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Multiplicity of constant scalar curvature metrics in Tk×M
Nonlinear Analysis 109 (2014) 103–112
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Multiplicity of constant scalar curvature metrics in Tk × M Héctor Fabián Ramírez-Ospina ∗ Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
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Article history: Received 27 March 2014 Accepted 11 June 2014 Communicated by Enzo Mitidieri MSC: 58E11 58J55 58E09
abstract We use bifurcation theory to prove the existence of uncountably many unit volume constant scalar curvature metrics in manifolds of the form Tk × M, k ≥ 2, with M a compact manifold (without boundary), that are conformal, but not isometric, to a product metric g flat ⊕ g, where g flat is a flat metric on the k-torus Tk , and g is a fixed constant scalar curvature metric on M. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Bifurcation theory Yamabe problem
1. Introduction Bifurcation theory provides a powerful technique for determining the existence of multiple solutions of several geometric variational problems. In particular, great attention has been given recently to the question of multiplicity of (unit volume) constant scalar curvature metrics in conformal classes of metrics on compact manifolds. For instance, in [1] spectral theory of the linearized Yamabe problem is used to determine the existence of multiple solutions in manifolds given by the product of a round sphere and an arbitrary compact manifold. In [2] the authors prove the existence of a countable number of bifurcating branches of solutions of the Yamabe problem in product manifolds M1 × M2 stemming from a trivial path of solutions given by (a unit volume normalization of) ]0, +∞[ ∋ t → g1 ⊕ tg2 , where g1 and g2 are constant scalar curvature metrics on M1 and M2 respectively. It is important to observe that, for the existence of bifurcation, one must assume that at least one of the two factors has positive scalar curvature. A somewhat more general topological situation is considered in [3], where the authors study bifurcation of solutions of the Yamabe problem in manifolds that are the total space of a Riemannian fibration, when the size of the fibers shrinks to 0 and the manifold collapses onto its base. On the opposite side, spectral theory is used in [4] to determine local rigidity for families of constant scalar curvature metrics, under suitable topological assumptions. In [5], bifurcation theory is employed to determine multiple solutions forming branches issuing from families of homogeneous metrics on spheres. In [6], bifurcation theory is applied in order to obtain existence of degenerate solutions of a certain nonlinear elliptic solution on the sphere Sn , yielding also multiplicity results for scalar curvature metrics on the product of two spheres. Bifurcation techniques for constant scalar curvature are also important in General Relativity, see for instance [7]. In all the above results, the main idea was to consider bifurcation of constant scalar curvature metrics issuing from paths of product metrics obtained by rescaling the volume of the two factors, or rescaling the volume of the fibers of a fibration.
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http://dx.doi.org/10.1016/j.na.2014.06.014 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
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In the present paper we consider a different situation, by taking the product of a flat k-torus Tk and a unit volume fixed constant scalar curvature Riemannian manifold (M , g ). We consider paths of metrics in Tk × M of the form s → gs ⊕ g, where s → gs is a path of unit volume flat metrics on Tk . Recall that, when k = 2, the space of isometry classes of unit volume flat metrics on T2 has dimension 2, and it can be naturally identified with the hyperbolic plane H2 (see [8, see Prop 10.2]). More generally, the space of isometry classes of flat metrics on the k-torus is identified with the set of isometry classes of (co-compact) lattices in Rk . By studying the spectral properties of the linearized scalar curvature operator along this type of paths, we obtain a proof of the following result. Theorem. Let k ≥ 2 be a fixed integer, and let (M m−k , g ) be a unit volume compact Riemannian manifold with m ≥ k + 2, having positive constant scalar curvature Scalg . Then, there are uncountably many branches of unit volume constant scalar curvature metrics in the product M = Tk × M, each of which belongs to the conformal class of (but is not isometric to) a product metric of the form g flat ⊕ g, with g flat a unit volume flat metric on Tk . Each bifurcating branch contains sequences of metrics (g n )n∈N whose (constant) scalar curvature tends to Scalg , and g n tends to a product metric g flat ⊕ g in the C ∞ -topology of metrics as n → ∞. Let us recall here that a well known result by Gromov and Lawson (see Section 4) says that every metric with non-negative scalar curvature on a torus Tk must be flat. Our result above implies in particular the existence of positive constant scalar curvature on Tk × M, which is not a product metric, for ‘‘arbitrarily small’’ compact differentiable manifold M admitting a Riemannian metric of positive scalar curvature. A proof of the theorem will be given in Section 6. The key point of the proof is the fact that, in order to produce bifurcation, one can find continuous paths of unit volume flat metrics on the torus Tk where one of the two endpoints is a metric whose Laplacian has an arbitrary number of eigenvalues that are arbitrarily small. For the case k = 2, a more precise description of the flat metrics that produce bifurcating branches will be given in Section 7, see Proposition 10 and Theorem 11. Motivation for the above result was given by the recent paper [9], where a similar bifurcation problem is studied in the context of singular Yamabe solutions in spheres that blowup on an equator. In that case, the authors consider products of the type Σ × M, where Σ is an orientable closed surface of genus greater than 1, endowed with a metric varying in the set of hyperbolic metrics (i.e., with curvature equal to −1). When M is a round sphere Sm , m ≥ 3, then constant scalar curvature metrics on Σ × M correspond to solutions of the singular Yamabe problem in Sm+2 \ S1 that are conformal to the round metric. The paper is organized as follows. In Section 2 we recall the analysts’ approach to the Yamabe problem, using a Sobolev H 1 -framework, which is suitable for the purposes of the present paper. This approach is based on a fundamental regularity result proved by Trudinger, see [10, Theorem 3]. We will recall here some basic facts on the first and the second variation formula for the Hilbert–Einstein functional, see (3). In Section 3 we discuss a convergence result for bifurcating branches of solutions of the Yamabe problem, which was somewhat overlooked in the existing literature. This is based on suitable Lp estimates, with p > 1, on the solution of a second order elliptic equation, see for instance [11, Theorem 9.14, p. 240]. In Section 4 we study the spectrum of the Jacobi operator for the Yamabe problem in the case of product metrics of the form gsflat ⊕ g as above. In Section 5 we prove that a necessary condition for bifurcation for paths of the form gsflat ⊕ g is that Scal(g ) > 0 (Corollary 7), we give an abstract result on bifurcation of metrics in T2 × M (Proposition 9), and we prove that bifurcating branches necessarily consist of non product metrics, see Proposition 8. Section 6 contains the proof of our Theorem for general k ≥ 2. Section 7 contains a more detailed analysis of the case k = 2, where more explicit computations are made, and concrete examples of paths containing bifurcation points are exhibited (Proposition 10 and Theorem 11). Finally, Section 8 contains a brief discussion on a further possible generalization of the results of the present paper to the case of products F × M, where F is a compact manifold admitting a flat metric. 2. Variational formulation of the Yamabe problem In this section we will recall the original variational framework for constant scalar curvature metrics in a conformal class, see the classical paper [10], or [12]. 2.1. The variational problem in H 1 Let N be a closed (i.e., compact and without boundary) smooth manifold with dim(N ) = n ≥ 3, and let g be a (smooth) Riemannian metric on N having volume vol(g ) = 1; for x ∈ N, denote by Scalg (x) the scalar curvature of g at x. Given a 4
positive smooth function u on N, let us consider the conformal metric g ′ = u n−2 g. It is easily computed that vol(g ′ ) = vol(g ) if and only if:
2n
u n−2 volg = 1. N
(1)
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Consider the Sobolev1 space H 1 (N ), endowed with the Hilbert norm:
∥φ∥2H 1 =
2 |φ| + g (∇φ, ∇φ) volg , N
and we denote by ⟨·, ·⟩H 1 the corresponding inner product. By the Gagliardo–Nirenberg–Sobolev inequality, there is a 2n
continuous inclusion H 1 (N ) ⊂ L n−2 (N ) (see for instance [13, Theorem 2.6, p. 32]), and the subset
[g ]1 := u ∈ H (N ) : u > 0, 1
u
2n n−2
volg = 1
N
is a smooth submanifold of H 1 (N ). Indeed, the number 1 is a regular value of the map I
{u ∈ H 1 (N ) : u > 0} ∋ u −→
2n
u n−2 volg . N
The differential of this map is dI (u)φ =
2n n−2
n+2
u n−2 φ volg . N
φ volg . Thus, at u0 ≡ 1, the tangent space Tu0 [g ]1 is given by φ volg = 0 .
At u0 ≡ 1, this gives dI (u0 )φ = n2n −2
Tu0 [g ]1 =
φ ∈ H 1 (N ) :
N
(2)
N
Consider the functional A : [g ]1 → R defined by:
A(u) :=
−1 g (∇ u, ∇ u) + scalg (x)u2 volg . 4 nn− 2
N
(3)
Proposition 1. The following statements hold: (a) A is smooth on [g ]1 ; (b) u ∈ [g ]1 is a critical point of A if and only if u is smooth, and g ′ = u · g has constant scalar curvature. Moreover, if g has constant scalar curvature (i.e., if u0 ≡ 1 is a critical point of A), then: (c) the second derivative of A at u0 ≡ 1 is the symmetric bilinear form: d A(u0 )(φ1 , φ2 ) = 2
(n−1)(n−2) 2
N
Scal
g (∇φ1 , ∇φ2 ) − n−1g φ1 φ2 volg .
