An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane
aa r X i v : . [ m a t h . DG ] M a y AN ISOPERIMETRIC INEQUALITY FOR THESECOND NON-ZERO EIGENVALUE OF THELAPLACIAN ON THE PROJECTIVE PLANE
NIKOLAI S. NADIRASHVILI AND ALEXEI V. PENSKOI
Abstract.
We prove an isoperimetric inequality for the secondnon-zero eigenvalue of the Laplace-Beltrami operator on the realprojective plane. For a metric of unit area this eigenvalue is notgreater than 20 π. This value is attained in the limit by a sequenceof metrics of area one on the projective plane. The limiting metricis singular and could be realized as a union of the projective planeand the sphere touching at a point, with standard metrics and theratio of the areas 3 : 2 . It is also proven that the multiplicity of thesecond non-zero eigenvalue on the projective plane is at most 6 . Introduction
Let M be a closed surface and g be a Riemannian metric on M. Letus consider the Laplace-Beltrami operator ∆ : C ∞ ( M ) −→ C ∞ ( M )associated with the metric g, ∆ f = − p | g | ∂∂x i (cid:18)p | g | g ij ∂f∂x j (cid:19) , and its eigenvalues(1) 0 = λ ( M, g ) < λ ( M, g ) λ ( M, g ) λ ( M, g ) . . . Let us denote by m ( M, g, λ i ) the multiplicity of the eigenvalue λ i ( M, g ) , i.e. how many times the value of λ i ( M, g ) appears in the sequence (1).Let us consider a functional¯ λ i ( M, g ) = λ i ( M, g ) Area(
M, g ) , where Area( M, g ) is the area of M with respect to the Riemannianmetric g. This functional is sometimes called an eigenvalue normalizedby the area or simply a normalized eigenvalue.
Mathematics Subject Classification.
Yang and Yau proved in the paper [56] that if M is an orientablesurface of genus γ then ¯ λ ( M, g ) π ( γ + 1) . Actually, the arguments of Yang and Yau imply a stronger estimate,¯ λ ( M, g ) π (cid:20) γ + 32 (cid:21) , see the paper [18] and also [41]. Here [ · ] denotes the integer part of anumber.Later Korevaar proved in the paper [36] that there exists a constant C, such that for any i > M of genus γ thefollowing upper bound holds:¯ λ i ( M, g ) C ( γ + 1) i. Recently this upper bound was improved by Hassannezhad [23]. Sheproved that there exists a constant C, such that for any i > M of genus γ , the following upper bound holds:¯ λ i ( M, g ) C ( γ + i ) . It follows that the functionals ¯ λ i ( M, g ) are bounded from above and itis a natural question to find for a given compact surface M and number i ∈ N the quantity Λ i ( M ) = sup g ¯ λ i ( M, g ) , where the supremum is taken over the space of all Riemannian metrics g on M. Let us remark that the functional ¯ λ i ( M, g ) is invariant under rescal-ing of the metric g tg, where t ∈ R + . It follows that it is equivalent tothe problem of finding sup λ i ( M, g ) , where the supremum is taken overthe space of all Riemannian metrics g of area 1 on M. That’s why thisproblem is sometimes called the isoperimetric problem for eigenvaluesof the Laplace-Beltrami operator.
Definition 1.1.
Let M be a closed surface. A metric g on M is calledmaximal for the functional ¯ λ i ( M, g ) if Λ i ( M ) = ¯ λ i ( M, g )If a maximal metric exists, it is defined up to multiplication by apositive constant due to the rescaling invariance of the functional.Surprisingly, the list of known results is quite short.Hersch proved in 1970 in the paper [24] that the standard metric onthe sphere is the unique maximal metric for ¯ λ ( S , g ) andΛ ( S ) = 8 π. N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 3
Li and Yau proved in 1982 in the paper [38] that the standard metricon the projective plane is the unique maximal metric for ¯ λ ( RP , g ) andΛ ( RP ) = 12 π. The first author proved in 1996 in the paper [41] that the stan-dard metric on the equilateral torus is the unique maximal metric for¯ λ ( T , g ) and Λ ( T ) = 8 π √ . It is not always that a maximal metric exists. As it was proved bythe first author in 2002 in the paper [42] and later with a differentargument by Petrides [51], Λ ( S ) = 16 π. However, there is no maximal metric. The supremum is attained as alimit on a sequence of smooth metrics on the sphere converging to asingular metric on two spheres of the same radius touching in a point.The functional ¯ λ i ( M, g ) depends continuously on the metric g. How-ever, when ¯ λ i ( M, g ) is a multiple eigenvalue this functional is not ingeneral differentiable. If we consider an analytic variation g t of the met-ric g = g , then it was proved by Berger [5], Bando and Urakawa [2],El Soufi and Ilias [20] that the left and right derivatives of the func-tional ¯ λ i ( M, g t ) with respect to t exist. This leads us to the followingdefinition given by the first author in the paper [41] and by El Soufiand Ilias in the papers [19, 20]. Definition 1.2.
