A UNIQUE EXTREMAL METRIC FOR THE LEAST EIGENVALUE OF THE LAPLACIAN ON THE KLEIN BOTTLE AHMAD EL SOUFI,HECTOR GIACOMINI ANDMUSTAPHAJAZAR 7 0 0 Abstract. We prove the following conjecture recently formulated by 2 Jakobson, Nadirashvili and Polterovich [15]: on theKlein bottle K, the n metric of revolution a 9+(1+8cos2v)2 dv2 J g0 = 1+8cos2v „du2+ 1+8cos2v«, 6 2 0 ≤ u < π2, 0 ≤ v < π, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all ] Riemannianmetricsofgivenarea. TheproofleadsustostudyaHamil- G tonian dynamical system which turns out to be completely integrable M byquadratures. . h t a m [ 1. Introduction and statement of main results 1 Among all the possible Riemannian metrics on a compact differentiable v 3 manifold M, the most interesting ones are those which extremize a given 7 Riemannian invariant. In particular, many recent works have been devoted 7 to the metrics which maximize the fundamental eigenvalue λ (M,g) of the 1 1 Laplace-Beltrami operator ∆ under various constraints (see, for instance, 0 g 7 [4,14, 16, 18,19]). Notice that, since λ (M,g) is not invariant underscaling 1 0 (λ (M,kg) = k 1λ (M,g)), such constraints are necessary. 1 − 1 / h In [22], Yang and Yau proved that on any compact orientable surface t M, the first eigenvalue λ (M,g) is uniformly bounded over the set of Rie- a 1 m mannian metrics of fixed area. More precisely, one has, for any Riemannian : metric g on M, v i X λ (M,g)A(M,g) 8π(genus(M)+1), 1 ≤ r a where A(M,g) stands for the Riemannian area of (M,g) (see [7] for an improvement of this upper bound). In the non-orientable case, the follow- ing upper bound follows from Li and Yau’s work [16]: λ (M,g)A(M,g) 1 ≤ 24π(genus(M) +1). On the other hand, if the dimension of M is greater than 2, then λ (M,g) is never bounded above over the set of Riemannian 1 metrics of fixed volume, see [5]. 2000 Mathematics Subject Classification. primary : 58J50; 58E11; 35P15; secondary 37C27. Key words and phrases. eigenvalue; Laplacian; Klein bottle; extremal metric, Hamil- tonian system; integrable system. 1 2 A.ELSOUFI,H.GIACOMINIANDM.JAZAR Hence, one obtains a relevant topological invariant of surfaces by setting, for any compact 2-dimensional manifold M, Λ(M) = supλ (M,g)A(M,g) = sup λ (M,g), 1 1 g g (M) ∈R where (M) denotes the set of Riemannian metrics of area 1 on M. R On the other hand, in spite of the non-differentiability of the functional g λ (M,g) with respect to metric deformations, a natural notion of ex- 1 7→ tremal (or critical) metric can be introduced. Indeed, for any smooth defor- mation g of a metric g, the function ε λ (M,g ) always admits left and ε 1 ε 7→ right derivatives at ε= 0 with d d λ (M,g ) λ (M,g ) 1 ε 1 ε dε ε=0+ ≤ dε ε=0− (cid:12) (cid:12) (see [9, 11] for details). The m(cid:12)etric g is then said(cid:12)to be extremal for the (cid:12) (cid:12) functional λ under volume preserving deformations if, for any deformation 1 g with g = g and vol(M,g ) = vol(M,g), one has ε 0 ε d d λ (M,g ) 0 λ (M,g ) . 1 ε 1 ε dε ε=0+ ≤ ≤ dε ε=0− (cid:12) (cid:12) This last condition can also b(cid:12)e formulated as follows:(cid:12) (cid:12) (cid:12) λ (M,g ) λ (M,g)+o(ε) as ε 0. 