(4)
(d) There exists a self-adjoint operator Fg : Tu0 [g ]1 → Tu0 [g ]1 , given by a compact perturbation of the identity, such that: d2 A(u0 )(φ1 , φ2 ) =
(n−1)(n−2) 2
⟨Fg (φ1 ), φ2 ⟩H 1
(5)
for all φ1 , φ2 ∈ Tu0 [g ]1 . Proof. See [10] for a proof of (a) and (b) (especially [10, Theorem 3] for the smoothness of critical points); see [14] for (c). A proof of (d) is obtained immediately observing that:
Scal
d2 A(1)(φ1 , φ2 ) = ⟨φ1 , φ2 ⟩H 1 − 1 + n−1g ⟨φ1 , φ2 ⟩L2 (N ) , (n−1)(n−2) 2
and using the fact that the inclusion H 1 (N ) ↩→ L2 (N ) is compact. Indeed, the bilinear form (φ1 , φ2 ) → ⟨φ1 , φ2 ⟩H 1 is obviously represented by the identity of H 1 (N ), and the bilinear form H 1 × H 1 ∋ (φ1 , φ2 ) → ⟨φ1 , φ2 ⟩L2 has a continuous extension to L2 × L2 . Since the inclusion i : H 1 ↩→ L2 is compact, the latter bilinear form is represented by a compact operator on H 1 , according to Lemma 2. Lemma 2. Assume that H1 , H2 are Hilbert spaces, and that i : H1 → H2 is a compact inclusion with dense image. If B : H1 × H1 → R is a bounded (symmetric) bilinear form that admits a continuous extension to H2 , then B is represented by a compact (self-adjoint) operator TB : H1 → H1 , i.e., B(x, y) = ⟨TB (x), y⟩1 for all x, y ∈ H1 . Proof. Let B be the extension of B to H2 , and denote by T : H2 → H2 the (self-adjoint) bounded operator that represents B. Then, TB = i∗ ◦ T ◦ i, where i∗ : H2 → H1 is the adjoint of i. Since i is compact, then TB is compact (and self-adjoint). Remark 3. Using the local representation for C 1 -maps having Fredholm derivative as given in [15, Theorem 1.7, p. 4] and a classical argument (see for instance [16]), from part (d) of Proposition 1 one obtains that A satisfies a local (PS) condition around u0 . 1 See [13] for the definition and the main properties of Lebesgue and Sobolev spaces of functions on Riemannian manifolds.
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2.2. Jacobi operator and Morse index Let us now assume that g is a metric on N with Scalg constant, so that u0 ≡ 1 is a critical point of A. Determining an explicit formula for the Fredholm operator in (5) is rather awkward, and in fact unnecessary. The second derivative d2 A(u0 ) can be easily described in terms of Fredholm operators if one uses the standard L2 -pairing, rather than the H 1 -inner product. Namely, for smooth φ1 , partial integration in (4) gives: d2 A(u0 )(φ1 , φ2 ) =
(n−1)(n−2)
2
Jg (φ1 ) · φ2 volg ,
(6)
N
where Jg is the linear elliptic operator:
Jg = ∆g −
Scalg n−1
,
(7)
and ∆g is the (positive) Laplace–Beltrami operator of g. One can look at Jg either as an unbounded self-adjoint Fredholm operator on L2 (N ), or as a bounded symmetric Fredholm operator from H 2 (N ) to L2 (N ). In what follows, we argue that the nullity and Morse index of (Proposition 5). It must be observed that Jg preserves the space L2∗ (N ) both of these operators coincide 2 2 of L -maps φ satisfying N φ volg = 0 (it is the L -orthogonal of the space of constant functions, which is clearly invariant by Jg ). The spectrum of the restriction of Jg to L2∗ (N ) consists of an unbounded non decreasing sequence of finite multiplicity eigenvalues {λi (Jg )}i≥1 : Scal
− n−1g < λ1 (Jg ) ≤ λ2 (Jg ) ≤ · · · ≤ λk (Jg ) −→ +∞ as k → ∞; in the above list eigenvalues are repeated according to their multiplicities. The corresponding normalized eigenfunctions e1 , . . . , ek are smooth, and they form an orthonormal basis of L2∗ (N ). Scal
Remark 4. Observe that, from (7), the spectrum of Jg is the spectrum of ∆g shifted by − n−1g , and the eigenfunctions of Jg Scal
and ∆g are the same. Observe also that λ0 = − n−1g is not included, because we restrict Jg to L2∗ (N ). Proposition 5. Assume that u0 ≡ 1 is a critical point of A in [g ]1 . Then: (i) the Morse index of u0 is finite, and it is given by: max k ≥ 1 : λk (Jg ) < 0 ;
(ii) ker d2 A(u0 ) = ker(Jg ), and the nullity of u0 , denoted with nul(u0 ) and defined as the dimension of this space, is the multiplicity2 of 0 as an eigenvalue of Jg .
Proof. Everything follows easily from (6), keeping in mind that the eigenspaces of Jg in L2∗ (N ) are contained in Tu0 [g ]1 , that such eigenspaces are d2 A(u0 )-orthogonal, and that Tu0 [g ]1 is dense in L2∗ (N ). 3. Convergence of bifurcating branches in the Yamabe problem In this section we want to clarify the topology of convergence of a bifurcating branch of solutions of the Yamabe problem when using an H 1 -approach as above. Let us consider the following setup: (a) N n is a compact manifold (without boundary), n ≥ 3; (b) [a, b] ∋ t → gt is a continuous map relatively to the C s -topology in the set of metrics, s ≥ 2, and it takes values in the set of smooth unit volume Riemannian metrics on N having constant scalar curvature equal to Scalt ; (c) t∗ ∈ ]a, b[ is a bifurcation instant for the family (gt )t ∈[a,b] , i.e., the next two conditions hold; • tr ∈ [a, b] is a sequence with limr →∞ tr = t∗ ; • ur is a sequence of smooth positive functions on N satisfying: 4
– the metric gr = urn−2 · gtr has unit volume and constant scalar curvature Scalr ; (here gr ̸= gtr for all r) – limr →∞ ur = 1 in the H 1 -topology. Proposition 6. In the above situation, the following holds: (a) limr →∞ Scalr = Scalt∗ ; (b) ur tends to 1 in W s+1,p , with p = n2n . +2 In particular, if s > m + 21 n, then ur tends to 1 in the C m -topology. g 2 or, equivalently, the multiplicity of in the spectrum of ∆g when Scalg > 0. If Scalg ≤ 0, then nul(u0 ) = 0, i.e., u0 is a nondegenerate critical point n−1 of A.
Scal
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Proof. We observe that the ur satisfies the elliptic PDE: n+2
4(n−1) ∆ r ur n−2
+ Scaltr ur = Scalr urn−2 ,
(8)
and that Scalr is given by: Scalr = Ar (ur ) =
n−1 4
N
Here, ∇
(r )
n−2
gtr ∇ (r ) ur , ∇ (r ) ur + Scaltr u2r dvolgtr .
and ∆r are respectively the Laplacian and the gradient operator relative to the metric gtr :
∆r = −
∂ ∂ xi
ij
gtr
∂ ∂ xj
,
∇ (r ) = gtijr
∂ . ∂ xi
(9)
Let us show first part (a): lim Scalr = Scalt∗ .
(10)
r →∞
To this aim, we observe:
• limr →∞ Scaltr = Scalt∗ , because gtr tends to gt∗ in the C 2 -topology; • dvolgtr = ψr · dvolgt∗ , where ψr tends to 1 in the C 2 -topology as r → ∞; • gtr ∇ (r ) ur , ∇ (r ) ur = gtijr ∂∂uxri ∂∂uxrj tends to 0 as r → ∞ in L1 (N , volgt∗ ). Equality (10) follows readily from these observations. From (8), we can write:
∆r ur = ar ur + br uqr −1 ,
(11)
with ar = 4(2n−−n1) Scaltr ,
br = 4(nn−−21) Scalr
and q = n2n ; −2
moreover: lim ar = a := 4(2n−−n1) Scalt∗ ,
r →∞
lim br = −a.
r →∞
Recalling that we have a continuous inclusion H 1 ↩→ Lq , since ur → 1 in H 1 as r → ∞, from (11) we obtain that: r →∞
∆r ur = fr := ar ur + br uqr −1 −→ 0 in Lp
(12)
with p = q−1 = > 1. Last inequality follows from n ≥ 3. Then, we claim that ur tends to the constant function 1 in W 2,p . This follows from standard Lp estimates, with p > 1, on the solution of a second order elliptic equation, see for instance [11, Theorem 9.14, p. 240]: q
2n n+2
∥ur − 1∥W 2,p ≤ Cr · ∥fr − σr (ur − 1)∥Lp ,
(13)
where the constants Cr and σr depend on:
• the L∞ -norm of the coefficients of the elliptic operator ∆r ; • the moduli of continuity of the coefficients of ∆r ; • the ellipticity constant of ∆r . ij
The coefficients of ∆r are expressed in terms of the metric coefficients gtr , see (9), and they tend uniformly to the coefficients of the elliptic operator ∆t∗ . Hence, in (13) we can choose coefficients Cr ≡ C and σr ≡ σ that do not depend on r, and obtain that ur tends to 1 in W 2,p , which proves our claim.3 We can now use induction in Eq. (11). We observe that the derivatives ∂s u satisfy an equation of the form (11), with the ij addition of a first order linear term in Du involving the first and second derivatives of the metric coefficients gtr . This says that the procedure can be iterated exactly (s − 1)-times, where s is the order of differentiability in the space of metrics, obtaining that ur tends to 1 in W 1+s,p . This proves (b). As to the last statement of the Proposition, it follows easily from the continuous inclusion W s+1,p ⊂ C m , which holds when s > m − 1 + np = m + 12 n, see for instance [13, Theorem 2.7, p. 34].
3 Observe that the Sobolev inclusion W 2,p ⊂ Lq holds precisely when q = 2n , see [13, Theorem 2.6, p. 32], so that (12) cannot be improved. n−2
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4. The basic geometrical setup Let us recall the following result of M. Gromov and H.B. Lawson, which characterizes the metric of nonnegative scalar curvature on the torus: Theorem (Gromov–Lawson). Any metric of non-negative scalar curvature on the torus Tk is flat. Proof. See [17].
Our aim is to show that, on the other hand, given any compact manifold M admitting a positive scalar curvature metric, then the product Tk × M admits uncountably many positive constant scalar curvature metrics that are not product metrics. Let (M m−k , g ) be a unit volume Riemannian manifold, with m ≥ k + 2, having constant scalar curvature Scalg . Let g flat be a unit volume flat metric on Tk . The product M = Tk × M is an m-dimensional manifold, and when it is endowed with the product metric g = g flat ⊕ g, it has unit volume and constant scalar curvature Scalg = Scalg , i.e., g is a critical point of the Hilbert–Einstein functional defined in the set of unit volume metrics in its conformal class. The second variation at g of the Hilbert–Einstein functional is represented by the self-adjoint Fredholm operator
Jg = ∆g −
Scalg m−1
,
(14)
defined in the space of functions f : M → R satisfying M f dvolg = 0. The spectrum of Jg consists of a countable set of finite multiplicity eigenvalues, and using (14) we get:
Spec Jg = λi + µj −
Scalg m−1
:i+j>0 ,
(15)
where: 0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λn → +∞ is the sequence of eigenvalues of the Laplace–Beltrami operator of (Tk , g flat ), and 0 = µ0 < µ1 ≤ µ2 ≤ · · · ≤ µn → +∞ is the sequence of eigenvalues of the Laplace–Beltrami operator of (M , g ). The Morse index of g as a critical point of the Hilbert–Einstein functional, is the number of elements in Spec Jg , counted with multiplicity, that are less than 0. If λi + µj − Hilbert–Einstein functional.