A Riemannian metric g on a closed surface M iscalled extremal metric for the functional ¯ λ i ( M, g ) if for any analyticdeformation g t such that g = g one has ddt ¯ λ i ( M, g t ) (cid:12)(cid:12)(cid:12) t =0+ ddt ¯ λ i ( M, g t ) (cid:12)(cid:12)(cid:12) t =0 − . Jakobson, the first author and Polterovich proved in 2006 in the pa-per [27] that the metric on the Klein bottle realized as so called bipolarLawson surface ˜ τ , , is extremal for ¯ λ ( KL , g ) . It was conjectured in thispaper that this metric is the maximal one. El Soufi, Giacomini andJazar proved in the same year in the paper [21] that this metric on˜ τ , is the unique extremal metric for the ¯ λ ( KL , g ) . It follows from theresults of [39] that there exists a smooth (up to at most a finite numberof conical points) metric g K on the Klein bottle such that sup ¯ λ ( KL , g )is attained on g K . It could be then shown (a detailed exposition of this
NIKOLAI S. NADIRASHVILI AND ALEXEI V. PENSKOI argument could be found in [13]) that the metric on ˜ τ , is the maximalone and, hence,Λ ( KL ) = ¯ λ ( KL , g ˜ τ , ) = 12 πE √ ! , where E is the complete elliptic integral of the second kind and g ˜ τ , isthe metric on ˜ τ , . More results on extremal metrics on tori and Klein bottles could befound in the papers [19, 28, 30, 31, 37, 47, 48, 50]. A review of theseresults is given by the second author in the paper [49].It was shown in [26] using a combination of analytic and numericaltools that the maximal metric for the first eigenvalue on the surfaceof genus two is the metric on the Bolza surface P induced from thecanonical metric on the sphere using the standard covering P −→ S . The authors stated this result as a conjecture, because the argument ispartly based on a numerical calculation. The proof of this conjecturewas given in a recent preprint [46].The first author and Sire proved in 2015 in the paper [45] the equalityΛ ( S ) = 24 π. It turns out that there is no maximal metric but the supremum couldbe obtained as a limit on a sequence of metrics on the sphere convergingto a singular metric on three touching spheres of the same radius. Itwas conjectured in the paper [42, 45] thatΛ k ( S ) = 8 πk. This conjecture was proven in the recent paper [33] by Karpukhin,Polterovich and the authors.The main goal of the present paper is to prove the following result.
Theorem 1.3.
The supremum of the normalized second nonzero eigen-value on the projective plane over the space of all Riemannian metricson RP is given by (2) Λ ( RP ) = 20 π. There is no maximal metric, even among metrics with a finite numberof conical singularities. The supremum is attained in the limit by asequence of metrics of area one on the projective plane. The limitingmetric is singular and could be realized as a union of the projectiveplane and the sphere touching at a point, with standard metrics andthe ratio of the areas . We postpone the definition of metrics with conical singularities tillSection 5.
N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 5
Remark 1.4.
This Theorem could be stated as an isoperimetric in-equality λ ( RP , g ) π for any metric g of area . Remark 1.5.
It would be interesting to check whether the equalityin (2) could be attained in the limit only by a sequence of metrics con-verging to a union of touching projective plane and sphere with standardmetrics and the ratio of the areas , or there exist other maximizingsequences.
Remark 1.6.
The degenerating sequence of metrics in Theorem 1.3illustrates the ”bubbling phenomenon” arising in the maximization ofhigher eigenvalues, see [44] for details.
Remark 1.7.