1 ε 1 ≤ → Given a compact surface M, the natural questions related to the func- tional λ are : 1 (1) What are the extremal metrics on M? (2) Is the supremum Λ(M) achieved and, if so, by what extremal met- rics? (3) How does Λ(M) depend on (the genus of) M? Concerningthelastquestion,itfollowsfrom[6]thatΛ(M)isanincreasing functionofthegenuswithalineargrowthrate. Explicitanswerstoquestions (1) and (2) are only known for the sphere S2, the real projective plane RP2 and the torus T2. Indeed, the standard metric gS2 (resp. gRP2) is, up to a dilatation, the only extremal metric on S2 (resp. RP2) (see [8, 9, 17]) and one has (see [14] and [16]) Λ(S2) = λ1(S2,gS2)A(S2,gS2) = 8π and Λ(RP2) = λ1(RP2,gRP2)A(RP2,gRP2) = 12π. Concerning the torus, the flat metrics g and g associated respectively sq eq with the square lattice Z2 and the equilateral lattice Z(1,0) Z(1, √3) are, ⊕ 2 2 up to dilatations, the only extremal metrics on T2(see [9]). Nadirashvili [18] has proved the existence of a regular global maximizer of the functional g λ (T2,g), which then implies that 1 7→ 8π2 Λ(T2) = λ (T2,g )A(T2,g ) = . 1 eq eq √3 However, some steps in Nadirashvili’s proof need to be completed as dis- cussed in the recent work of Girouard [13]. The metric g corresponds to a sq saddle point of the functional λ . 1 EXTREMAL METRIC FOR λ1 ON THE KLEIN BOTTLE 3 What about the Klein bottle K? Nadirashvili [18] observed that an extremal metric on K cannot be a flat metric. Recently, Jakobson, Nadirashvili and Polterovich [15] proved that a metric of revolution 9+(1+8cos2v)2 dv2 g = du2+ , 0 1+8cos2v 1+8cos2v (cid:18) (cid:19) 0 u< π, 0 v < π, is an extremal metric on K and conjectured that this ≤ 2 ≤ metric is, up to a dilatation, the unique extremal metric on K. The main purpose of this paper is to prove this conjecture. Indeed, we will prove the following Theorem 1.1. The Riemannian metric g is, up to a dilatation, the unique 0 extremal metric of the functional λ under area preserving deformations of 1 metrics on the Klein bottle K. Remark 1.1. Nadirashvili [18] has given a sketch of proof of the fact that the supremum Λ(K) is necessarily achieved by a regular (real analytic) Rie- mannian metric. An immediate consequence of such a result and Theorem 1.1 would be Λ(K) = λ (K,g )A(K,g ) = 12πE(2√2/3) 13.365π, 1 0 0 ≃ where E(2√2/3) is the complete elliptic integral of the second kind evaluated at 2√2. 3 It is worth noticing that the metric g does not maximize the systole 0 functional g sys(g) (where sys(g) denotes the length of the shortest non- 7→ contractible loop) over the set of metrics of fixed area on the Klein bottle (see [3]), while on RP2 and T2, the functionals λ and sys are maximized 1 by the same Riemannian metrics. The proof of Theorem 1.1 relies on the characterization of extremal met- rics in terms of minimal immersions into spheres by the first eigenfunctions. Indeed, a metric g is extremal for λ with respect to area preserving defor- 1 mations if and only if there exists a family h , ,h of first eigenfunctions 1 d ··· of∆ satisfying dh dh = g (see[9,10]). Thislastcondition actually g i d i⊗ i means that the ma≤p (h , ,h ) : (M,g) Rd is an isometric immersion P 1 ··· d → whose image is a minimal immersed submanifold of a sphere. As noticed in [15], the surface (K,g ) is isometrically and minimally im- 0 mersed in S4 as the bipolar surface of Lawson’s minimal torus τ defined 3,1 as the image in S3 of the map (u,v) (cosvexp(3iu),sinvexp(iu)). 