Scalg m−1
= 0 for some i + j > 0, then g is a degenerate critical point of the
5. Bifurcation of constant scalar curvature metrics in Tk × M Our aim is to consider a continuous path I ∋ s → gsflat , with I ⊂ R an interval and gsflat a unit volume flat metric on Tk , and to establish under which conditions bifurcation issuing from the corresponding path of metrics s → g s := gsflat ⊕ g occurs. More precisely, given a curve I ∋ s → g s , we say that an instant s∗ ∈ I is a bifurcation instant if there exists a sequence (sj )j≥1 in I and a sequence of unit volume constant scalar curvature metrics (g j )j≥1 in Tk × M, such that: (i) limj→∞ sj = s∗ and limj→∞ g j = g s ; (ii) g j is conformal to g sj for all j; (iii) g j ̸= g sj for all j. In this situation, we say that the metrics (g j )j≥1 form the bifurcating branch of solutions of the Yamabe problem issuing from the curve s → g s at the instant s∗ . If s∗ is not a bifurcation instant, then we say that the family g s is locally rigid at s∗ ; if there are no bifurcation instants in the interval I, we say that the family g s is locally rigid. A necessary condition for bifurcation at s = s∗ is that g s∗ be degenerate; this follows easily from the Implicit Function Theorem, see for instance [2, Proposition 3.1]. As an immediate corollary of this observation, we get that a necessary condition for the existence of bifurcation instant for a path of the form s → gsflat ⊕ g is that Scal(g ) > 0: Corollary 7. If Scal(g ) ≤ 0, then any path of constant scalar curvature metrics s → g s = gsflat ⊕ g on Tk × M is locally rigid. Scal(g )
Proof. If Scal(g ) ≤ 0, then, using (15), any eigenvalue λi + µj − m−1 , i + j > 0, of Jg s is positive. This implies that every g s is a nondegenerate critical point of the Hilbert–Einstein functional. We will henceforth assume that Scal(g ) > 0, and we will prove that in this situation there exist uncountably many paths of the form s → gsflat ⊕ g admitting bifurcation instants.
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Initially, it is important to point out that bifurcating branches produce non trivial solutions of the Yamabe problem on Tk × M. More precisely: Proposition 8. In the above bifurcation setup, any metric g j in a bifurcating branch issuing from the curve s → g s = gsflat ⊕ g is not a product metric on Tk × M. Proof. This result follows easily from the observation that two product metrics are conformal if and only if they are homothetic. If g j = h1 ⊕ h2 , then h1 ⊕ h2 is homothetic to gsflat ⊕ g. Since both metrics have unit volume, they must be j isometric, and therefore h1 is isometric to gsflat and h2 is isometric to g. This is impossible, because by definition of bifurcation, j g j is not isometric to g sj .
For the result of this paper, we will use the following bifurcation criterion for constant scalar curvature metrics. m
Proposition 9. Let M be a compact manifold without boundary, with m ≥ 3, and let [s∗ − ε, s∗ + ε ] ∋ s → g s be a path of unit volume constant scalar curvature metrics on M, which is continuous with respect to the C r -topology of metrics, with r ≥ 2. Denote by iMorse (g s ) the number of eigenvalues of ∆g s , with multiplicity, that are less than Scalg s /(m − 1), cf. Proposition 5(i). Assume the following: (a) for all s ̸= s∗ , g s is nondegenerate (i.e., either Scalg s = 0 or Scalg s /(m − 1) does not belong to the spectrum of ∆g s ); (b) iMorse (g s∗ −ε ) ̸= iMorse (g s∗ +ε ). Then, s∗ is a bifurcation instant for the family (g s )s . Proof. The proof of this criterion is known in the literature, and implicitly assumed in several articles, as for instance [3,5,2]. 6. Proof of theorem Using the bifurcation criterion given in Proposition 9 and formula (15), a proof of our Theorem will be obtained by proving the existence of uncountably many continuous paths [0, 1] ∋ s → gsflat of unit volume flat metrics on Tk , with endpoints g0flat and g1flat satisfy: (a)
Scal(g ) m−1
̸∈ Spec(∆g flat )
Spec(∆g flat ); 1
0
(b) the number (with multiplicity) of eigenvalues of ∆g flat that are less than 0
multiplicity) of eigenvalues of ∆g flat that are less than 1
Scal(g ) m−1
is different from the number (with
Scal(g ) . m−1
Namely, (a) corresponds to the nondegeneracy assumption at the endpoints of the path gsflat ⊕ g, and (b) corresponds to the jump of Morse index assumption, in Proposition 9. The space of (unit volume) flat metrics of Tk is a connected real-analytic manifold, and the existence of continuous paths as above will be easily deduced from the following two facts:
• for every positive real number λ, the set of unit volume flat metrics on Tk whose Laplacian does not contain λ in its spectrum is dense in the set of unit volume flat metrics on Tk ;
• there exist unit volume flat metrics on Tk whose Laplacian has arbitrarily small eigenvalues, i.e., for all ε > 0 and N ∈ N, there exists g flat such that λi (Tk , g flat ) < ε for all i = 0, . . . , N. In order to give a proof of these facts, let us recall a few things about the spectrum of a flat metric on Tk . Given a lattice Γ ⊂ Rk , the spectrum of the flat k-torus Rk /Γ is computed in terms of the dual lattice Γ ∗ , see [18, p. 146]. Let B = (v1 , . . . , vk ) ∈ GL(k, R) be a matrix whose columns form a basis of Rk and let Γ be the lattice associated to B, that is,
Γ = spanZ B =
ai v i : ai ∈ Z .
i
The columns of the matrix (BT )−1 , the inverse of the transpose of B, give a basis of the dual lattice, i.e, Γ ∗ = spanZ (BT )−1 . The condition that the flat torus Rk /Γ has unit volume is encoded by det B = 1. The spectrum of Rk /Γ is given by all numbers of the form 4π 2 ∥w∥2 , where w ∈ Γ ∗ . Setting, e.g., Bt := (t v1 , 1t v2 , . . . , vk ), t ≥ 1, we obtain a 1-parameter family of lattices Γt = spanZ Bt starting from Γ1 = Γ , with dual lattices Γt∗ = spanZ (BTt )−1 . Let wt ∈ Γt∗ be the first column vector of (BTt )−1 ; it satisfies limt →+∞ ∥wt ∥ = 0. Thus, given ε > 0 and N ∈ N, there exists t > 1 such that 4π 2 b2 ∥wt ∥2 < ε for all b = −N , . . . , N, i.e., Rk /Γt has arbitrarily small eigenvalues as t → +∞. As to the density of unit volume flat metrics on Tk whose Laplacian does not contain a fixed positive number λ in its spectrum, this can be argued as follows. In the first place, one observes that it suffices to show just one unit volume flat metric on Tk whose Laplacian does not contain λ in its spectrum. This follows from the fact that given any real-analytic connected
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manifold S of Riemannian metrics on a compact manifold, then for every λ > 0, the subset S ′ = g ∈ S : λ ̸∈ Spec(∆g ) is either empty or dense in S. Now, given λ > 0, a unit volume flat metric on Tk whose Laplacian does not contain λ in its spectrum is given by considering the lattice Γ ⊂ Rk whose dual lattice is spanned by the vectors:
v1 = (a, 0, . . . , 0),
v2 = (0, a, 0, . . . , 0), . . . vk−1 = (0, . . . , a, 0), 4π 2 2 ̸∈ λ m : m ∈ N .
vk = (0, . . . , 0, a1−k ),
where a2 > 4πλ 2 and a2k−2 This concludes the proof. 7. The case k = 2
In the case k = 2, one can make more explicit calculations, and exhibit concrete examples of paths of flat metrics in T2 producing bifurcation of constant scalar curvature metrics in T2 × M. For (r , θ ) ∈ ]0, +∞[×]0, π[, let T2r ,θ denote the unit area flat torus R2 /Γr ,θ , where Γr ,θ is the lattice in R2 generated by:
√
1 r sin θ
(1, 0)
and
1
√
r sin θ
(r cos θ , r sin θ ) .
The dual lattice Γr∗,θ is generated by
√
1 r sin θ
(r sin θ , −r cos θ ),
and
√
1 r sin θ
(0, 1).
Denote by gr ,θ the flat metric on T2 (r , θ ) induced by R2 . It is well known that every unit volume flat metric on T2 is isometric to exactly one gr ,θ . Denote by g r ,θ the product metric gr ,θ ⊕ g on T2 × M, and by J (r , θ ) the corresponding Jacobi operator, as in (14). As in (15), the spectrum of J (r , θ ) is now computed explicitly:
: yi ∈ Γr∗,θ and µj ∈ Spec(M , g ) := λa,b,j (r , θ ) : a, b ∈ Z, j ∈ N ∪ {0}, |a| + |b| + j > 0
Spec J (r , θ ) = 4π 2 |yi |2 + µj −
Scalg m−1
where
λa,b,j (r , θ ) = 4π 2
a2 r 2 +b2 r sin θ
− 2ab cot θ + µj −
Scalg m−1
,
and again 0 = µ0 < µ1 ≤ µ2 ≤ · · · ≤ µj −→ +∞ is the set of eigenvalues of the Laplace–Beltrami operator of the metric g. The Morse index of g r ,θ , denoted by iMorse (r , θ ) is the number of such eigenvalues, counted with multiplicity, that are less than 0. We will now exhibit paths s → (rs , θs ) ∈ ]0, +∞[ × ]0, π[ where a change of sign of the eigenvalue functions s → λa,b,j (rs , θs ) occur. Towards this goal, it will be useful to study the level curves λa,b,j (r , θ ) = 0 in the half-plane (r , θ ) ∈ ]0, +∞[ × ]0, π[. For all j ≥ 0, let us set:
αj =
1 4π 2
Scalg m−1
− µj ;
(16)
since µ0 = 0 and limj→∞ µj = +∞, there exists j0 ≥ 0 such that αj > 0 for j ∈ {0, . . . , j0 } and αj ≤ 0 for j > j0 , i.e.: j0 = max j ≥ 0 : αj > 0 .