It was conjectured in the paper [33] , written after thefirst version of the present paper, that the equality Λ k ( RP ) = 4 π (2 k + 1) holds for any k > . The paper is organized in the following way. In Section 2 we recall therelation between extremal metrics and minimal immersions into spheresand explain the importance of upper bounds on the multiplicities ofeigenvalues.In Section 3 we recall the basics of the theory of nodal graphs andthe Courant Nodal Domain Theorem. We use them in Section 4 inorder to obtain an upper bound for the multiplicity m ( RP , g, λ ) . Letus remark that bounds on multiplicity of eigenvalues of the Laplace-Beltrami operator on surfaces were subject of numerous papers, seee.g. [7, 12, 25, 32, 40].In Section 5 we pass from minimal immersions to harmonic immer-sions, extend our considerations to harmonic immersions with branchpoints and metrics with conical singularities and explain why the re-sults from the previous sections also hold in this case.In Section 6 we recall the Calabi-Barbosa Theorem about harmonicimmersions with branch points S −→ S n and apply it to our situation.Section 7 contains the description of the space of harmonic immer-sions with branch points S −→ S due to Bryant and results aboutsingularities of these maps.Section 8 deals with the question of existence of maximal metrics.Finally, in Section 9 we prove Theorem 1.3. NIKOLAI S. NADIRASHVILI AND ALEXEI V. PENSKOI
Acknowledgements.
The authors are very indebted to Mikhail Kar-pukhin, Iosif Polterovich and the referee for useful remarks and sugges-tions. The second author is very grateful to the Institut de Math´ema-tiques de Marseille (I2M, UMR 7373) for the hospitality. The secondauthor is very indebted to Pavel Winternitz for fruitful discussions.2.
Extremal metrics and minimal immersions into spheres
In this Section we recall the relation between extremal metrics andminimal immersions into spheres and explain the importance of upperbounds on the multiplicities of eigenvalues.Let us recall the definition of a minimal map, see e.g. [16, 17].Let (
M, g ) be a Riemannian manifold of dimension m. Let α be asymmetric bilinear 2-form on T M.
Let σ k be the k -th elementary sym-metric function. Let σ k ( α ) = σ k ( λ , . . . , λ m ) , where λ i are eigenvaluesof α related to g, i.e. the roots of the polynomial det( α ij − λg ij ) = 0 . Definition 2.1.
Let ( M, g ) and ( N, h ) be Riemannian manifolds. Asmooth map f : M −→ N is called minimal if f is an extremal for thevolume functional V [ f ] = Z M p | σ m ( f ∗ h ) | dV ol g , where m = dim M. It is well-known that a surface M R is minimal if and only ifthe coordinate functions x i are harmonic with respect to the Laplace-Beltrami operator on M. A similar result holds for a minimal subman-ifold in R n . Since harmonic functions are eigenfunctions with eigen-value 0 , it is natural to ask what is an analogue of this statement for anon-zero eigenvalue. The answer was given by Takahashi in 1966. Theorem 2.2 (Takahashi [55]) . If an isometric immersion f : M R n +1 , f = ( f , . . . , f n +1 ) , is defined by eigenfunctions f i of the Laplace-Beltrami operator ∆ witha common eigenvalue λ, ∆ f i = λf i , then (i) the image f ( M ) lies on the sphere S nR of radius R with thecenter at the origin such that (3) λ = dim MR , (ii) the immersion f : M S nR is minimal. N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 7 If f : M S nR ⊂ R n +1 , f = ( f , . . . , f n +1 ) , is a minimal isometric immersion of a manifold M into the sphere S nR of radius R, then f i are eigenfunctions of the Laplace-Beltrami opera-tor ∆ , ∆ f i = λf i , with the same eigenvalue λ given by formula (3) . We introduce the eigenvalue counting function N ( λ ) = { λ i | λ i < λ } . Takahashi’s Theorem 2.2 implies that if M is isometrically minimallyimmersed in the sphere S nR , then among the eigenvalues of M thereare at least n + 1 eigenvalues equal to dim MR . It is easy to see that λ N ( dim MR ) is the first eigenvalue equal to dim MR . This is important dueto the following theorem.
Theorem 2.3 (Nadirashvili [41], El Soufi, Ilias [20]) . Let M S nR bean immersed minimal compact submanifold. Then the metric inducedon M by this immersion is extremal for the functional ¯ λ N ( dim MR )( M, g ) . If a metric on a compact manifold M is extremal for some eigenvaluethen there exists an isometric minimal immersion to the sphere M S nR by eigenfunctions with eigenvalue dim MR of the Laplace-Beltrami op-erator corresponding to this metric. If a metric is extremal for ¯ λ i ( M, g ) , then there exist a minimalimmersion of M by corresponding eigenfunctions into S n ⊂ R n +1 . Ifthe image is not contained in some hyperplane then one should haveat least n + 1 linearly indepenent eigenfunctions. This means that n + 1 m ( M, g, λ i ) . If follows that if we have an upper bound on the multiplicity of aneigenvalue then we have an upper bound on the dimension of the spherewhere M is minimally immersed by the corresponding eigenfunctions.We later use Theorem 2.3 for M = RP . In this case dim M = 2 . Using rescaling one can consider only the case of R = 1 . Remark thatTheorem 2.3 holds also for a non-orientable M. Since we are interested in the functional ¯ λ ( RP , g ) , we need an upperbound for m ( RP , g, λ ) in order to bound the dimension of the spherewhich is sufficient to consider. NIKOLAI S. NADIRASHVILI AND ALEXEI V. PENSKOI Nodal graphs and Courant Nodal Domain Theorem
In this Section we recall the basics of the theory of nodal graphs andthe Courant Nodal Domain Theorem that we need in order to obtainin Section 4 an upper bound m ( RP , g, λ ) . Let us now recall the following theorem due to Bers.
