A new bound on the Morse index of constant mean curvature tori of revolution in S3

A NEW BOUND ON THE MORSE INDEX OF CONSTANT MEAN CURVATURE TORI OF REVOLUTION IN S3 ANTONIO CA ˜ NETE ABSTRACT. In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere S3, by computing some explicit negative eigenvalues for the corresponding Jacobi operator. INTRODUCTION Given a closed surface Mwith constant mean curvature (CMC), immersed in a three-dimensional manifold N, it is well known that Mis a critical point of the area functional, when we consider variations preserving the enclosed volume [3], [4]. For this kind of surfaces, we can discuss the stability by studying the second variation of the area, which can be expressed by means of an useful and classical functional operator L; more precisely, denoting by f∈ C∞(M,R)to the normal component of the vector field associated to a variation of M, the second variation formula will be given by (1) −ZMf L f da, where L=∆+R+|σ|2,∆is the Laplacian operator of M,Ris the Ricci curvature of N, and σis the second fundamental form of M. Such operator L:C∞(M,R)→ C∞(M,R)is usually called the Jacobi (or stability) operator. For CMC surfaces, Mis said to be stable if and only if the above expression (1) is greater than or equal to zero, for any volume-preserving variation [3, §2]. In this setting, a usual way of measuring the instability of a surface Mis given by the Morse index of M(see for instance [2], [20]), which is defined as the number of negative eigenvalues, counted with their multiplicities, of the Jacobi operator L(throughout this paper, a value λ∈Rwill be an eigenvalue of Lif there exists a function fλ∈ C∞(M,R)such that L fλ+λfλ=0). This intrinsic relation with stability (see [13] for further details) has stimulated the study of the Morse index of CMC surfaces in several works. The main approaches have focused on the minimal case, that is, surfaces with zero mean curvature (see [14], [8], [15], [23], [6]). In the euclidean space R3, for instance, planes have zero index [7], meanwhile catenoids and Enneper’s surfaces have index one [10], [13]. In the sphere S3, an interesting result from J. Simons [22] states that the index of any compact minimal surface Mis always greater than or equal to one, with equality if and only if Mis a totally geodesic 2-sphere (in fact, such a result was stated in general dimension). Later on, F. Urbano [24] proved that any compact minimal surface (not totally geodesic) in S3has index greater than or equal to five, with equality 2000 Mathematics Subject Classification. 53C42, 53A10. Key words and phrases. Morse index, Jacobi operator, stability, constant mean curvature. This work was partially supported by MEC research project MTM2007-61919. 2 A. CA ˜ NETE uniquely for the Clifford torus. The analogous result in the n-dimensional case has been partially demonstrated in [9], [11], [16]. A nice review of these results can be found in [1]. However, in the family of nonminimal CMC surfaces, the Morse index has been less discussed in literature, and the main results refer to bounds for the index. In R3, apart from the spheres (which have index one), only lower bounds for tori of revolution [19], and upper and lower bounds for the Wente tori [18] are known. Moreover, for CMC immersions of revolution in S3, W. Rossman and N. Sultana [20] have recently computed the index of flat tori of revolution (in terms of the mean curvature), and have also found a lower bound for non-flat tori, giving as well numerical methods for explicit calculations [21]. It follows from their work that the index of these tori is at least five, in any case. In this work, we focus on the index of CMC tori of revolution immersed in S3. By using an approach different from the one used in [20], we shall explicitly find some negative eigenvalues for the Jacobi operator of these surfaces, obtaining new bounds for the Morse index. Our technique implies a suitable arrangement of the metric of the surface. This will lead us to have a nice expression of the Jacobi operator, which allows to determine some specific eigenfunctions and eigenvalues. We remark that these eigenvalues will depend on some geometric quantities associated to the surface (as the energy and the mean curvature), and we will discuss analytically their sign, improving in most of the cases the previous known bounds for the Morse index (see Theorem 2.14). For instance, we obtain that when the mean curvature of the torus is less than −1 or greater than 3/2, the Morse index is, at least, eight. We point out that an interesting question, in this CMC setting, is finding a similar result to that of F. Urbano, who characterized Clifford tori in S3by means of a minimum value of the Morse index [24]. Some partial progresses have been made in this direction [2]. Acknowledgments. The author would like to thank Manuel Ritor´e for his continuous support and kind help during the elaboration of these notes. 1. PRELIMINARIES Consider a torus of revolution Mwith constant mean curvature H, immersed in the 3-dimensional unit sphere S3⊂R4. Let us fix a geodesic curve ℓin S3, given by ℓ={(cos(t), sin(t), 0, 0),t∈R}. Then, the torus can be seen as generated by the rotation about ℓof a given curve parameterized by arc-length γ:[0, t0]−→ S3 defined by γ(t) = (γ1(t),γ2(t),γ3(t), 0),t∈[0, t0]. We can further assume that γ3(t)>0 for all t. Remark 1.1.The description of the generating curve of a CMC torus of revolution can be found in [12, Th. 3] (see also [20, §1]), being unduloidal or nodoidal. Hence, an immersion ψ:M→S3of the torus into S3will be given by ψ(t,θ) = (γ1(t),γ2(t),γ3(t)cos θ,γ3(t)sin θ),t∈[0, t0],θ∈[0, 2π]. MORSE INDEX OF CMC TORI OF REVOLUTION IN S33 Then we have that the tangent space to ψ(t,θ)is generated by the vectors ∂t= (γ′ 1(t),γ′ 2(t),γ′ 3(t)cos θ,γ′ 3(t)sin θ),(2) ∂θ= (0, 0, −γ3(t)sin θ,γ3(t)cos θ), with |∂t|=1,|∂θ|=γ3(t). Moreover, it is straightforward checking that the unit normal vector to ψ(t,θ)will be given by (3) N(t,θ) =     γ2(t)γ′ 3(t)−γ′ 2(t)γ3(t) γ3(t)γ′ 1(t)−γ′ 3(t)γ1(t) (γ1(t)γ′ 2(t)−γ′ 1(t)γ2(t)) cos θ (γ1(t)γ′ 2(t)−γ′ 1(t)γ2(t)) sin θ     =    a(t) b(t) c(t)cos θ c(t)sin θ     . Since Mis rotationally symmetric, we can consider in Ma metric of type ds2=dt2+f(t)2dθ2, for certain positive function f:M→R. We are interested in computing the Hopf differential of our immersion ψ, and so we shall need conformal (isothermal) coordinates in M. In order to have this, we use the following change of coordinates: consider a new coordinate t′defined by t=G(t′), where Gis a diffeomorphism of R determined by G′(t′) = f(G(t′)) and with initial condition G(0) = 0. Defining now g(t′,θ) = (G(t′),θ), it is not difficult to check that the new immersion of Mgiven by (ψ◦g)(t′,θ) = ψ((G(t′),θ)) is conformal, with ds2=f(G(t′))2(dt′)2+dθ2. Observe that we have properly replaced coordinates (t,θ)with (t′,θ). It also follows from above that (4) |∂θ|=f(t) = γ3(t), for all t. Now, by considering the flat metric ds2 0associated to the Hopf differential of the immersion ψ◦g, it follows that (5) ds2=exp(2w)b−2ds2 0, where b2=4(1+H2), and w:M→Ris a smooth function defined on the torus (see [17, §1]). Remark 1.2.We point out that (6) ds2 0=β((dt′)2+dθ2), for certain β∈R, due to the flatness of the metric ds2 0. From the above expressions (5), (6) we get (7) β=exp(−2w(t′)) b2f(t)2>0, or equivalently f(t) = exp(w(t′)) b−1β1/2. 