(17)
Let us set: F (a, b, r , θ ) :=
a2 r 2 + b 2 r sin θ
− 2ab cot θ ,
and let us study the level curves F (a, b, r , θ ) = αj , j ∈ {0, . . . , j0 }, with (a, b) ̸= (0, 0) fixed; let us use coordinates (x, y) ∈ R × R+ in the half-plane. We observe that F (a, b, r , θ ) = F (−a, −b, r , θ ); moreover: F (a, −b, r , θ ) = F (a, b, r , π − θ ).
(18)
For a = 0, we obtain a sequence of horizontal lines, denoted by Lb,j , with b ∈ Z \ {0} and j ∈ {0, . . . , j0 }, given by the 2
2
equation y = bα ; as |b| → +∞, this lines diverge to infinity. The eigenvalue λ0,b,j is negative in the half-space y > bα , and j j 2
positive in the strip 0 < y < bα . j
H.F. Ramírez-Ospina / Nonlinear Analysis 109 (2014) 103–112
111
a2 r 2 +b2
α
α
When a ̸= 0, the curve r sin θ − 2ab cot θ = αj is a circumference Ca,b,j with center in a , 2aj2 and of radius 2aj2 ; it is tangent to the x axis. The eigenvalue λa,b,j is negative inside this circumference, and positive outside. Observe that Ca,b,j = C−a,−b,j , and that C−a,b,j is symmetric to Ca,b,j with respect to the y-axis. Observe also that Ca,b,j is contained in the strip 0 < y ≤ α0 .
b
Proposition 10. Let s → (rs , θs ) be any C 1 -path that crosses transversally one of the curves Lb,j , with b ∈ Z \ {0} and j ∈ {0, . . . , j0 }, or Ca,b,j , with a ∈ Z \ {0}, b ∈ Z and j ∈ {0, . . . , j0 } at the instant s∗ . Then, s∗ is a bifurcation instant for the corresponding path of metrics s → g rs ,θs . Proof. By our analysis above, each instant of transversal intersection with the given curves corresponds to the change of sign of one of the eigenvalues λa,b,j (rs , θs ), i.e., a jump of the Morse index iMorse (rs , θs ). The conclusion follows from Proposition 9. Examples. As examples, let us consider the following family of paths in the upper half-plane:
• half-lines ]0, +∞[ ∋ s → (s, θ0 ), with θ0 ∈ ]0, π[ fixed; • half-circles ]0, π[ ∋ s → (r0 , s), withr0 ∈ ]0, +∞[ fixed; • horizontal lines ]0, π[ ∋ s → sinc s , s , with c ∈ ]α0 , +∞[ fixed. Theorem 11. Assume Scalg > 0. (a) For every θ0 ∈ ]0, π[ fixed, there exists an unbounded increasing sequence of bifurcation instants (sk )k≥1 for the curve of metrics ]0, +∞[ ∋ s → g s,θ0 on T2 × M; (b) For every r0 ∈ ]0, +∞[ fixed, the set Br0 ⊂ ]0, π[ of bifurcation instants for the path of metrics ]0, π[ ∋ θ → g r0 ,θ is symmetric with respect to π2 . Set
ℓr 0 = If
ℓr0 r0
<
inf
(a,b)∈Z2 \{(0,0)}
Scalg 4π 2 (m−1)
2 2 a r − b2 = 0.
(19)
0
, then Br0 ̸= ∅. If ℓr0 = 0, then Br0 is infinite, and its closure contains 0 and π .
(c) For all but a countable number of values c ∈ ]α0 , +∞[, the path of metrics ]0, π[ ∋ s → g
c ,s sin s
is locally rigid.
α
Proof. For the proof of (a), it suffices to observe that, for s > sin0θ , the unique intersections between the curve s → (s, θ0 ) 0 and the level curves λa,b,j = 0 occur when a = 0, j ∈ {0, . . . , j0 }, and all b sufficiently large. As we have observed, these level curves form a family of horizontal line diverging to ∞. At each such intersection there is a (positive) jump of Morse index, and therefore it corresponds to a bifurcation instant. The first statement in (b) follows readily from (18). A direct computation gives the following:
inf
a2 r 2 + b 2 r sin θ
θ∈]0,π [
|a2 r 2 − b2 | − 2ab cot θ = ; r
(20)
the infimum being actually a minimum when a2 r 2 − b2 ̸= 0, and
sup
θ∈]0,π [
a2 r 2 + b 2 r sin θ
− 2ab cot θ = +∞.
(21)
It is also easy to check that the function θ → F (a, b, r , θ ) is strictly convex in ]0, π[. Using (20), (21) and the convexity ℓr0
Scal
< 4π 2 (mg−1) , and that when ℓr0 = 0 there are infinitely many instants θ ∈ ]0, π[ where some eigenvalue function θ → λa,b,j (r0 , θ ) changes its sign. For the proof of (c), observe that the curve ]0, π[ ∋ s → sinc s , s in the upper half plane is the horizontal line y = c. |b|2 When c is greater than α0 and it does not belong to the countable set α , b ∈ Z, j ∈ {0, . . . , j0 } , then there are j no intersections between the line y = c and the level curves λa,b,j = 0. Thus, g c ,s is never degenerate, hence locally sin s property, ones sees that Br0 ̸= ∅ when
rigid.
r0
8. Further developments A natural generalization of the bifurcation results discussed in this paper would be to consider products of the form F × M, where F is a compact k-manifold admitting a flat metric. By a classic result of Bieberbach, such a manifold F admits a finite locally isometric covering by a flat torus Tk ; in this situation, the spectrum of the Laplacian of F is a subset of the spectrum of the flat torus. We observe however that multiplicity of constant scalar curvature metrics on Tk × M does not yield multiplicity of constant scalar curvature metrics in F ×M, since it may not be the case that a non-product constant scalar curvature metric on Tk × M can be found invariant by the group of deck transformation of the covering Tk × M → F × M.
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As it has been discussed in Section 6, an extension of the bifurcation results for constant scalar curvature metrics in a product F × M would require a proof of the existence of unit volume flat metrics on F with arbitrarily small eigenvalues. Since F admits a finite covering by a torus, then its spectrum is contained in the spectrum of a flat torus Rk /Γ , see [19]. Again, it is not clear whether deformations of Γ that produce arbitrarily small eigenvalues of the Laplacian of Rk /Γ can be made invariant under the group of deck transformations of this covering. Also here the case k = 2 is special, and it provides a concrete situation where all computations can be made explicitly. Besides the flat torus T2 , the only other compact manifold that admits a flat metric is the Klein bottle K . One has a double covering T2 → K ; for each x, y > 0, there is a flat Klein bottle K (x, y) such that R2 /Γ → K (x, y) is a Riemannian (i.e., locally 2 isometric) covering, where Γ is the lattice givenby spanZ {xe1 , ye2 }, being {e1 , e2 } the canonical basis of R . The volume of 2 R /Γ is equal to xy, hence if xy = 2, vol K (x, y) = 1. The spectrum of the Laplacian of K (x, y) can be computed explicitly (see [18, p. 153]):
Spec K (x, y) = 4π 2
a2 x2
+
b2 y2
: a, b ∈ Z, b ̸= 0 if a is odd .
By setting x = t, y = 1t , t ≥ 1, we obtain a path of unit volume flat metrics gt on K with arbitrarily small eigenvalues. Thus, given any unit volume compact manifold (M , g ) with positive scalar curvature, one has infinitely many bifurcation instants along the path of metrics g t = gt ⊕ g on K × M. Acknowledgments This work was partially supported by MINECO (Ministerio de Economía y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037. An original version of this paper only dealt with the case k = 2. The natural extension of the result to general products of the form Tk × M, with k ≥ 2, as well as a number of very interesting observations, were suggested in a detailed and very instructive report by an anonymous referee, whose support is very thankfully acknowledged by the author. Among other things, the material presented in Section 6 and in Section 8 has been suggested by the referee. References [1] J. Petean, Metrics of constant scalar curvature conformal to Riemannian products, Proc. Amer. Math. Soc. 138 (8) (2010) 2897–2905. [2] L.L. de Lima, P. Piccione, M. Zedda, On bifurcation of solutions of the Yamabe problem in product manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2) (2012) 261–277. [3] R.G. Bettiol, P. Piccione, Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions, Pacific J. Math. 266 (1) (2013) 1–21. [4] L.L. de Lima, P. Piccione, M. Zedda, A note on the uniqueness of solutions for the Yamabe problem, Proc. Amer. Math. Soc. 140 (12) (2012) 4351–4357. [5] R.G. Bettiol, P. Piccione, Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres, Calc. Var. Partial Differential Equations 47 (3–4) (2013) 789–807. [6] J. Petean, Degenerate solutions of a nonlinear elliptic equation on the sphere, Nonlinear Anal. 100 (2014) 23–29. [7] M. Holst, C. Meier, An alternative between non-unique and negative Yamabe solutions to the conformal formulation of the Einstein constraint equations, arXiv:1306.1210. [8] B. Farb, D. Margalit, A Primer on Mapping Class Groups, Princeton University Press, Princeton, NJ, 2012. [9] R.G. Bettiol, P. Piccione, B. Santoro, Bifurcation of periodic solutions to the singular Yamabe problem on spheres, preprint 2014, arXiv:1401.7071. [10] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (1968) 265–274. [11] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, in: Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. [12] D. Fortunato, G. Palmieri, Remarks on the Yamabe problem and the Palais–Smale condition, Rend. Semin. Mat. Univ. Padova 75 (1986) 47–65. [13] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, in: Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [14] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in: Topics in Calculus of Variations (Montecatini Terme, 1987), in: Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. [15] R. Abraham, J. Robbin, Transversal Mappings and Flows, W. A. Benjamin Inc., 1967. [16] A. Marino, G. Prodi, Metodi perturbativi nella teoria di Morse, Boll. Unione Mat. Ital. (4), Suppl. fasc. 11 (3) (1975) 1–32. [17] M. Gromov, H.B. Lawson, Spin and scalar curvature in the presence of a fundamental group I, Ann. Math. 11 (2) (1980) 209–230. [18] M. Berger, P. Gauduchon, E. Mazet, Le spectre dune varit riemannienne, 194, Springer-Verlag, Berlin-New, 1971. [19] T. Sunada, Spectrum of a compact flat manifold, Comment Math. Helv 53 (4) (1978) 613–621.