Theorem 3.1 (L. Bers [6]) . Let ( M, g ) be a compact 2-dimensionalclosed Riemannian manifold and x is a point on M. Then there existits neighbourhood chart U with coordinates x = ( x , x ) ∈ U ⊂ R cen-tered at x such that for any eigenfunction u of the Laplace-Beltramioperator on M there exists an integer n > and a non-trivial homoge-neous harmonic polynomial P n ( x ) of degree n on the Euclidean plane R such that u ( x ) = P n ( x ) + O ( | x | n +1 ) , where x ∈ U . The integer n from Bers’s Theorem 3.1 is called an order of zero ofan eigenfunction u at a point x . Let us denote it by ord x u. Consider the sets N l ( u ) = { x ∈ M | ord x u > l } . Definition 3.2.
The set N ( u ) is called a nodal set of u. Connectedcomponents of its complement M \ N ( u ) are called the nodal domainsof u. It is well-known that in the polar coordinates r, ϕ in R any homo-geneous harmonic polynomial P n of degree n has the form(4) P n ( r, ϕ ) = r n ( A cos nϕ + B sin nϕ ) . The zeroes of such polynomials form n straight lines intersecting atorigin at equal angles.It follows that the nodal set N ( u ) is a graph such that the points of N ( u ) are its vertices and the connected components of N ( u ) \ N ( u )are its edges. Definition 3.3.
This graph is called a nodal graph of an eigenfunc-tion u. Let us remark that if x is a vertex of the nodal graph then it is azero of u and there is 2 ord x u edges emanating from x in a sufficientlysmall neighborhood of x . Globally some of these edges could form loopsstarting and ending at x . Let us remark that locally in a neighborhood of zero x of order n the nodal graph N ( u ) looks like a star consisting of n rays with equal N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 9 angles between adjacent rays . Let us give the following definition inorder to be more precise.
Definition 3.4.
A star S x ( N ( u )) at the vertex x of the nodal graph N ( u ) of an eigenfunction u consists of n unitary tangent vectors toedges emanating from x , where n is the order of zero of u at x . It follows from formula (4) that in coordinates given by the BersTheorem 3.1 the angles between adjacent vectors in S x ( N ( u )) areequal.If one has a triangulation of a surface M with V vertices, E edgesand F faces, then one has the well-known formula for the Euler char-acteristic,(5) χ ( M ) = V − E + F. Let us consider an eigenfunction u. If we consider the vertices of anodal graph, the edges of a nodal graph and the nodal domains of afunction u, then the formula (5) does not in general hold since thenodal domains are not in general homeomorphic to a disc. As a result,we obtain in this case only the Euler inequality(6) χ ( M ) V − E + F that implies the following well-known Lemma. Lemma 3.5.
Let u be an eigenfunction. Let x j , j = 1 , . . . , n, be zeroesof u of order m j > . Let Ω j , j = 1 , . . . , s, be nodal domains of thefunction u. Then s > χ ( M ) − n + n X j =1 m j . Proof.
One can immediately see that V = n, F = s. Since 2 ord x j u =2 m j edges emanate from x j and each edge connects two vertices, onehas E = n P j =1 m j . It is sufficient now to apply inequality (6). (cid:3)
Let us now recall the following theorem (remark that we count eigen-values starting from λ ). Theorem 3.6 (Courant Nodal Domain Theorem [15]) . An eigenfunc-tion corresponding to the eigenvalue λ i has at most i +1 nodal domains. Lemma 3.5 and Courant Nodal Domain Theorem 3.6 imply immedi-ately the following Proposition.
Proposition 3.7.
Let u be an eigenfunction corresponding to the eigen-value λ i . Let x j , j = 1 , . . . , n, be zeroes of u of order m j > . Then (7) i + 1 > χ ( M ) − n + n X j =1 m j . Multiplicity of the second non-zero eigenvalue of theLaplace-Beltrami operator on the projective plane
It was proven by the first author in the paper [40] that the followingupper bound for the multiplicities of the eigenvalues of the Laplace-Beltrami operator on the projective plane holds, m ( RP , g, λ i ) i + 3 . For the first eigenvalue this means m ( RP , g, λ ) , which is a sharp inequality and was proved first by Besson [7].For the second eigenvalue we have m ( RP , g, λ ) . The main goal of this Section is to improve the last upper bound.