4 A. CA ˜ NETE 1.1. On the value of the constant β.The value of the constant βcan be computed in terms of the principal curvatures kt,kθof the immersion ψ, taking into account the construction of the Hopf differential. More precisely, denoting by σ′the second fundamental form of the conformal immersion ψ◦g, we have (see [17, §1]) (8) β=σ′(∂t′,∂t′)−σ′(∂θ,∂θ) = f(t)2(kt−kθ), where, taking into account (2), (3), the principal curvatures are given by kθ=σ(∂θ,∂θ) = γ′ 1(t)γ2(t)−γ1(t)γ′ 2(t) γ3(t)=−c(t) γ3(t),(9) kt=σ(∂t,∂t) = − γ1(t)γ2(t)γ3(t) γ′ 1(t)γ′ 2(t)γ′ 3(t) γ′′ 1(t)γ′′ 2(t)γ′′ 3(t) . In order to calculate explicitly the value of β(which we shall need later), we will take into account some particular parametrization of the points in S3appearing in [12, §1]. This will provide, in particular, new useful expressions for the principal curvatures kt,kθ. For any point p∈M⊂S3, there exists a point q∈ℓsuch that pq is a geodesic arc whose length is equal to the distance between pand ℓ. Fixing a base point in ℓ and considering the arc length on ℓwith respect to the base point, we can assign to p new coordinates (x,y), where xis the coordinate of qin ℓ, and yis the length of the geodesic pq. This procedure allows to express the generating curve γ⊂S3in terms of coordinates x,y. Straightforward computations yields γ1(t) = cos(x(t)) cos(y(t)),(10) γ2(t) = sin(x(t)) cos(y(t)), γ3(t) = sin(y(t)). Moreover, since the mean curvature His constant, the following relations hold ([12, eq. (4)]): (11) x′(t) = sin(α(t)) cos(y(t)),y′(t) = cos(α(t)), where α(t)is the angle between the tangent vector to γ(t)and the vertical direction. Then, from (9), (10) we get that kθ=−sin(α(t)) cot(y(t)),(12) kt=sin(α(t)) tan(y(t)) −α′(t). We recall that (13) 2H=kt+kθ=−sin(α(t)) cot(y(t)) + sin(α(t)) tan(y(t)) −α′(t). On the other hand, it is easy to check that (14) E=sin(y(t)) cos(y(t)sin α(t) + Hsin(y(t)) is constant (just compute the derivative with respect to t, using the above relations (11)). MORSE INDEX OF CMC TORI OF REVOLUTION IN S35 Finally, taking into account previous expressions (8), (12), (13) and (14), it follows that (15) β=f(t)2(kt−kθ) = sin2(y(t)) 2 sin α(t)cot(y(t)) + 2H=2E. Remark 1.3.In fact, the above constant Eis a first integral associated to the system of equations (11) of the generating curve γ(see [12, Th. 1]). Remark 1.4.In the previous computations of kt,kθ, we have considered the normal vector N(t,θ)defined by (3). If we consider that normal vector with opposite sign, the values of kt,kθwill change the sign too. The appropriate choice of N(t,θ)will be determined by the positivity of β, in view of (8). Remark 1.5.It follows from (4), (10) that (16) f(t) = γ3(t) = sin(y(t)),t∈[0, t0]. 1.2. On the function w.The function warising in (5) establishes the relation between the metric ds2in Mand the flat metric ds2 0associated to the Hopf differential of the immersion. It only depends on the variable tsince Mis rotationally symmetric, and it satisfies the sinh-Gordon equation for the laplacian associated to ds2 0([17, eq. (2)]); equivalently, by using (6) we have (17) (w′′ +βsinh(w)cosh(w) = 0, w(0) = c,w′(0) = 0, where c∈R, and the derivatives are taken with respect to the standard flat metric (dt′)2+dθ2. Remark 1.6.The above value c∈Rin (17) is related with the length of the parallel S1× {0} ⊂ M, since that length is equal to 2πf(0) = 2πexp(c)b−1β1/2. On the other hand, by using [12, eq. (1)] and (16), we have that L(S3× {0}) = 2πcos(y(0)) = 2πp1−f(0)2. Thus, the value cequals log bq1−γ2 3(0) β1/2 !