, INDEX OF COMPACT MINIMAL SUBMANIFOLDS OF THE BERGER SPHERES 35 where {e1,e2}is an orthonormal reference on M. Now, from (5.1), the Gauss equation of Φ, (2.4) and (5.6) we obtain that Q(X) = ZM(τ2−4)|X|2+ 2 ∑ i=1{|ˆ σ(X,ei)|2+|σ(X,ei)|2dv =ZM(1−3τ2)|X|2+1−τ2 τ2hX,ξi2−(1−τ2)|(JX)>|2+1−τ2 τ2hX,ξi2ν2dv≤0. Moreover, if Q(X) = 0, then τ2=1/3, (JX)>=0 and hX,ξiν=0. In this case it is not difficult to get that ν=0, which implies that Mis the Clifford surface. This is imposible because we are assuming that the genus g≥5. So Q(X)<0 for any non-null harmonic vector field Xon M. Suppose that Ind(M) = mand let {f1, . . . , fm}the eigenfunctions of Lcorresponding to the mnegative eigenvalues. If H(M)denotes the linear space of harmonic vector fields on M, we define a map F:H(M)→R8mby F(X) = ZMf1X, . . . , ZMfmX). If X∈ker F, then Q(X)≥0, and so X=0. This means that 2g=dim Img F≤ 8m, which proves (ii).  Appendix: The Tai embedding Let HM(n+1) = {A∈gl(n+1, C):A=At}be the space of Hermitian matrices of order n+1, endowed with the Euclidean metric hA,Bi=tr AB. Let I∈HM(n+1)be the identity matrix. Then HM1(n+1) = {A∈HM(n+1):hA,Ii=1 √2(1−τ2)}(5.5) is an affine hyperplane of HM(n+1). Let π:S2n+1(1 √1−τ2)→CPn(4(1−τ2)) be the Hopf fibration of the sphere of radius 1 √1−τ2over the complex projective space of constant holomorphic curvature 4(1−τ2). The Tai map [20]T:CPn(4(1−τ2)) →HM(n+1)is given by T([z]) = √1−τ2 √2ztz,z∈S2n+1(1 √1−τ2)⊂Cn+1. Proposition 10.The Tai map T verifies the following properties: (i) It is an isometric embedding. (ii) Its image is contained in the sphere Sn2+2n−1(c,r)of HM1(n+1)with center c= ( 1 (n+1)√2(1−τ2))I and radius r =√n √(n+1)2(1−τ2). (iii) The second fundamental form σof T :CPn(4(1−τ2)) →HM(n+1)is parallel and σ(JX,JY) = σ(X,Y). Moreover, for any x,y,v,w∈T[z]CPn(4(1−τ2)) hσ(x,y),σ(v,w)i= (1−τ2)2hx,yihv,wi+hx,wihy,vi+hx,vihy,wi +(1−τ2)hx,Jwihy,Jvi+hx,Jvihy,Jwi.(5.6) (iv) T :CPn(4(1−τ2)) →Sn2+2n−1(c,r)is a minimal embedding. 36 FRANCISCO TORRALBO AND FRANCISCO URBANO Proof. This is a well-known result and it was proven in [16, §1] for CPn(1). 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