Proposition 4.1.
The following upper bound for the multiplicity ofthe second eigenvalue of the Laplace-Beltrami operator on the projectiveplane holds, (8) m ( RP , g, λ ) . For the purposes of the present paper the upper bound (8) is suffi-cient. However, this bound is further improved and generalized in thepaper [4].Let us postpone the proof and start with several lemmas.
Lemma 4.2.
Let u , . . . , u be linearly independent eigenfunctions cor-responding to the second eigenvalue λ ( RP , g ) . Then for any point x ∈ RP there exists a non-trivial linear combination v = P i =1 α i u i suchthat the eigenfunction v has a zero of order at least at the point x . Proof.
This lemma is a particular case of Lemma 4 from paper [40]. Infact, the proof is an easy corollary of Bers Theorem 3.1 and formula (4). (cid:3)
N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 11
Lemma 4.3.
Let u be an eigenfunction corresponding to the secondeigenvalue λ ( RP , g ) such that at a point x this eigenfunction has azero of order at least . Then x is the only zero of u of order greaterthan and the order of zero at x is exactly . Proof.
Since i = 2 , χ ( RP ) = 1 , inequality (7) implies in this case theinequality 2 > n X j =1 ( m j − . Since m > m i > i > , we have m − > , m i − > i > . It follows that m = 3 and n = 1 . (cid:3) Let us fix a point x ∈ RP and consider the space V of eigenfunc-tions of ∆ corresponding to the second eigenvalue λ ( RP , g ) with zeroof order at least 3 at x . Let us suppose that dim V > . Then thereexist two linearly independent eigenfunctions u , u ∈ V. Consider thefamily of eigenfunctions(9) v τ = cos τ u + sin τ u . Lemma 4.4.
The star S x ( N ( v τ )) defines the eigenfunction v τ fromformula (9) completely up to a sign, i.e. if S x ( N ( v τ )) = S x ( N ( v τ )) then v τ = ± v τ . Proof.
Since S x ( N ( v τ )) = S x ( N ( v τ )) , the homogeneous harmonicpolynomials P τ and P τ corresponding by Bers Theorem 3.1 to v τ and v τ are proportional. But then formula (9) implies that either P τ = P τ or P τ = − P τ . In the first case we have v τ − v τ = O ( | x | ) . Then v τ − v τ is an eigenfunction of ∆ corresponding to the secondeigenvalue λ ( RP , g ) with zero of order at least 4 at x . It followsfrom Lemma 4.3 that v τ − v τ ≡ v τ = v τ . A similarargument shows that in the second case we have v τ = − v τ . (cid:3) Lemma 4.5.
Let x ∈ RP and V be the space of eigenfunctions of ∆ corresponding to the second eigenvalue λ ( RP , g ) with a zero of orderat least at x . Then dim V . Proof.
Let us suppose that dim V > . Then there exist two lin-early independent eigenfunctions u , u ∈ V. Consider the family ofeigenfunctions v τ ∈ V defined by equation (9) and the family of nodalgraphs N ( v τ ) . Let p : S −→ RP be the standard projection. Let us considerthe eigenfunction u ◦ p on the sphere S . It follows from the above mentioned arguments that the nodal graph N ( u ◦ p ) on the spherehas the following properties: • there are exactly two vertices p − ( x ) that we call N and S ,they are antipodal, • locally exactly 6 edges emanate from each vertex. Claim 1.
All nodal domains are topological disks.
Indeed, the Euler inequality (6) for the nodal graph of u on RP implies that there is at least 3 nodal domains. In the same time, theCourant Nodal Domain Theorem 3.6 implies that there are at most 3nodal domains. As a result, there are exactly 3 nodal domains for u on RP . Now remark that it follows that the Euler inequality (6) turnsinto an equality. It is possible if and only if all nodal domains of u are topological disks. Let us consider now the nodal graph of u ◦ p onthe sphere S . Since there are no non-trivial coverings of a disk, andthe nodal domains of u ◦ p are preimages of the nodal domains of u , there are exactly 6 nodal domains of u ◦ p on S and all are topologicaldisks. Claim 2.