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Multiplicity of constant scalar curvature metrics in Tk × M Héctor Fabián Ramírez-Ospina ∗ Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
article
info
Article history: Received 27 March 2014 Accepted 11 June 2014 Communicated by Enzo Mitidieri MSC: 58E11 58J55 58E09
abstract We use bifurcation theory to prove the existence of uncountably many unit volume constant scalar curvature metrics in manifolds of the form Tk × M, k ≥ 2, with M a compact manifold (without boundary), that are conformal, but not isometric, to a product metric g flat ⊕ g, where g flat is a flat metric on the k-torus Tk , and g is a fixed constant scalar curvature metric on M. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Bifurcation theory Yamabe problem
1. Introduction Bifurcation theory provides a powerful technique for determining the existence of multiple solutions of several geometric variational problems. In particular, great attention has been given recently to the question of multiplicity of (unit volume) constant scalar curvature metrics in conformal classes of metrics on compact manifolds. For instance, in [1] spectral theory of the linearized Yamabe problem is used to determine the existence of multiple solutions in manifolds given by the product of a round sphere and an arbitrary compact manifold. In [2] the authors prove the existence of a countable number of bifurcating branches of solutions of the Yamabe problem in product manifolds M1 × M2 stemming from a trivial path of solutions given by (a unit volume normalization of) ]0, +∞[ ∋ t → g1 ⊕ tg2 , where g1 and g2 are constant scalar curvature metrics on M1 and M2 respectively. It is important to observe that, for the existence of bifurcation, one must assume that at least one of the two factors has positive scalar curvature. A somewhat more general topological situation is considered in [3], where the authors study bifurcation of solutions of the Yamabe problem in manifolds that are the total space of a Riemannian fibration, when the size of the fibers shrinks to 0 and the manifold collapses onto its base. On the opposite side, spectral theory is used in [4] to determine local rigidity for families of constant scalar curvature metrics, under suitable topological assumptions. In [5], bifurcation theory is employed to determine multiple solutions forming branches issuing from families of homogeneous metrics on spheres. In [6], bifurcation theory is applied in order to obtain existence of degenerate solutions of a certain nonlinear elliptic solution on the sphere Sn , yielding also multiplicity results for scalar curvature metrics on the product of two spheres. Bifurcation techniques for constant scalar curvature are also important in General Relativity, see for instance [7]. In all the above results, the main idea was to consider bifurcation of constant scalar curvature metrics issuing from paths of product metrics obtained by rescaling the volume of the two factors, or rescaling the volume of the fibers of a fibration.
∗
Tel.: +34 868883666. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.na.2014.06.014 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
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In the present paper we consider a different situation, by taking the product of a flat k-torus Tk and a unit volume fixed constant scalar curvature Riemannian manifold (M , g ). We consider paths of metrics in Tk × M of the form s → gs ⊕ g, where s → gs is a path of unit volume flat metrics on Tk . Recall that, when k = 2, the space of isometry classes of unit volume flat metrics on T2 has dimension 2, and it can be naturally identified with the hyperbolic plane H2 (see [8, see Prop 10.2]). More generally, the space of isometry classes of flat metrics on the k-torus is identified with the set of isometry classes of (co-compact) lattices in Rk . By studying the spectral properties of the linearized scalar curvature operator along this type of paths, we obtain a proof of the following result. Theorem. Let k ≥ 2 be a fixed integer, and let (M m−k , g ) be a unit volume compact Riemannian manifold with m ≥ k + 2, having positive constant scalar curvature Scalg . Then, there are uncountably many branches of unit volume constant scalar curvature metrics in the product M = Tk × M, each of which belongs to the conformal class of (but is not isometric to) a product metric of the form g flat ⊕ g, with g flat a unit volume flat metric on Tk . Each bifurcating branch contains sequences of metrics (g n )n∈N whose (constant) scalar curvature tends to Scalg , and g n tends to a product metric g flat ⊕ g in the C ∞ -topology of metrics as n → ∞. Let us recall here that a well known result by Gromov and Lawson (see Section 4) says that every metric with non-negative scalar curvature on a torus Tk must be flat. Our result above implies in particular the existence of positive constant scalar curvature on Tk × M, which is not a product metric, for ‘‘arbitrarily small’’ compact differentiable manifold M admitting a Riemannian metric of positive scalar curvature. A proof of the theorem will be given in Section 6. The key point of the proof is the fact that, in order to produce bifurcation, one can find continuous paths of unit volume flat metrics on the torus Tk where one of the two endpoints is a metric whose Laplacian has an arbitrary number of eigenvalues that are arbitrarily small. For the case k = 2, a more precise description of the flat metrics that produce bifurcating branches will be given in Section 7, see Proposition 10 and Theorem 11. Motivation for the above result was given by the recent paper [9], where a similar bifurcation problem is studied in the context of singular Yamabe solutions in spheres that blowup on an equator. In that case, the authors consider products of the type Σ × M, where Σ is an orientable closed surface of genus greater than 1, endowed with a metric varying in the set of hyperbolic metrics (i.e., with curvature equal to −1). When M is a round sphere Sm , m ≥ 3, then constant scalar curvature metrics on Σ × M correspond to solutions of the singular Yamabe problem in Sm+2 \ S1 that are conformal to the round metric. The paper is organized as follows. In Section 2 we recall the analysts’ approach to the Yamabe problem, using a Sobolev H 1 -framework, which is suitable for the purposes of the present paper. This approach is based on a fundamental regularity result proved by Trudinger, see [10, Theorem 3]. We will recall here some basic facts on the first and the second variation formula for the Hilbert–Einstein functional, see (3). In Section 3 we discuss a convergence result for bifurcating branches of solutions of the Yamabe problem, which was somewhat overlooked in the existing literature. This is based on suitable Lp estimates, with p > 1, on the solution of a second order elliptic equation, see for instance [11, Theorem 9.14, p. 240]. In Section 4 we study the spectrum of the Jacobi operator for the Yamabe problem in the case of product metrics of the form gsflat ⊕ g as above. In Section 5 we prove that a necessary condition for bifurcation for paths of the form gsflat ⊕ g is that Scal(g ) > 0 (Corollary 7), we give an abstract result on bifurcation of metrics in T2 × M (Proposition 9), and we prove that bifurcating branches necessarily consist of non product metrics, see Proposition 8. Section 6 contains the proof of our Theorem for general k ≥ 2. Section 7 contains a more detailed analysis of the case k = 2, where more explicit computations are made, and concrete examples of paths containing bifurcation points are exhibited (Proposition 10 and Theorem 11). Finally, Section 8 contains a brief discussion on a further possible generalization of the results of the present paper to the case of products F × M, where F is a compact manifold admitting a flat metric. 2. Variational formulation of the Yamabe problem In this section we will recall the original variational framework for constant scalar curvature metrics in a conformal class, see the classical paper [10], or [12]. 2.1. The variational problem in H 1 Let N be a closed (i.e., compact and without boundary) smooth manifold with dim(N ) = n ≥ 3, and let g be a (smooth) Riemannian metric on N having volume vol(g ) = 1; for x ∈ N, denote by Scalg (x) the scalar curvature of g at x. Given a 4
positive smooth function u on N, let us consider the conformal metric g ′ = u n−2 g. It is easily computed that vol(g ′ ) = vol(g ) if and only if:
2n
u n−2 volg = 1. N
(1)
H.F. Ramírez-Ospina / Nonlinear Analysis 109 (2014) 103–112
105
Consider the Sobolev1 space H 1 (N ), endowed with the Hilbert norm:
∥φ∥2H 1 =
2 |φ| + g (∇φ, ∇φ) volg , N
and we denote by ⟨·, ·⟩H 1 the corresponding inner product. By the Gagliardo–Nirenberg–Sobolev inequality, there is a 2n
continuous inclusion H 1 (N ) ⊂ L n−2 (N ) (see for instance [13, Theorem 2.6, p. 32]), and the subset
[g ]1 := u ∈ H (N ) : u > 0, 1
u
2n n−2
volg = 1
N
is a smooth submanifold of H 1 (N ). Indeed, the number 1 is a regular value of the map I
{u ∈ H 1 (N ) : u > 0} ∋ u −→
2n
u n−2 volg . N
The differential of this map is dI (u)φ =
2n n−2
n+2
u n−2 φ volg . N
φ volg . Thus, at u0 ≡ 1, the tangent space Tu0 [g ]1 is given by φ volg = 0 .
At u0 ≡ 1, this gives dI (u0 )φ = n2n −2
Tu0 [g ]1 =
φ ∈ H 1 (N ) :
N
(2)
N
Consider the functional A : [g ]1 → R defined by:
A(u) :=
−1 g (∇ u, ∇ u) + scalg (x)u2 volg . 4 nn− 2
N
(3)
Proposition 1. The following statements hold: (a) A is smooth on [g ]1 ; (b) u ∈ [g ]1 is a critical point of A if and only if u is smooth, and g ′ = u · g has constant scalar curvature. Moreover, if g has constant scalar curvature (i.e., if u0 ≡ 1 is a critical point of A), then: (c) the second derivative of A at u0 ≡ 1 is the symmetric bilinear form: d A(u0 )(φ1 , φ2 ) = 2
(n−1)(n−2) 2
N
Scal
g (∇φ1 , ∇φ2 ) − n−1g φ1 φ2 volg .