7→ In fact, we will prove the following Theorem 1.2. The minimal surface (K,g ) ֒ S4 is, up to isometries, the 0 → only isometrically and minimally immersed Klein bottle into a sphere by its first eigenfunctions. In [8], Ilias and the first author gave a necessary condition of symmetry for a Riemannian metric to admit isometric immersions into spheres by the first eigenfunctions. On the Klein bottle, this condition amounts to the 4 A.ELSOUFI,H.GIACOMINIANDM.JAZAR invariance of the metric under the natural S1-action on K. Taking into ac- count this symmetry property and the fact that any metric g is conformally equivalent to a flat one, for which the eigenvalues and the eigenfunctions of the Laplacian are explicitly known, it is of course expected that the ex- istence problem of minimal isometric immersions into spheres by the first eigenfunctions reduces to a second order system of ODEs (see Proposition 2.1). Actually, the substantial part of this paper is devoted to the study of the following second order nonlinear system: ϕ = (1 2ϕ2 8ϕ2)ϕ , ′1′ − 1 − 2 1 (1)  ϕ = (4 2ϕ2 8ϕ2)ϕ ,  ′2′ − 1 − 2 2 for which we look for periodic solutions satisfying  ϕ is odd and has exactly two zeros in a period, (2) 1 ϕ is even and positive everywhere; 2 (cid:26) and the initial conditions ϕ (0) = ϕ (0) = 0(from parity conditions (2)), 1 ′2 (3) 1 ϕ (0) = ϕ (0) =:p (0,1]. ( 2 2 ′1 ∈ Notice that a similar approach is used in [12] where the construction of S1-equivariant minimal tori in S4 and S1-equivariant Willmore tori in S3 is related to a completely integrable Hamiltonian system. In [15], Jakobson, Nadirashvili and Polterovich proved that the initial value p = ϕ (0) = 3/8 corresponds to a periodic solution of (1)-(3) satis- 2 fying (2). Based on numerical evidence, they conjectured that this value of p p is the only one which corresponds to a periodic solution satisfying (2). As mentionedbythem,acomputer-assistedproofofthisconjectureisextremely difficult, due to the lack of stability of the system. In Section 3, we provide a complete analytic study of System (1). First, we show that this system admits two independent first integrals (one of them has been already found in [15]). Using a suitable linear change of vari- ables, weshowthatthesystembecomes Hamiltonian and,hence, integrable. The general theory of integrable Hamiltonian systems tells us that bounded orbits correspond to periodic or quasi-periodic solutions (see [2]). How- ever, to distinguish periodic solutions from non-periodic ones is not easy in general. Fortunately, our first integrals turn out to be quadratic in the mo- menta which enables us to apply the classical Bertrand-Darboux-Whittaker Theorem and, therefore, to completely decouple the system by means of a parabolic type change of coordinates (ϕ ,ϕ ) (u,v). We show that, for 1 2 7→ any p = √3/2, the solutions u and v of the decoupled system are periodic. 