The nodal graph of u ◦ p is invariant under rotation by ± π around the axis going though N and S. Let us emphasize that “invariant” here and below means “invariantup to a homotopy preserving tangent vectors at the point N and S ”.The proof of the Claim 2 is as follows. Since v = − v π , the nodalgraph N ( v τ ) is deformed continuously when τ changes from 0 to π and the result coincides with the initial graph, N ( v ) = N ( v π ) . Since N ( v ) = N ( v π ) , when τ changes from 0 to π the 6-raystar S x ( N ( v τ )) rotates by angle k π . But then k = ± . Indeed, if k = ± < τ < π such that S x ( N ( v τ )) isobtained from S x ( N ( v )) by the rotation by angle (sgn k ) π . Then S x ( N ( v τ )) = S x ( N ( v )) and Lemma 4.4 implies that v τ = ± v , but this contradicts the inequality 0 < τ < π. Let us change the direction of counting the angle in such a way thatthe angle of rotation is π . Then we have the following result: when τ changes from to π, the star S x ( N ( v τ )) rotates exactly by π . Claim 3.
There are no loops in the nodal graph, i.e. all edges join N and S. Indeed, let us consider an edge γ emanating from N such that an-other endpoint of γ is also N. Let us numerate the vectors from thestar S N ( N ( u ◦ p )) in consecutive order as v , . . . , v in such a way that N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 13 the edge γ emanates from N with the tangent vector v . Then there istwo cases.In case I the tangent vector at the endpoint N of γ is − v . In thiscase the nodal graph is clearly not invariant under the rotation by π . Remark that the tangent vector − v at the endpoint N could beconsidered as − v with another numeration order of the vectors fromthe S N ( N ( u ◦ p )) . In case II the tangent vector at the endpoint N of γ is − v k , where k = 2 or k = 3 . Since the nodal graph is invariant under the rotationby π , the edge emanating from N with tangent vector v has − v k +1 asits tangent vector at its endpoint N. This implies that there are twoloops on a sphere intersecting transversally at exactly one point N butthis is impossible.Remark that the tangent vector − v at the endpoint N could beconsidered as − v with another numeration order of the vectors from S N ( N ( u ◦ p )) . In both cases we obtain a contradiction with the assumption thatan edge can start and end at the same vertex. Hence, all edges join N to S. Let us now finish the proof of Lemma 4.5. Consider the nodal graphof u ◦ p on S . A small neighbourhood of N is divided by the graphin 6 sectors, where the signs of u ◦ p alternate. By Claim 3, thesesectors lie in different nodal domains. Since there are exactly 6 nodaldomains, there are three of them where u ◦ p is positive and three ofthem where u ◦ p is negative.Let us consider the action of the antipodal map σ on nodal domains.It is well-defined. Indeed, suppose x and y belong to the same nodaldomain. Then one can join x and y by a path inside their nodal domain.Applying σ we obtain a path joining σ ( x ) and σ ( y ) , on which u ◦ p does not change sign. Thus, σ ( x ) and σ ( y ) belong to the same nodaldomain.Since u ◦ p is obtained from the eigenfunction u on RP , the an-tipodal map preserves the sign of u ◦ p. Since there are three nodaldomains of the same sign, there is at least one nodal domain that ismapped by σ to itself. At the same time, by Claim 1 each nodal do-main is a topological disk. Since σ is a free involution, it can not mapa disk into itself by Brouwer’s theorem. This completes the proof ofLemma 4.5. (cid:3) Proof of Proposition 4.1. Let us suppose that m ( RP , g, λ ) > . Thenthere exist 7 linearly independent eigenfunctions ϕ , . . . , ϕ correspond-ing to the second eigenvalue λ ( RP , g ) . Let us fix a point x ∈ RP . Let us apply Lemma 4.2 to ϕ , . . . , ϕ and obtain an eigenfunction u = P i =1 α i ϕ i with zero of order at least3 at the point x . Then by Lemma 4.3 the point x is a zero of orderexactly 3 . We can suppose without loss of generality that α = 0 . Let us thenapply Lemma 4.2 to the eigenfunctions ϕ , . . . , ϕ and obtain an eigen-function u = P i =2 β i ϕ i with zero of order at least 3 at the point x . Then by Lemma 4.3 the point x is a zero of order exactly 3 . Let us remark that u and u are linearly independent since α = 0 . This contradicts Lemma 4.5. (cid:3) Harmonic maps with branch points and metrics withconical singularities
Let us recall the definition of a harmonic map, see e.g. the re-view [16].
Definition 5.1.
Let ( M, g ) and ( N, h ) be Riemannian manifolds. Asmooth map f : M −→ N is called harmonic if f is an extremal forthe energy functional (10) E [ f ] = Z M | df ( x ) | dV ol g . The following theorem (see, e.g. the paper [17]) explains the re-lation between minimal and harmonic maps in the class of isometricimmersions.