(4)
(d) There exists a self-adjoint operator Fg : Tu0 [g ]1 → Tu0 [g ]1 , given by a compact perturbation of the identity, such that: d2 A(u0 )(φ1 , φ2 ) =
(n−1)(n−2) 2
⟨Fg (φ1 ), φ2 ⟩H 1
(5)
for all φ1 , φ2 ∈ Tu0 [g ]1 . Proof. See [10] for a proof of (a) and (b) (especially [10, Theorem 3] for the smoothness of critical points); see [14] for (c). A proof of (d) is obtained immediately observing that:
Scal
d2 A(1)(φ1 , φ2 ) = ⟨φ1 , φ2 ⟩H 1 − 1 + n−1g ⟨φ1 , φ2 ⟩L2 (N ) , (n−1)(n−2) 2
and using the fact that the inclusion H 1 (N ) ↩→ L2 (N ) is compact. Indeed, the bilinear form (φ1 , φ2 ) → ⟨φ1 , φ2 ⟩H 1 is obviously represented by the identity of H 1 (N ), and the bilinear form H 1 × H 1 ∋ (φ1 , φ2 ) → ⟨φ1 , φ2 ⟩L2 has a continuous extension to L2 × L2 . Since the inclusion i : H 1 ↩→ L2 is compact, the latter bilinear form is represented by a compact operator on H 1 , according to Lemma 2. Lemma 2. Assume that H1 , H2 are Hilbert spaces, and that i : H1 → H2 is a compact inclusion with dense image. If B : H1 × H1 → R is a bounded (symmetric) bilinear form that admits a continuous extension to H2 , then B is represented by a compact (self-adjoint) operator TB : H1 → H1 , i.e., B(x, y) = ⟨TB (x), y⟩1 for all x, y ∈ H1 . Proof. Let B be the extension of B to H2 , and denote by T : H2 → H2 the (self-adjoint) bounded operator that represents B. Then, TB = i∗ ◦ T ◦ i, where i∗ : H2 → H1 is the adjoint of i. Since i is compact, then TB is compact (and self-adjoint). Remark 3. Using the local representation for C 1 -maps having Fredholm derivative as given in [15, Theorem 1.7, p. 4] and a classical argument (see for instance [16]), from part (d) of Proposition 1 one obtains that A satisfies a local (PS) condition around u0 . 1 See [13] for the definition and the main properties of Lebesgue and Sobolev spaces of functions on Riemannian manifolds.
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2.2. Jacobi operator and Morse index Let us now assume that g is a metric on N with Scalg constant, so that u0 ≡ 1 is a critical point of A. Determining an explicit formula for the Fredholm operator in (5) is rather awkward, and in fact unnecessary. The second derivative d2 A(u0 ) can be easily described in terms of Fredholm operators if one uses the standard L2 -pairing, rather than the H 1 -inner product. Namely, for smooth φ1 , partial integration in (4) gives: d2 A(u0 )(φ1 , φ2 ) =
(n−1)(n−2)
2
Jg (φ1 ) · φ2 volg ,
(6)
N
where Jg is the linear elliptic operator:
Jg = ∆g −
Scalg n−1
,
(7)
and ∆g is the (positive) Laplace–Beltrami operator of g. One can look at Jg either as an unbounded self-adjoint Fredholm operator on L2 (N ), or as a bounded symmetric Fredholm operator from H 2 (N ) to L2 (N ). In what follows, we argue that the nullity and Morse index of (Proposition 5). It must be observed that Jg preserves the space L2∗ (N ) both of these operators coincide 2 2 of L -maps φ satisfying N φ volg = 0 (it is the L -orthogonal of the space of constant functions, which is clearly invariant by Jg ). The spectrum of the restriction of Jg to L2∗ (N ) consists of an unbounded non decreasing sequence of finite multiplicity eigenvalues {λi (Jg )}i≥1 : Scal
− n−1g < λ1 (Jg ) ≤ λ2 (Jg ) ≤ · · · ≤ λk (Jg ) −→ +∞ as k → ∞; in the above list eigenvalues are repeated according to their multiplicities. The corresponding normalized eigenfunctions e1 , . . . , ek are smooth, and they form an orthonormal basis of L2∗ (N ). Scal
Remark 4. Observe that, from (7), the spectrum of Jg is the spectrum of ∆g shifted by − n−1g , and the eigenfunctions of Jg Scal
and ∆g are the same. Observe also that λ0 = − n−1g is not included, because we restrict Jg to L2∗ (N ). Proposition 5. Assume that u0 ≡ 1 is a critical point of A in [g ]1 . Then: (i) the Morse index of u0 is finite, and it is given by: max k ≥ 1 : λk (Jg ) < 0 ;
(ii) ker d2 A(u0 ) = ker(Jg ), and the nullity of u0 , denoted with nul(u0 ) and defined as the dimension of this space, is the multiplicity2 of 0 as an eigenvalue of Jg .
Proof. Everything follows easily from (6), keeping in mind that the eigenspaces of Jg in L2∗ (N ) are contained in Tu0 [g ]1 , that such eigenspaces are d2 A(u0 )-orthogonal, and that Tu0 [g ]1 is dense in L2∗ (N ). 3. Convergence of bifurcating branches in the Yamabe problem In this section we want to clarify the topology of convergence of a bifurcating branch of solutions of the Yamabe problem when using an H 1 -approach as above. Let us consider the following setup: (a) N n is a compact manifold (without boundary), n ≥ 3; (b) [a, b] ∋ t → gt is a continuous map relatively to the C s -topology in the set of metrics, s ≥ 2, and it takes values in the set of smooth unit volume Riemannian metrics on N having constant scalar curvature equal to Scalt ; (c) t∗ ∈ ]a, b[ is a bifurcation instant for the family (gt )t ∈[a,b] , i.e., the next two conditions hold; • tr ∈ [a, b] is a sequence with limr →∞ tr = t∗ ; • ur is a sequence of smooth positive functions on N satisfying: 4
– the metric gr = urn−2 · gtr has unit volume and constant scalar curvature Scalr ; (here gr ̸= gtr for all r) – limr →∞ ur = 1 in the H 1 -topology. Proposition 6. In the above situation, the following holds: (a) limr →∞ Scalr = Scalt∗ ; (b) ur tends to 1 in W s+1,p , with p = n2n . +2 In particular, if s > m + 21 n, then ur tends to 1 in the C m -topology. g 2 or, equivalently, the multiplicity of in the spectrum of ∆g when Scalg > 0. If Scalg ≤ 0, then nul(u0 ) = 0, i.e., u0 is a nondegenerate critical point n−1 of A.
Scal
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Proof. We observe that the ur satisfies the elliptic PDE: n+2
4(n−1) ∆ r ur n−2
+ Scaltr ur = Scalr urn−2 ,
(8)
and that Scalr is given by: Scalr = Ar (ur ) =
n−1 4
N
Here, ∇
(r )
n−2
gtr ∇ (r ) ur , ∇ (r ) ur + Scaltr u2r dvolgtr .
and ∆r are respectively the Laplacian and the gradient operator relative to the metric gtr :
∆r = −
∂ ∂ xi
ij
gtr
∂ ∂ xj
,
∇ (r ) = gtijr
∂ . ∂ xi
(9)
Let us show first part (a): lim Scalr = Scalt∗ .
(10)
r →∞
To this aim, we observe:
• limr →∞ Scaltr = Scalt∗ , because gtr tends to gt∗ in the C 2 -topology; • dvolgtr = ψr · dvolgt∗ , where ψr tends to 1 in the C 2 -topology as r → ∞; • gtr ∇ (r ) ur , ∇ (r ) ur = gtijr ∂∂uxri ∂∂uxrj tends to 0 as r → ∞ in L1 (N , volgt∗ ). Equality (10) follows readily from these observations. From (8), we can write:
∆r ur = ar ur + br uqr −1 ,
(11)
with ar = 4(2n−−n1) Scaltr ,
br = 4(nn−−21) Scalr
and q = n2n ; −2
moreover: lim ar = a := 4(2n−−n1) Scalt∗ ,
r →∞
lim br = −a.
r →∞
Recalling that we have a continuous inclusion H 1 ↩→ Lq , since ur → 1 in H 1 as r → ∞, from (11) we obtain that: r →∞
∆r ur = fr := ar ur + br uqr −1 −→ 0 in Lp
(12)
with p = q−1 = > 1. Last inequality follows from n ≥ 3. Then, we claim that ur tends to the constant function 1 in W 2,p . This follows from standard Lp estimates, with p > 1, on the solution of a second order elliptic equation, see for instance [11, Theorem 9.14, p. 240]: q
2n n+2
∥ur − 1∥W 2,p ≤ Cr · ∥fr − σr (ur − 1)∥Lp ,
(13)
where the constants Cr and σr depend on:
• the L∞ -norm of the coefficients of the elliptic operator ∆r ; • the moduli of continuity of the coefficients of ∆r ; • the ellipticity constant of ∆r . ij
The coefficients of ∆r are expressed in terms of the metric coefficients gtr , see (9), and they tend uniformly to the coefficients of the elliptic operator ∆t∗ . Hence, in (13) we can choose coefficients Cr ≡ C and σr ≡ σ that do not depend on r, and obtain that ur tends to 1 in W 2,p , which proves our claim.3 We can now use induction in Eq. (11). We observe that the derivatives ∂s u satisfy an equation of the form (11), with the ij addition of a first order linear term in Du involving the first and second derivatives of the metric coefficients gtr . This says that the procedure can be iterated exactly (s − 1)-times, where s is the order of differentiability in the space of metrics, obtaining that ur tends to 1 in W 1+s,p . This proves (b). As to the last statement of the Proposition, it follows easily from the continuous inclusion W s+1,p ⊂ C m , which holds when s > m − 1 + np = m + 12 n, see for instance [13, Theorem 2.7, p. 34].
3 Observe that the Sobolev inclusion W 2,p ⊂ Lq holds precisely when q = 2n , see [13, Theorem 2.6, p. 32], so that (12) cannot be improved. n−2
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4. The basic geometrical setup Let us recall the following result of M. Gromov and H.B. Lawson, which characterizes the metric of nonnegative scalar curvature on the torus: Theorem (Gromov–Lawson). Any metric of non-negative scalar curvature on the torus Tk is flat. Proof. See [17].
Our aim is to show that, on the other hand, given any compact manifold M admitting a positive scalar curvature metric, then the product Tk × M admits uncountably many positive constant scalar curvature metrics that are not product metrics. Let (M m−k , g ) be a unit volume Riemannian manifold, with m ≥ k + 2, having constant scalar curvature Scalg . Let g flat be a unit volume flat metric on Tk . The product M = Tk × M is an m-dimensional manifold, and when it is endowed with the product metric g = g flat ⊕ g, it has unit volume and constant scalar curvature Scalg = Scalg , i.e., g is a critical point of the Hilbert–Einstein functional defined in the set of unit volume metrics in its conformal class. The second variation at g of the Hilbert–Einstein functional is represented by the self-adjoint Fredholm operator
Jg = ∆g −
Scalg m−1
,
(14)
defined in the space of functions f : M → R satisfying M f dvolg = 0. The spectrum of Jg consists of a countable set of finite multiplicity eigenvalues, and using (14) we get:
Spec Jg = λi + µj −
Scalg m−1
:i+j>0 ,
(15)
where: 0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λn → +∞ is the sequence of eigenvalues of the Laplace–Beltrami operator of (Tk , g flat ), and 0 = µ0 < µ1 ≤ µ2 ≤ · · · ≤ µn → +∞ is the sequence of eigenvalues of the Laplace–Beltrami operator of (M , g ). The Morse index of g as a critical point of the Hilbert–Einstein functional, is the number of elements in Spec Jg , counted with multiplicity, that are less than 0. If λi + µj − Hilbert–Einstein functional.