=log bp1−f(0)2 β1/2 !. In this setting, we recall that the Gauss curvature Kof M, with respect to the metric ds2, only depends on the t-coordinate and is given by (18) K= (b2/4)(1−exp(−4w)). Remark 1.7.By integrating equality (17), multiplied by 2 w′, we obtain (19) (w′)2+βcosh2(w) = βcosh2(c), which is a first integral of equation (17), where the derivative is with respect to the flat metric (dt′)2+dθ2. 1.3. Index form and Jacobi operator. Recall that Ndenotes the normal vector field of M, and Kis the Gauss curvature. Then, it is well known that the second variation formula of the area, for variations preserving the volume enclosed by M, is given in general by [3, Prop. 2.5] I(f,f) = −ZMf∆f+ (R+|σ|2)f2da(20) =−ZMf∆f+ (4+4H2−2K)f2da, 6 A. CA ˜ NETE where ∆is the laplacian operator associated to the metric ds2,fis the normal component of the vector field associated to the variation, R=2Ric(N)is the Ricci curvature of the ambient space S3, and |σ|2is the square of the norm of the second fundamental form σ. We shall refer to the quadratic form defined by (20) as the index form of M. The associated Jacobi operator is thus given by L f =∆f+ (4+4H2−2K)f, for any C∞(M)function f, and so I(f,f) = −ZMf L f da. Taking into account expression (18) we have that 4+4H2−2K=b2−2K= (b2/2) (1+exp(−4w)) =b2exp(−2w)cosh(2w) =b2exp(−2w) (cosh2(w) + sinh2(w)).(21) In view of expression (21), we now define a new operator L0by L0f=exp(2w)b−2L f ,f∈C∞(M). Then, it is clear, in view of (5) and (6), that L0f=∆0f+ (cosh2(w) + sinh2(w)) f(22) =1 β∂2f ∂t2+∂2f ∂θ2+ (cosh2(w) + sinh2(w)) f, where ∆0represents the laplacian of ds2 0. An important fact that we will use later is that both operators L,L0only differ in a positive scalar factor. 1.4. Morse index of CMC surfaces. The Morse index of a closed constant mean curvature (CMC) surface Sis defined by means of the Jacobi operator and, as indicated in [20], provides a degree of the instability of Swith respect to the area. We first recall the following definition. Definition 1. Given a function f :M→R, we shall say that f is an eigenfunction of the Jacobi operator L, with associated eigenvalue λ∈R, if L f +λf=0. It is known that the set of eigenvalues {λi}i∈Nof the Jacobi operator Lconsists of an increasing sequence, diverging to +∞, and that the first eigenvalue λ1has multiplicity one (see [5] for further details). We can now define the Morse index of a CMC surface. Definition 2. Given a closed CMC surface S, the Morse index Ind(S) is the number of negative eigenvalues of the Jacobi operator L, each one counted with its multiplicity. Our purpose is giving a lower bound for the Morse index for CMC tori of revolution immersed in S3. To do that, we will focus on the operator L0, since due to its definition, it will have the same number of negative eigenvalues that the Jacobi operator L. MORSE INDEX OF CMC TORI OF REVOLUTION IN S37 2. EXPLICIT COMPUTATION OF SOME EIGENVALUES OF L0 In this Section we shall compute directly some (negative) eigenvalues of the operator L0, taking into account its expression (22), by using certain functions on Mwith independent variables. We first define the operator Lt, on the set of functions of real variable, as Ltf=1 