6 The couple (u,v) is then periodic if and only if the periods of u and v are commensurable. Weexpress theperiodsof uandv interms of hyper-elliptic integrals and study their ratio as a function of p. The following fact (Propo- sition 3.1) gives an idea about the complexity of the situation: there exists a countable dense subset (0,√3/2) such that the solution of (1)-(3) P ⊂ corresponding to p (0,√3/2) is periodic if and only if p . ∈ ∈ P In conclusion, we show that the solution associated with p = 3/8 is the only periodic one to satisfy Condition (2). p EXTREMAL METRIC FOR λ1 ON THE KLEIN BOTTLE 5 2. Preliminaries: reduction of the problem Accordingto[9,10],anecessaryandsufficientconditionforaRiemannian metricgonacompactmanifoldM tobeextremalforthefunctionalλ under 1 area-preserving metric deformations is that there exists a family h , ,h 1 d ··· of first eigenfunctions of ∆ satisfying g (4) dh dh = g, i i ⊗ i d X≤ which means that the map h = (h , ,h ) is an isometric immersion from 1 d ··· (M,g) to Rd. Since h , ,h are eigenfunctions of ∆ , the image of h is 1 d g ··· a minimal immersed submanifold of the Euclidean sphere Sd 1 2 − λ1(M,g) of radius 2/λ (M,g) (Takahashi’s theorem [20]). In particula(cid:16)r,qwe have(cid:17) 1 p 2 (5) h2 = . i λ (M,g) 1 i d X≤ In [8], Ilias and the first author have studied conformal properties of Riemannian manifolds (M,g) admitting such minimal isometric immersions into spheres. It follows from their results that, if g is an extremal metric of λ under area preserving deformations, then 1 (i) g is, up to a dilatation, the unique extremal metric in its conformal class, (ii) g maximizes the restriction of λ to the set of metrics conformal to 1 g and having the same volume, (iii) the isometry group of (M,g) contains the isometry groups of all the metrics g conformal to g. ′ For any positive real numbera, we denote by Γ therectangular lattice of a R2 generated by the vectors (2π,0) and (0,a) and byg˜ the flat Riemannian a metric of the torus T2 R2/Γ associated with the rectangular lattice a ≃ a Γ . The Klein bottle K is then diffeomorphic to the quotient of T2 by the a a involution s :(x,y) (x+π, y). We denote by g the flat metric induced a 7→ − on K by such a diffeomorphism. It is well known that any Riemannian metric on K is conformally equivalent to one of the flat metrics g . a Letg = fg beaRiemannian metricon K. From theproperty (iii) above, a if g is an extremal metric of λ under area preserving deformations, then 1 Isom(K,g ) Isom(K,g), which implies that the function f is invariant a ⊂ under the S1-action (x,y) (x+t,y), t [0,π], on K, and then, f (or its 7→ ∈ lift to R2) does not depend on the variable x. Proposition 2.1. Let a be a positive real number and f a positive periodic function of period a. The following assertions are equivalent (I) The Riemannian metric g = f(y)g on K is an extremal metric of a the functional λ under area preserving deformations. 