Scalg m−1
= 0 for some i + j > 0, then g is a degenerate critical point of the
5. Bifurcation of constant scalar curvature metrics in Tk × M Our aim is to consider a continuous path I ∋ s → gsflat , with I ⊂ R an interval and gsflat a unit volume flat metric on Tk , and to establish under which conditions bifurcation issuing from the corresponding path of metrics s → g s := gsflat ⊕ g occurs. More precisely, given a curve I ∋ s → g s , we say that an instant s∗ ∈ I is a bifurcation instant if there exists a sequence (sj )j≥1 in I and a sequence of unit volume constant scalar curvature metrics (g j )j≥1 in Tk × M, such that: (i) limj→∞ sj = s∗ and limj→∞ g j = g s ; (ii) g j is conformal to g sj for all j; (iii) g j ̸= g sj for all j. In this situation, we say that the metrics (g j )j≥1 form the bifurcating branch of solutions of the Yamabe problem issuing from the curve s → g s at the instant s∗ . If s∗ is not a bifurcation instant, then we say that the family g s is locally rigid at s∗ ; if there are no bifurcation instants in the interval I, we say that the family g s is locally rigid. A necessary condition for bifurcation at s = s∗ is that g s∗ be degenerate; this follows easily from the Implicit Function Theorem, see for instance [2, Proposition 3.1]. As an immediate corollary of this observation, we get that a necessary condition for the existence of bifurcation instant for a path of the form s → gsflat ⊕ g is that Scal(g ) > 0: Corollary 7. If Scal(g ) ≤ 0, then any path of constant scalar curvature metrics s → g s = gsflat ⊕ g on Tk × M is locally rigid. Scal(g )
Proof. If Scal(g ) ≤ 0, then, using (15), any eigenvalue λi + µj − m−1 , i + j > 0, of Jg s is positive. This implies that every g s is a nondegenerate critical point of the Hilbert–Einstein functional. We will henceforth assume that Scal(g ) > 0, and we will prove that in this situation there exist uncountably many paths of the form s → gsflat ⊕ g admitting bifurcation instants.
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Initially, it is important to point out that bifurcating branches produce non trivial solutions of the Yamabe problem on Tk × M. More precisely: Proposition 8. In the above bifurcation setup, any metric g j in a bifurcating branch issuing from the curve s → g s = gsflat ⊕ g is not a product metric on Tk × M. Proof. This result follows easily from the observation that two product metrics are conformal if and only if they are homothetic. If g j = h1 ⊕ h2 , then h1 ⊕ h2 is homothetic to gsflat ⊕ g. Since both metrics have unit volume, they must be j isometric, and therefore h1 is isometric to gsflat and h2 is isometric to g. This is impossible, because by definition of bifurcation, j g j is not isometric to g sj .
For the result of this paper, we will use the following bifurcation criterion for constant scalar curvature metrics. m
Proposition 9. Let M be a compact manifold without boundary, with m ≥ 3, and let [s∗ − ε, s∗ + ε ] ∋ s → g s be a path of unit volume constant scalar curvature metrics on M, which is continuous with respect to the C r -topology of metrics, with r ≥ 2. Denote by iMorse (g s ) the number of eigenvalues of ∆g s , with multiplicity, that are less than Scalg s /(m − 1), cf. Proposition 5(i). Assume the following: (a) for all s ̸= s∗ , g s is nondegenerate (i.e., either Scalg s = 0 or Scalg s /(m − 1) does not belong to the spectrum of ∆g s ); (b) iMorse (g s∗ −ε ) ̸= iMorse (g s∗ +ε ). Then, s∗ is a bifurcation instant for the family (g s )s . Proof. The proof of this criterion is known in the literature, and implicitly assumed in several articles, as for instance [3,5,2]. 6. Proof of theorem Using the bifurcation criterion given in Proposition 9 and formula (15), a proof of our Theorem will be obtained by proving the existence of uncountably many continuous paths [0, 1] ∋ s → gsflat of unit volume flat metrics on Tk , with endpoints g0flat and g1flat satisfy: (a)
Scal(g ) m−1
̸∈ Spec(∆g flat )
Spec(∆g flat ); 1
0
(b) the number (with multiplicity) of eigenvalues of ∆g flat that are less than 0
multiplicity) of eigenvalues of ∆g flat that are less than 1
Scal(g ) m−1
is different from the number (with
Scal(g ) . m−1
Namely, (a) corresponds to the nondegeneracy assumption at the endpoints of the path gsflat ⊕ g, and (b) corresponds to the jump of Morse index assumption, in Proposition 9. The space of (unit volume) flat metrics of Tk is a connected real-analytic manifold, and the existence of continuous paths as above will be easily deduced from the following two facts:
• for every positive real number λ, the set of unit volume flat metrics on Tk whose Laplacian does not contain λ in its spectrum is dense in the set of unit volume flat metrics on Tk ;
• there exist unit volume flat metrics on Tk whose Laplacian has arbitrarily small eigenvalues, i.e., for all ε > 0 and N ∈ N, there exists g flat such that λi (Tk , g flat ) < ε for all i = 0, . . . , N. In order to give a proof of these facts, let us recall a few things about the spectrum of a flat metric on Tk . Given a lattice Γ ⊂ Rk , the spectrum of the flat k-torus Rk /Γ is computed in terms of the dual lattice Γ ∗ , see [18, p. 146]. Let B = (v1 , . . . , vk ) ∈ GL(k, R) be a matrix whose columns form a basis of Rk and let Γ be the lattice associated to B, that is,
Γ = spanZ B =
ai v i : ai ∈ Z .
i
The columns of the matrix (BT )−1 , the inverse of the transpose of B, give a basis of the dual lattice, i.e, Γ ∗ = spanZ (BT )−1 . The condition that the flat torus Rk /Γ has unit volume is encoded by det B = 1. The spectrum of Rk /Γ is given by all numbers of the form 4π 2 ∥w∥2 , where w ∈ Γ ∗ . Setting, e.g., Bt := (t v1 , 1t v2 , . . . , vk ), t ≥ 1, we obtain a 1-parameter family of lattices Γt = spanZ Bt starting from Γ1 = Γ , with dual lattices Γt∗ = spanZ (BTt )−1 . Let wt ∈ Γt∗ be the first column vector of (BTt )−1 ; it satisfies limt →+∞ ∥wt ∥ = 0. Thus, given ε > 0 and N ∈ N, there exists t > 1 such that 4π 2 b2 ∥wt ∥2 < ε for all b = −N , . . . , N, i.e., Rk /Γt has arbitrarily small eigenvalues as t → +∞. As to the density of unit volume flat metrics on Tk whose Laplacian does not contain a fixed positive number λ in its spectrum, this can be argued as follows. In the first place, one observes that it suffices to show just one unit volume flat metric on Tk whose Laplacian does not contain λ in its spectrum. This follows from the fact that given any real-analytic connected
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manifold S of Riemannian metrics on a compact manifold, then for every λ > 0, the subset S ′ = g ∈ S : λ ̸∈ Spec(∆g ) is either empty or dense in S. Now, given λ > 0, a unit volume flat metric on Tk whose Laplacian does not contain λ in its spectrum is given by considering the lattice Γ ⊂ Rk whose dual lattice is spanned by the vectors:
v1 = (a, 0, . . . , 0),
v2 = (0, a, 0, . . . , 0), . . . vk−1 = (0, . . . , a, 0), 4π 2 2 ̸∈ λ m : m ∈ N .
vk = (0, . . . , 0, a1−k ),
where a2 > 4πλ 2 and a2k−2 This concludes the proof. 7. The case k = 2
In the case k = 2, one can make more explicit calculations, and exhibit concrete examples of paths of flat metrics in T2 producing bifurcation of constant scalar curvature metrics in T2 × M. For (r , θ ) ∈ ]0, +∞[×]0, π[, let T2r ,θ denote the unit area flat torus R2 /Γr ,θ , where Γr ,θ is the lattice in R2 generated by:
√
1 r sin θ
(1, 0)
and
1
√
r sin θ
(r cos θ , r sin θ ) .
The dual lattice Γr∗,θ is generated by
√
1 r sin θ
(r sin θ , −r cos θ ),
and
√
1 r sin θ
(0, 1).
Denote by gr ,θ the flat metric on T2 (r , θ ) induced by R2 . It is well known that every unit volume flat metric on T2 is isometric to exactly one gr ,θ . Denote by g r ,θ the product metric gr ,θ ⊕ g on T2 × M, and by J (r , θ ) the corresponding Jacobi operator, as in (14). As in (15), the spectrum of J (r , θ ) is now computed explicitly:
: yi ∈ Γr∗,θ and µj ∈ Spec(M , g ) := λa,b,j (r , θ ) : a, b ∈ Z, j ∈ N ∪ {0}, |a| + |b| + j > 0
Spec J (r , θ ) = 4π 2 |yi |2 + µj −
Scalg m−1
where
λa,b,j (r , θ ) = 4π 2
a2 r 2 +b2 r sin θ
− 2ab cot θ + µj −
Scalg m−1
,
and again 0 = µ0 < µ1 ≤ µ2 ≤ · · · ≤ µj −→ +∞ is the set of eigenvalues of the Laplace–Beltrami operator of the metric g. The Morse index of g r ,θ , denoted by iMorse (r , θ ) is the number of such eigenvalues, counted with multiplicity, that are less than 0. We will now exhibit paths s → (rs , θs ) ∈ ]0, +∞[ × ]0, π[ where a change of sign of the eigenvalue functions s → λa,b,j (rs , θs ) occur. Towards this goal, it will be useful to study the level curves λa,b,j (r , θ ) = 0 in the half-plane (r , θ ) ∈ ]0, +∞[ × ]0, π[. For all j ≥ 0, let us set:
αj =
1 4π 2
Scalg m−1
− µj ;
(16)
since µ0 = 0 and limj→∞ µj = +∞, there exists j0 ≥ 0 such that αj > 0 for j ∈ {0, . . . , j0 } and αj ≤ 0 for j > j0 , i.e.: j0 = max j ≥ 0 : αj > 0 .