βf′′(t) + (cosh2(w) + sinh2(w)) f(t),f:R→R. It is clear that, for functions defined on M, we have that L0=1 β ∂2 ∂θ2+Lt. We begin with the following key result. Lemma 2.1. Let u =u(t)be an eigenfunction of Lt, with associated eigenvalue λ∈R, and let v =v(θ)be an eigenfunction of the laplacian ∂2 ∂θ2, with associated eigenvalue µ∈R. Then, the function f :M→Rgiven by f (t,θ) = u(t)v(θ)is an eigenfunction of L0, with associated eigenvalue λ+µ/β. Proof. Applying the operator L0to f, we have that L0f=1 β ∂2v ∂θ2u+v Ltu=−(λ+µ β)u v =−(λ+µ β)f, and so the result follows.  We now proceed to find convenient functions for applying Lemma 2.1. It is clear that v(θ)can be taken equal to a constant, or equal to cos(θ), sin(θ), which are eigenfunctions of the laplacian (with µ=0 and µ=1 as eigenvalues, respectively). The next result gives some eigenfunctions for the operator Lt. Proposition 2.2. The functions u1,u2:[0, t0]→Rdefined by u1(t) = cosh(w(t)),u2(t) = sinh(w(t)), are independent eigenfunctions of Lt, with associated eigenvalues λ1=−cosh2(c),λ2=1−cosh2(c), respectively. Proof. The proof is straightforward, taking into account (17) and (19).  Remark 2.3.We point out that when wis identically zero (which corresponds to flat metric ds2), previous Proposition 2.2 only shows that constant functions will be eigenfunctions of Lt, with eigenvalue λ=−1. Our idea consists of using functions with independent variables. By combining the functions u1(t),u2(t)from Proposition 2.2 with a constant function, with sin(θ), or with cos(θ), we can apply Lemma 2.1 to obtain some eigenvalues for the operator L0. Theorem 2.4. Let M be a CMC torus of revolution immersed in S3and L0the operator defined previously. Then, i) the first eigenfunction of L0is f1(t,θ) = cosh(w(t)), with associated eigenvalue λ1=−cosh2(c)<0, ii) f2(t,θ) = sinh(w(t)) is an eigenfunction of L0, with associated eigenvalue 1− cosh2(c)<0, 8 A. CA ˜ NETE iii) f3(t,θ) = cosh(w(t)) sin(θ),f3(t,θ) = cosh(w(t)) cos(θ)are two eigenfunctions of L0, with associated eigenvalue −cosh2(c) + 1/β, iv) f4(t,θ) = sinh(w(t)) sin(θ),f4(t,θ) = sinh(w(t)) cos(θ)are two eigenfunctions of L0, with associated eigenvalue 1−cosh2(c) + 1/β. Furthermore, these six eigenfunctions are independent, and Ind(M)⩾2. Proof. Just apply Lemma 2.1 with the corresponding functions in order to obtain the six independent eigenfunctions. Moreover, f1is the first eigenfunction of L0since it does not vanish. Finally, λ1=−cosh2(c)is always negative, and since c6=0 (otherwise, (17) yields w=0 and so fis constant), we have that 1 −cosh2(c)is also a negative eigenvalue for L0. Another negative eigenvalue of L0can be obtained by following some ideas from [20]. For a given geodesic curve ℓ′in S3, orthogonal to ℓ, we can consider the Killing vector field Kassociated to the rotations about ℓ′in S3. Then, the normal component f=hK,Niof Ksatisfies L0(f) = 0 (that is, fis a Jacobi function for L0), and can be expressed as f(t,θ) = u(t)cos θ(see [20, Lemma 4.1]). In this situation, it is easy to check that uis an eigenfunction of L0, with associated eigenvalue −1/β(which is negative since β>0). Therefore, as in [20, Lemma 4.2], by taking the two geodesic curves orthogonal to ℓ, we get two independent eigenfunctions of L0(depending only on variable t) with eigenvalue −1/β. Straightforward computations show that these two eigenfunctions are given by u(t,θ) = −γ3(t)b(t) + γ2(t)c(t) = −γ′ 1(t),(23) u(t,θ) = −γ3(t)a(t) + γ1(t)c(t) = γ′ 2(t), where a(t),b(t),c(t)are the real functions provided by (3). Lemma 2.5 summarizes these properties. Lemma 2.5. ([20, Lemmata 4.1 and 4.2])The functions u,u defined by (23) are two independent eigenfunctions of L0with −1/βas associated (negative) eigenvalue. From above lemma we have the following interesting consequence related with Theorem 2.4. Lemma 2.6. The eigenvalue −cosh2(c) + 1/βis negative. Proof. From Theorem 2.4 we have that λ1=−cosh2(c)is the first eigenvalue of L0. Then, since −1/βis another eigenvalue of L0, we necessarily have −cosh2(c)< −1/β, and so −cosh2(c) + 1/β<0.  