1 (II) There exists a homothetic minimal immersion h = (h , ,h ) : 1 d ··· (K,g) Sd 1 such that, i d, h is first eigenfunction of ∆ . − i g → ∀ ≤ (III) The function f is proportional to ϕ2+4ϕ2, where ϕ and ϕ are two 1 2 1 2 periodic functions of period a satisfying the following conditions: 6 A.ELSOUFI,H.GIACOMINIANDM.JAZAR (a) (ϕ ,ϕ ) is a solution of the equations 1 2 ϕ = (1 2ϕ2 8ϕ2)ϕ , ′1′ − 1− 2 1  ϕ = (4 2ϕ2 8ϕ2)ϕ ;  ′2′ − 1− 2 2 (b) ϕ1 is odd, ϕ2 is even and ϕ′1(0) = 2ϕ2(0); (c) ϕ admits two zeros in a period and ϕ is positive everywhere; 1 2 (d) ϕ2 +ϕ2 1 and the equality holds at exactly two points in a 1 2 ≤ period. From the results [9, 10] mentioned above, it is clear that (I) and (II) are equivalent. Most of the arguments of the proof of “(II) implies (III)” can be found in [18] and [15]. For the sake of completeness, we will recall the main steps. The proof of “(III) implies (II)” relies on the fact that the system (1) admits two independent first integrals. Proof of Proposition 2.1. TheLaplacian∆ associatedwiththeRiemannian g metric g = f(y)g on K can be identified with the operator 1 ∂2+∂2 a −f(y) x y acting on Γ -periodic and s-invariant functions on R2. Using separation of a (cid:0) (cid:1) variables and Fourier expansions, one can easily show that any eigenfunc- tion of ∆ is a linear combination of functions of the form ϕ (y)coskx and g k ϕ (y)sinkx, where, k, ϕ is a periodic function with period a satisfying k k ∀ ϕ ( y) = ( 1)kϕ (y) and ϕ = (k2 λf)ϕ . Since a first eigenfunction al- k − − k ′k′ − k waysadmitsexactlytwonodaldomains,thefirsteigenspaceof∆ isspanned g by ϕ (y), ϕ (y)cosx, ϕ (y)sinx, ϕ (y)cos2x, ϕ (y)sin2x , 0 1 1 2 2 { } where, unless they are identically zero, ϕ does not vanish while ϕ and ϕ 2 0 1 admit exactly two zeros in [0,a). In particular, the multiplicity of λ (K,g) 1 is at most 5. Let us suppose that g is an extremal metric of λ under area preserving 1 deformations and let h , ,h be a family of first eigenfunctions satisfying 1 d ··· the equations (4) and (5) above. Without loss of generality, we may assume thatλ (K,g) = 2andthath , ,h arelinearlyindependent,whichimplies 1 1 d ··· that d 5. Since h = (h , ,h ) : K Sd 1 is an immersion, one has 1 d − ≤ ··· → d 4. If d = 4, then using elementary algebraic arguments like in the proof ≥ of Proposition 5 of [17], one can see that there exists an isometry ρ O(4) ∈ such that ρ h = (ϕ (y)eix,ϕ (y)e2ix) with ϕ2 + ϕ2 = 1 (eq. (5)) and ◦ 1 2 1 2 ϕ2 +ϕ2 = ϕ2 +4ϕ2 = f (eq. (4)) which is impossible since ϕ2 +ϕ2 = 1 ′1 ′2 1 2 1 2 implies that ϕ and ϕ admit a common critical point. Therefore, d = 1 2 multiplicity of λ (K,g) = 5 and there exists ρ O(5) such that ρ h = 1 ∈ ◦ (ϕ (y),ϕ (y)eix,ϕ (y)e2ix), with ϕ2 +ϕ2 +ϕ2 = 1 and ϕ2 +ϕ2 +ϕ2 = 0 1 2 0 1 2 ′0 ′1 ′2 ϕ2+4ϕ2 = f. Sincethe linear components of ρ h are firsteigenfunctions of 1 2 ◦ (K,g), one should has, k = 0,1,2, ϕ = (k2 λ (K,g)f)ϕ = (k2 2ϕ2 ∀ ′k′ − 1 k − 1 − 8ϕ2)ϕ . Now, it is immediate to check that one of the couples of functions 2 k ( ϕ , ϕ ) satisfies the Conditions (a), ..., (d) of the statement. Indeed, 1 2 ± ± the parity condition ϕ ( y) = ( 1)kϕ (y) implies that ϕ (0) = ϕ (0) = k − − k 1 ′0 ϕ (0) = 0 and, then, ϕ2(0) = 4ϕ2(0). Conditions (c) and (d) follow from ′2 ′1 2 the fact that a first eigenfunction has exactly two nodal domains in K. EXTREMAL METRIC FOR λ1 ON THE KLEIN BOTTLE 7 Conversely, let ϕ and ϕ be two periodic functions of period a satisfying 1 2 Conditions (a), ..., (d) of (III) and consider the Riemannian metric g = f(y)g on K, with