(17)
Let us set: F (a, b, r , θ ) :=
a2 r 2 + b 2 r sin θ
− 2ab cot θ ,
and let us study the level curves F (a, b, r , θ ) = αj , j ∈ {0, . . . , j0 }, with (a, b) ̸= (0, 0) fixed; let us use coordinates (x, y) ∈ R × R+ in the half-plane. We observe that F (a, b, r , θ ) = F (−a, −b, r , θ ); moreover: F (a, −b, r , θ ) = F (a, b, r , π − θ ).
(18)
For a = 0, we obtain a sequence of horizontal lines, denoted by Lb,j , with b ∈ Z \ {0} and j ∈ {0, . . . , j0 }, given by the 2
2
equation y = bα ; as |b| → +∞, this lines diverge to infinity. The eigenvalue λ0,b,j is negative in the half-space y > bα , and j j 2
positive in the strip 0 < y < bα . j
H.F. Ramírez-Ospina / Nonlinear Analysis 109 (2014) 103–112
111
a2 r 2 +b2
α
α
When a ̸= 0, the curve r sin θ − 2ab cot θ = αj is a circumference Ca,b,j with center in a , 2aj2 and of radius 2aj2 ; it is tangent to the x axis. The eigenvalue λa,b,j is negative inside this circumference, and positive outside. Observe that Ca,b,j = C−a,−b,j , and that C−a,b,j is symmetric to Ca,b,j with respect to the y-axis. Observe also that Ca,b,j is contained in the strip 0 < y ≤ α0 .
b
Proposition 10. Let s → (rs , θs ) be any C 1 -path that crosses transversally one of the curves Lb,j , with b ∈ Z \ {0} and j ∈ {0, . . . , j0 }, or Ca,b,j , with a ∈ Z \ {0}, b ∈ Z and j ∈ {0, . . . , j0 } at the instant s∗ . Then, s∗ is a bifurcation instant for the corresponding path of metrics s → g rs ,θs . Proof. By our analysis above, each instant of transversal intersection with the given curves corresponds to the change of sign of one of the eigenvalues λa,b,j (rs , θs ), i.e., a jump of the Morse index iMorse (rs , θs ). The conclusion follows from Proposition 9. Examples. As examples, let us consider the following family of paths in the upper half-plane:
• half-lines ]0, +∞[ ∋ s → (s, θ0 ), with θ0 ∈ ]0, π[ fixed; • half-circles ]0, π[ ∋ s → (r0 , s), withr0 ∈ ]0, +∞[ fixed; • horizontal lines ]0, π[ ∋ s → sinc s , s , with c ∈ ]α0 , +∞[ fixed. Theorem 11. Assume Scalg > 0. (a) For every θ0 ∈ ]0, π[ fixed, there exists an unbounded increasing sequence of bifurcation instants (sk )k≥1 for the curve of metrics ]0, +∞[ ∋ s → g s,θ0 on T2 × M; (b) For every r0 ∈ ]0, +∞[ fixed, the set Br0 ⊂ ]0, π[ of bifurcation instants for the path of metrics ]0, π[ ∋ θ → g r0 ,θ is symmetric with respect to π2 . Set
ℓr 0 = If
ℓr0 r0
<
inf
(a,b)∈Z2 \{(0,0)}
Scalg 4π 2 (m−1)
2 2 a r − b2 = 0.
(19)
0
, then Br0 ̸= ∅. If ℓr0 = 0, then Br0 is infinite, and its closure contains 0 and π .
(c) For all but a countable number of values c ∈ ]α0 , +∞[, the path of metrics ]0, π[ ∋ s → g
c ,s sin s
is locally rigid.
α
Proof. For the proof of (a), it suffices to observe that, for s > sin0θ , the unique intersections between the curve s → (s, θ0 ) 0 and the level curves λa,b,j = 0 occur when a = 0, j ∈ {0, . . . , j0 }, and all b sufficiently large. As we have observed, these level curves form a family of horizontal line diverging to ∞. At each such intersection there is a (positive) jump of Morse index, and therefore it corresponds to a bifurcation instant. The first statement in (b) follows readily from (18). A direct computation gives the following:
inf
a2 r 2 + b 2 r sin θ
θ∈]0,π [
|a2 r 2 − b2 | − 2ab cot θ = ; r
(20)
the infimum being actually a minimum when a2 r 2 − b2 ̸= 0, and
sup
θ∈]0,π [
a2 r 2 + b 2 r sin θ
− 2ab cot θ = +∞.
(21)
It is also easy to check that the function θ → F (a, b, r , θ ) is strictly convex in ]0, π[. Using (20), (21) and the convexity ℓr0
Scal
< 4π 2 (mg−1) , and that when ℓr0 = 0 there are infinitely many instants θ ∈ ]0, π[ where some eigenvalue function θ → λa,b,j (r0 , θ ) changes its sign. For the proof of (c), observe that the curve ]0, π[ ∋ s → sinc s , s in the upper half plane is the horizontal line y = c. |b|2 When c is greater than α0 and it does not belong to the countable set α , b ∈ Z, j ∈ {0, . . . , j0 } , then there are j no intersections between the line y = c and the level curves λa,b,j = 0. Thus, g c ,s is never degenerate, hence locally sin s property, ones sees that Br0 ̸= ∅ when
rigid.
r0
8. Further developments A natural generalization of the bifurcation results discussed in this paper would be to consider products of the form F × M, where F is a compact k-manifold admitting a flat metric. By a classic result of Bieberbach, such a manifold F admits a finite locally isometric covering by a flat torus Tk ; in this situation, the spectrum of the Laplacian of F is a subset of the spectrum of the flat torus. We observe however that multiplicity of constant scalar curvature metrics on Tk × M does not yield multiplicity of constant scalar curvature metrics in F ×M, since it may not be the case that a non-product constant scalar curvature metric on Tk × M can be found invariant by the group of deck transformation of the covering Tk × M → F × M.
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As it has been discussed in Section 6, an extension of the bifurcation results for constant scalar curvature metrics in a product F × M would require a proof of the existence of unit volume flat metrics on F with arbitrarily small eigenvalues. Since F admits a finite covering by a torus, then its spectrum is contained in the spectrum of a flat torus Rk /Γ , see [19]. Again, it is not clear whether deformations of Γ that produce arbitrarily small eigenvalues of the Laplacian of Rk /Γ can be made invariant under the group of deck transformations of this covering. Also here the case k = 2 is special, and it provides a concrete situation where all computations can be made explicitly. Besides the flat torus T2 , the only other compact manifold that admits a flat metric is the Klein bottle K . One has a double covering T2 → K ; for each x, y > 0, there is a flat Klein bottle K (x, y) such that R2 /Γ → K (x, y) is a Riemannian (i.e., locally 2 isometric) covering, where Γ is the lattice givenby spanZ {xe1 , ye2 }, being {e1 , e2 } the canonical basis of R . The volume of 2 R /Γ is equal to xy, hence if xy = 2, vol K (x, y) = 1. The spectrum of the Laplacian of K (x, y) can be computed explicitly (see [18, p. 153]):
Spec K (x, y) = 4π 2
a2 x2
+
b2 y2
: a, b ∈ Z, b ̸= 0 if a is odd .
By setting x = t, y = 1t , t ≥ 1, we obtain a path of unit volume flat metrics gt on K with arbitrarily small eigenvalues. Thus, given any unit volume compact manifold (M , g ) with positive scalar curvature, one has infinitely many bifurcation instants along the path of metrics g t = gt ⊕ g on K × M. Acknowledgments This work was partially supported by MINECO (Ministerio de Economía y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037. An original version of this paper only dealt with the case k = 2. The natural extension of the result to general products of the form Tk × M, with k ≥ 2, as well as a number of very interesting observations, were suggested in a detailed and very instructive report by an anonymous referee, whose support is very thankfully acknowledged by the author. Among other things, the material presented in Section 6 and in Section 8 has been suggested by the referee. References [1] J. Petean, Metrics of constant scalar curvature conformal to Riemannian products, Proc. Amer. Math. Soc. 138 (8) (2010) 2897–2905. [2] L.L. de Lima, P. Piccione, M. Zedda, On bifurcation of solutions of the Yamabe problem in product manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2) (2012) 261–277. [3] R.G. Bettiol, P. Piccione, Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions, Pacific J. Math. 266 (1) (2013) 1–21. [4] L.L. de Lima, P. Piccione, M. Zedda, A note on the uniqueness of solutions for the Yamabe problem, Proc. Amer. Math. Soc. 140 (12) (2012) 4351–4357. [5] R.G. Bettiol, P. Piccione, Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres, Calc. Var. Partial Differential Equations 47 (3–4) (2013) 789–807. [6] J. Petean, Degenerate solutions of a nonlinear elliptic equation on the sphere, Nonlinear Anal. 100 (2014) 23–29. [7] M. Holst, C. Meier, An alternative between non-unique and negative Yamabe solutions to the conformal formulation of the Einstein constraint equations, arXiv:1306.1210. [8] B. Farb, D. Margalit, A Primer on Mapping Class Groups, Princeton University Press, Princeton, NJ, 2012. [9] R.G. Bettiol, P. Piccione, B. Santoro, Bifurcation of periodic solutions to the singular Yamabe problem on spheres, preprint 2014, arXiv:1401.7071. [10] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (1968) 265–274. [11] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, in: Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. [12] D. Fortunato, G. Palmieri, Remarks on the Yamabe problem and the Palais–Smale condition, Rend. Semin. Mat. Univ. Padova 75 (1986) 47–65. [13] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, in: Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [14] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in: Topics in Calculus of Variations (Montecatini Terme, 1987), in: Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. [15] R. Abraham, J. Robbin, Transversal Mappings and Flows, W. A. Benjamin Inc., 1967. [16] A. Marino, G. Prodi, Metodi perturbativi nella teoria di Morse, Boll. Unione Mat. Ital. (4), Suppl. fasc. 11 (3) (1975) 1–32. [17] M. Gromov, H.B. Lawson, Spin and scalar curvature in the presence of a fundamental group I, Ann. Math. 11 (2) (1980) 209–230. [18] M. Berger, P. Gauduchon, E. Mazet, Le spectre dune varit riemannienne, 194, Springer-Verlag, Berlin-New, 1971. [19] T. Sunada, Spectrum of a compact flat manifold, Comment Math. Helv 53 (4) (1978) 613–621.