Theorem 5.2.
Let
M, N be Riemannian manifolds. If f : M N isan isometric immersion, then f is harmonic if and only if f is minimal. Theorem 2.3 and Theorem 5.2 imply the following Proposition.
Proposition 5.3.
The extremal metrics on a compact surface M areexactly the metrics induced on M by harmonic immersions M S n . It turns out however that it is useful to consider a wider class ofharmonic immersions with branch points.
Definition 5.4 (see e.g. [22]) . Let M be a manifold of dimension . A smooth map f : M −→ N has a branch point of order k at point p if there exist local coordinates u , u centered at p and defined ina neighborhood of p and x , . . . , x n centered at f ( p ) and defined in a N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 15 neighborhood of f ( p ) such that in these coordinates f is written as x + ix = w k +1 + σ ( w ) ,x k = χ k ( w ) , k = 3 , . . . , n,σ ( w ) , χ k ( w ) = o ( | w | k +1 ) ,∂σ∂u j ( w ) , ∂χ k ∂u j ( w ) = o ( | w | k ) , j = 1 , , k = 3 , . . . , n, where w = u + iu . If M is compact then a map f : M −→ N could have only finitenumber of branch points.However we have now a problem. If ( N, g ) is a Riemannian manifoldand f : M N is an immersion with branch points, then the inducedmetric f ∗ g is not a smooth metric. Definition 5.5 (see e.g. [34]) . A point p on a surface M is called aconical singularity of order α > − or angle π ( α + 1) of a metric g if in an appropriate local complex coordinate z centered at p the metrichas the form g ( z ) = | z | α ρ ( z ) | dz | in a neighborhood of p, where ρ (0) > . Then we obtain immediately the following Proposition.
Proposition 5.6. If M is a compact surface, ( N, h ) is a Riemannianmanifold and f : M N is an immersion with branch point, then g = f ∗ h is a smooth Riemannian metric except a finite number ofbranch points of the map f. At these points the metric g has conicalsingularities. The order of the conical singularity at a point p is equalto the order of p as a branch point. Thus, we switch to a setting larger than the initial one. We considernot only Riemannian metrics but also Riemannian metrics with a finitenumber of conical singularities and not only harmonic immersions butalso harmonic immersions with branch points. Then we should checkthat all key results from the previous sections hold.It is well-known that the eigenvalues of the Laplace-Beltrami opera-tor could be defined using a variational approach,(11) λ k = min V ⊂ H M ) dim V = k max u ∈ V u ⊥ R [ v ] , where R [ v ] = R M |∇ u | dV ol R M | u | dV ol is the Rayleigh quotient. This formula holds also in the case of metricswith conical singularities, see e.g. [34]. Proposition 5.7 ([35, Corollary 4.7]) . Theorem 2.3 holds if we con-sider metrics with conical singularities and harmonic maps with branchpoints.
The next problem is to prove that
V, E and F are finite and in-equality (6) holds. The problem is that in the case of surfaces withisolated conical singularities the points of N ( u ) can a priori accumu-late towards singularities. It turns out that it is not possible since thispossibility can be ruled out using resolution procedure used in the pa-pers [32, Lemma 3.1.1] and [34] in order to prove the finiteness of anodal graph in other contexts.Let us define the resolution procedure following the paper [32]. Let x ∈ N ( u ) be a vertex of nodal graph. If n = ord x ( u ) then the degreeof this vertex is 2 n. According to Bers’s Theorem 3.1 there exists aneighborhood U of x diffeomorphic to a disk such that U does notcontain other vertices and such that nodal arcs incident to x intersect U at 2 n points precisely. Let us denote these intersection points by y i , where i = 0 , . . . , n − , and assume that they are ordered consequentlyin the clockwise fashion. A new graph is obtained from the nodal graphby changing it inside U and removing possibly appeared edges withoutvertices. More precisely, we remove the nodal set inside U and round-off the edges on the boundary U by non-intersecting arcs in U joiningthe points y j and y j +1 . If there was an edge that starts and ends at x, then such a procedure may make it into a loop. If this occurs, thenwe remove this loop to obtain a genuine graph in the sufrace. The newgraph has one vertex less and at most as many faces as the originalgraph.We give now a short proof by Karpukhin (with his permission). Proposition 5.8 (Karpukhin, [29]) . A nodal graph of an eigenfunction u on a surface M with isolated conical singularities is finite. Proof.