f = ϕ2+4ϕ2. We set ϕ = 1 ϕ2 ϕ2 and define the a 1 2 0 − 1− 2 map h :K S4 by h= (ϕ (y),ϕ (y)eix,ϕ (y)e2ix). It suffices to check that → 0 1 2 p the components of h are first eigenfunctions of ∆ satisfying (4). g Indeed, in the next section we will see that the second order differential system satisfied by ϕ and ϕ (Condition (a)) admits the two following first 1 2 integrals: (ϕ2+4ϕ2)2 ϕ2 16ϕ2 +ϕ 2+4ϕ 2 = C, 1 2 − 1− 2 ′1 ′2 (6)   12ϕ22(ϕ22 −1)+3ϕ21ϕ22+ϕ22ϕ′12−2ϕ1ϕ′1ϕ2ϕ′2+(3+ϕ21)ϕ′22 = C, withC = 4ϕ (0)2(4ϕ (0)2 3)(notethatCondition(b)impliesthatϕ (0) =  2 2 1 − ϕ (0) = 0). Differentiating ϕ2+ϕ2+ϕ2 = 1 and using the second equation ′2 0 1 2 in (6), we get ϕ2ϕ 2 = ϕ2ϕ 2+ϕ2ϕ 2+2ϕ ϕ ϕ ϕ 0 ′0 1 ′1 2 ′2 1 ′1 2 ′2 = ϕ2ϕ 2+ϕ2ϕ 2+12ϕ2(ϕ2 1)+3ϕ2ϕ2+ϕ2ϕ 2+(3+ϕ2)ϕ 2 C 1 ′1 2 ′2 2 2 − 1 2 2 ′1 1 ′2 − = (ϕ2+ϕ2)ϕ 2+(3+ϕ2+ϕ2)ϕ 2+12ϕ2(ϕ2 1)+3ϕ2ϕ2 C 1 2 ′1 1 2 ′2 2 2 − 1 2− = (1 ϕ2)ϕ 2+(4 ϕ2)ϕ 2+12ϕ2(ϕ2 1)+3ϕ2ϕ2 C. − 0 ′1 − 0 ′2 2 2− 1 2− Therefore ϕ2 ϕ 2+ϕ 2+ϕ 2 = ϕ 2+4ϕ 2+12ϕ2(ϕ2 1)+3ϕ2ϕ2 C 0 ′0 ′1 ′2 ′1 ′2 2 2 − 1 2− (cid:16) (cid:17) = 1 ϕ2 ϕ2 ϕ2+4ϕ2 , − 1− 2 1 2 where the last equality follows from the first equation of (6). Hence, (cid:0) (cid:1)(cid:0) (cid:1) ∂ h 2 = ϕ 2+ϕ 2+ϕ 2 = ϕ2+4ϕ2 = ∂ h2 | y | ′0 ′1 ′2 1 2 | x | and, since ∂ h and ∂ h are orthogonal, the map h is isometric, which means x y that Equation (4) is satisfied. From Condition (a) one has ϕ = (1 2f)ϕ and ϕ = (4 2f)ϕ , ′1′ − 1 ′2′ − 2 which implies that the functions h = ϕ (y)cosx, h = ϕ (y)sinx, h = 1 1 2 1 3 ϕ (y)cos2x and h = ϕ (y)sin2x are eigenfunctions of ∆ associated with 2 4 2 g the eigenvalue λ = 2. Moreover, differentiating twice the identity ϕ2 + 0 ϕ2 +ϕ2 = 1 and using Condition (a) and the identity ϕ 2 +ϕ 2 +ϕ 2 = 1 2 ′0 ′1 ′2 ϕ2 +4ϕ2 = f, one obtains after an elementary computation, ϕ = 2fϕ . 1 2 ′0′ − 0 Hence, all the components of h are eigenfunctions of ∆ associated with the g eigenvalue λ = 2. Itremainstoprovethat2is thefirstpositiveeigenvalue of ∆ or, equivalently, for each k = 0,1,2, the function ϕ corresponds to the g k lowest positive eigenvalue of the Sturm-Liouville problem ϕ = (k2 λf)ϕ ′′ − subject to the parity condition ϕ( y) = ( 1)kϕ(y). As explained in the − − proof of Proposition 3.4.1 of [15], this follows from conditions (c) and (d) giving the number of zeros of ϕ , and the special properties of the zero sets k of solutions of Sturm-Liouville equations (oscillation theorems of Haupt and Sturm). (cid:3) Remark 2.1. Once the initial conditions ϕ (0) = 2ϕ (0) = p and ϕ (0) = ′1 2 1 ϕ (0) = 0 (since ϕ is odd and ϕ is even) are fixed, the solution of the ′2 1 2 systemgiveninassertion(III)ofProposition 2.1isclearlyunique. Hence, as 8 A.ELSOUFI,H.GIACOMINIANDM.JAZAR we have seen in the proof of this proposition, if a Kleinbottle (K,g = f(y)g ) a admits an isometric full minimal immersion h : (K,g) Sd 1 by the first − → eigenfunctions, then d =the multiplicity of λ (K,g) = 5 and there exists 1 ρ O(5) such that ∈ ρ h = 1 ϕ2(y) ϕ2(y),ϕ (y)eix,ϕ (y)e2ix , ◦ − 1 − 2 1 2 (cid:18)q (cid:19) where (ϕ ,ϕ ) is a unique solution of (III) (with ϕ (0) = 2ϕ (0) = f(0) 1 2 ′1 2 and ϕ (0) = ϕ (0) = 0). Recall that an immersion h into Sd 1 is said to 1 ′2 − p be full if its image does not lie in any hyperplane of Rd (i.e. its components h ,...,h are linearly independent). 