The Morse index is defined taking into account the multiplicities of the negative eigenvalues. In this sense, we have to study carefully whether uor ubelong to the eigenfunctions space associated to one of the eigenvalues described in Theorem 2.4 (if this is the case, they will not contribute to the Morse index). For having that, a necessary condition is that −1/βcoincides with one of the eigenvalues previously obtained. It is clear that it cannot be equal to λ1=−cosh2(c), otherwise the first eigenvalue of L0would have multiplicity greater than one, which is a contradiction (recall that u,uare independent). If −1/βcoincides with λ=−cosh2(c) + 1/β, as u,uonly depend on variable t, they will be independent from f3,f3, and so the multiplicity of λwill be greater than (or equal to) four (hence contributing to the Morse index, since λ<0 from Lemma 2.6). The same reasoning is valid if −1/β MORSE INDEX OF CMC TORI OF REVOLUTION IN S39 coincides with λ′=1−cosh2(c) + 1/β(observe that the sign of λ′has not been discussed yet). Finally, we have to study if −1/βcoincides with 1 −cosh2(c), equivalently 1 −cosh2(c) + 1/β=0. This case of equality is treated in Subsection 2.1. However, if this equality does not occur, we immediately have the following result, which establishes lower bounds for the Morse index. Theorem 2.7. Let M be a CMC torus of revolution immersed in S3. With the previous notation, assume that 1−cosh2(c) + 1/β6=0. Then, i) Ind(M)⩾8, if 1−cosh2(c) + 1/β<0. ii) Ind(M)⩾6, if 1−cosh2(c) + 1/β>0. Proof. From the hypothesis we have that the eigenfunctions shown in Theorem 2.4 and Lemma 2.5 are independent. An analysis of the eigenvalue 1 −cosh2(c) + 1/β yields the statement.  Remark 2.8.Note that Theorem 2.7 establishes bounds on the Morse index of CMC tori of revolution in S3which improve the ones given in [20, Th. 1.1]. Observe that above Theorem 2.7 yields a numerical criterion (based only on the sign of 1 −cosh2(c) + 1/β, which depends on the constant values of cand β) that provides lower bounds for the Morse index. Now, we shall express the eigenvalue 1−cosh2(c) + 1/βin terms of the mean curvature Hand the value f(0), in order to study its sign. From Remark 1.6 we have that cosh(c) = ec+e−c 2=b2(1−f(0)2) + β 2β1/2 bp1−f(0)2, and so (24) cosh2(c) = 4(1+H2)(1−f(0)2) + β2 16 β(1+H2)(1−f(0)2). On the other hand, the constant βcan be expressed only in terms of Hand f(0): in fact, its value can be obtained by taking t=0 in (14), and so β=2 sin(y(0))cos(y(0)) sin(α(0)) + Hsin(y(0)). Since f(t) = sin(y(t)), and taking t=0 such that α(0) = π/2, it follows that (25) β=2f(0)q1−f(0)2+H f (0). Using equalities (24) and (25), the eigenvalue 1 −cosh2(c) + 1/βcan be expressed only in terms of Hand f(0). Moreover, since β>0, it follows that p1−f(0)2+ H f (0)must be positive, and so necessarily f(0)<r1 1+H2, if H<0 (in the case H⩾0, then f(0)∈(0, 1)). Above equalities allow to calculate explicitly that eigenvalue, for each H∈Rand each considered value of f(0). Numerical computations show that both possibilities