1 d 3. Study of the dynamical system: proof of results According to Proposition 2.1, one needs to deal with thefollowing system of second order differential equations (Condition (a) of Prop. 2.1) ϕ = (1 2ϕ2 8ϕ2)ϕ , (7) ′1′ − 1− 2 1 ϕ = (4 2ϕ2 8ϕ2)ϕ , (cid:26) ′2′ − 1− 2 2 subject to the initial conditions (Condition (b) of Prop. 2.1) ϕ (0) = 0, ϕ (0) = p, (8) 1 2 ϕ (0) = 2p, ϕ (0) = 0, (cid:26) ′1 ′2 where p (0,1] (Condition (d) of Prop. 2.1). ∈ Notice that the system (7)-(8) is invariant under the transform (ϕ (y),ϕ (y)) ( ϕ ( y),ϕ ( y)). 1 2 1 2 7→ − − − Consequently, the solution (ϕ ,ϕ ) of (7)-(8) is such that ϕ is odd and ϕ 1 2 1 2 is even. We are looking for periodic solutions satisfying the following condition (Condition (c) of Prop. 2.1): ϕ has exactly two zeros in a period, (9) 1 ϕ is positive everywhere. 2 (cid:26) Our aim is to prove the following Theorem 3.1. There exists only one periodic solution of (7)-(8) satisfying Condition (9). It corresponds to the initial value ϕ (0) = p = 3/8. 2 In fact, this theorem follows from the qualitative behavior ofpsolutions, in terms of p, given in the following Proposition 3.1. Let (ϕ ,ϕ ) be the solution of (7)-(8). 1 2 (1) For all p (0,1], p = √3/2, (ϕ ,ϕ ) is periodic or quasi-periodic. 1 2 ∈ 6 (2) For p = √3, (ϕ ,ϕ ) tends to the origin as y (hence, it is 2 1 2 → ∞ neither periodic nor quasi-periodic). (3) For all p (√3/2,1], ϕ vanishes at least once in each period (of 2 ∈ ϕ ). Hence, Condition (9) is not satisfied. 2 (4) There exists a countable dense subset (0,√3/2), with 3/8 P ⊂ ∈ , such that the solution (ϕ ,ϕ ) corresponding to p (0,√3/2) is P 1 2 ∈ p periodic if and only if p . ∈P EXTREMAL METRIC FOR λ1 ON THE KLEIN BOTTLE 9 (5) For p = 3/8, (ϕ ,ϕ ) satisfies (9) and, for any p , p = 3/8, 1 2 ∈ P 6 ϕ admits at least 6 zeros in a period. 1 p p Notice that the assertions (2) and (3) of this proposition were also proved in [15] by other methods. The first fundamental step in the study of the system above is the exis- tence of the following two independent first integrals. 3.1. First integrals. The functions H (ϕ ,ϕ ,ϕ ,ϕ ):= (ϕ2 +4ϕ2)2 ϕ2 16ϕ2 +(ϕ )2+4(ϕ )2, 1 1 2 ′1 ′2 1 2 − 1− 2 ′1 ′2  (10) H (ϕ ,ϕ ,ϕ ,ϕ ):= 12ϕ2(ϕ2 1)+3ϕ2ϕ2+ϕ2(ϕ )2  2 1 2 ′1 ′2 2 2 − 1 2 2 ′1   2ϕ ϕ ϕ ϕ +(3+ϕ2)(ϕ )2, − 1 ′1 2 ′2 1 ′2   are two independent first integrals of (7), i.e. they satisfy the equation  ∂H ∂H ∂H ∂H i i i i ϕ +ϕ +ϕ +ϕ 0. ′1∂ϕ ′2∂ϕ ′1′∂ϕ ′2′∂ϕ ≡ 1 2 ′1 ′2 The first one, H , has been obtained by Jakobson et al. [15]. The orbit of 1 a solution of (7) is then contained in an algebraic variety defined by H (ϕ ,ϕ ,ϕ ,ϕ )= K , 1 1 2 ′1 ′2 1 (11)  H (ϕ ,ϕ ,ϕ ,ϕ )= K ,  2 1 2 ′1 ′2 2 where K and K are two constants. Taking into account the initial condi- 1 2  tions (8), one has K = K = 4p2(3 4p2). In other words, the solution 1 2 − − of (7)-(8) is also solution of (ϕ2+4ϕ2)2 ϕ2 16ϕ2 +(ϕ )2+4(ϕ )2+4p2(3 4p2) = 0, 1 2 − 1− 2 ′1 ′2 −  (12) 12ϕ2(ϕ2 1)+3ϕ2ϕ2+ϕ2(ϕ )2 2ϕ ϕ ϕ ϕ  2 2− 1 2 2 ′1 − 1 ′1 2 ′2   +(3+ϕ2)(ϕ )2+4p2(3 4p2)= 0, 1 ′2 −   with the initial conditions  ϕ (0) = 0, 1 (13)  ϕ (0) = p.  2 Notice that the parameter p appears in both the equations (12) and the  initial conditions (13). The system (12) gives rise to a “multi-valued” 2- dimensional dynamical system in the following way. 3.2. 2-dimensional dynamical systems. From (12) one can extract ex- plicit expressions of ϕ and ϕ in terms of ϕ and ϕ . For instance, elimi- ′1 ′2 1 2 nating ϕ , one obtains the following fourth degree equation in ϕ ′1 ′2 (14) d (ϕ ,ϕ )(ϕ )4 2d (ϕ ,ϕ )(ϕ )2+d (ϕ ,ϕ ) = 0, 4 1 2 ′2 − 2 1 2 ′2 0 1 2 where d , d and d are polynomials in ϕ , ϕ and p. The discriminant of 0 2 4 1 2 (14) is given by ∆ := 64ϕ2ϕ2w w w , − 1 2 1 2 3 10 A.ELSOUFI,H.GIACOMINIANDM.JAZAR with w (ϕ ,ϕ )= ϕ2+ϕ2 1, 1 1 2 1 2 − w (ϕ ,ϕ )= p2ϕ2 (3 4p2)ϕ2 +p2(3 4p2), 2 1 2 1− − 2 − w (ϕ ,ϕ )= (3 4p2)ϕ2+16p2ϕ2 4p2(3 4p2). 3 1 2 − − 1 2− − It is quite easy to show that, for any p, each one of the curves (w = 0) i contains the orbit of a particular solution of (12). Moreover, the unit circle (w = 0) represents the orbit of the solution of (12) satisfying the initial 1 conditions (13) with p = 1. For p = 3/8, we have w 4w and the 3 2 ≡ − curve (w = 0) contains the orbit of the solution of (12)-(13). 2 p These particular algebraic orbits suggest us searching solutions (ϕ ,ϕ ) 1 2 definedbyalgebraic relations oftheformw (ϕ ,ϕ ) = F(ϕ2,ϕ2)= 0, where 4 1 2 1 2 F is a polynomial of degree 4. Apart the three quadrics above, the only ≤ additional solution of this type we found is w =(ϕ2 +4ϕ2)2 12ϕ2 = 0. 4 1 2 − 2 Like (w = 0), the curve (w = 0) is independent of p and represents the 1 4 orbit of a particular solution of (12) for arbitrary values of p. Since w (ϕ ,ϕ )= (ϕ2 +4ϕ2 2√3ϕ )(ϕ2+4ϕ2 +2√3ϕ ), 4 1 2 1 2 − 2 1 2 2 the set (w = 0) is the union of two ellipses passing through the origin, each 4 one being symmetric to the other with respect to the ϕ -axis. The upper 1 ellipse (15) ϕ2+4ϕ2 2√3ϕ = 0 1 2− 2 corresponds to the orbit of the solution of (12)-(13) associated with p = √3. 2 √3 3.3. Proof of Proposition 3.1(2): case p = . In this case, the orbit 2 of the solution of (12)-(13) is given by (15). The only critical point of (12) lying on this ellipse is the origin, which is also a critical point of the system (7). Therefore, (ϕ (y),ϕ (y)) tends to the origin as y goes to infinity (see 1 2 also [15]). From now on, we will assume that p = √3. 6 2 3.4. A bounded region for the orbit. The orbit of the solution of (12)- (13) must lie in the region of the (ϕ ,ϕ )-plane where the discriminant ∆ of 1 2 (14) is nonnegative. This region, (∆ 0), is a bounded domain delimited ≥ by the unit circle (w = 0) and the quadrics (w = 0) and (w = 0). Its 1 2 3 shape depends on the values of p. For p (0,√3/2), (w = 0) and (w = 0) are hyperbolas. 2 3 • ∈ The case p = 3/8 is a special one since then, w 4w , and the 3 2 • ≡ − region (∆ 0) shrinks to the arc of the hyperbola (w = 0) lying 2 ≥ p inside the unit disk. For p (√3/2,1], w is positive and (w = 0) is an ellipse. 3 2 • ∈ From (14) and (12) one can express ϕ and ϕ in terms of ϕ , ϕ and ′1 ′2 1 2 p. Thus, we obtain a multi-valued 2-dimensional dynamical system param- eterized by p with the initial conditions ϕ (0) = 0 and ϕ (0) = p. How- 1 2 ever, the dynamics of such a multi-valued system is very complex to study.