Suppose that there are infinitely many points in N ( u ), it iseasy to see that in this case the set N ( u ) is countable. Then theonly possible accumulation points of N ( u ) are conical singularities.For each conical singularity p j let us choose a base of neighbourhoods V ( j ) i such that ¯ V ( j ) i +1 ⊂ V ( j ) i and N ( u ) ∩ ∞ S i =1 ∂V ( j ) i = ∅ . Hence forthe sets V i = S j V ( j ) i we have ¯ V i +1 ⊂ V i , N ( u ) ∩ ∞ S i =1 ∂V i = ∅ and M \ V i contains only finite quantity of elements of N ( u ). For any i for N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 17 the points of N ( u ) in V i \ ¯ V i +1 one can choose a collection of disjointneighbourhoods U ki such that ¯ U ki ⊂ V i \ ¯ V i +1 . Thus we constructed acollection of disjoint neighbourhoods of all points in N ( u ).Next we apply the resolution procedure at all but finite number ofvertices. Choosing this finite number big enough and applying Euler’sinequality we arrive at contradiction with Courant’s nodal domain the-orem. (cid:3) Thus, in the setting of metrics with conical singularities inequal-ity (6) and all results obtained with its help hold, including the keyupper bound m ( RP , g, λ ) Calabi-Barbosa theorem and its implications
Now we should study harmonic immersions with branched points RP S n . Since we have the upper bound m ( RP , g, λ ) λ are among immer-sions RP S . Let p : S −→ RP be the standard projection. We can lift a har-monic immersion with branch points f : RP S to a harmonicimmersion with branch points F = f ◦ p : S −→ S . The following theorem was proved by Calabi in 1967 and later refinedby Barbosa in 1975. Let g S n denote the standard metric on S n . Theradius of S n is 1 . Theorem 6.1 (Calabi [11], Barbosa [3]) . Let F : S −→ S n be aharmonic immersion with branch points such that the image is notcontained in a hyperplane. Then (i) the area of S with respect to the induced metric Area( S , F ∗ g S n ) is an integer multiple of π ;(ii) n is even, n = 2 m, and Area( S , F ∗ g S n ) > πm ( m + 1) . Definition 6.2. If Area( S , F ∗ g S n ) = 4 πd, then we say that F is ofharmonic degree d. We obtain immediately a lower bound for the harmonic degree.
Proposition 6.3.
Let F : S −→ S m be a harmonic immersion withbranch points such that the image is not contained in a hyperplane.Then d > m ( m +1)2 . Calabi-Barbosa Theorem 6.1 implies the following Proposition.
Proposition 6.4.
It is sufficient for our goals to consider harmonicimmersions with branch points F : S −→ S (such that the image is notcontained in a hyperplane) of harmonic degree d > and F : S −→ S . It follows that we should consider only harmonic immersions withbranch points RP −→ S and RP −→ S . However, the followingProposition permits to exclude maps RP −→ S . Proposition 6.5 (see e.g. [16]) . Every harmonic map RP −→ S isconstant. Harmonic maps from S to S and their singularities Let us recall the well-known Penrose twistor map T : CP −→ HP ∼ = S , T ([ z : z : z : z ]) = [ z + z j : z + z j ] . Let z be a conformal parameter on S . Definition 7.1.
Let us call a curve f : S −→ CP , f ( z ) = [ f ( z ) : f ( z ) : f ( z ) : f ( z )] , horizontal if f ′ f − f f ′ + f ′ f − f f ′ = 0 . In 1982, Bryant described in the paper [10] a very important rela-tion between harmonic immersions with branch points S −→ S and(anti)holomorphic horizontal curves in CP . Let A : S −→ S be the antipodal map. Theorem 7.2 (Bryant [10]) . For each harmonic immersion with branchpoints F : S −→ S there exists either a holomorphic or an antiholo-morphic horizontal curve f : S −→ CP , such that T ◦ f = F, CP T (cid:15) (cid:15) S F / / f = = ④④④④④④④④④ S For each (anti)holomorphic horizontal curve f : S −→ CP the map F = T ◦ f : S −→ S is a harmonic immersion with branch points.If a harmonic immersion F : S −→ S has a holomorphic (anti-holomorphic) horizontal curve f : S −→ CP , then A ◦ F : S −→ S has an antiholomorphic (holomorphic) horizontal curve. N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 19
Definition 7.3.
An (anti)holomorphic horizontal curve f appearing inBryant’s Theorem 7.2 is called the lift of an harmonic immersion F. Let us remark that F and A ◦ F induce the same metric on S . It follows that it is sufficient to consider harmonic immersions withholomorphic lifts.
Theorem 7.4 (Bryant [10]) . Let F : S −→ S be a harmonic immer-sion with branched points of harmonic degree d with holomorphic lift f : S −→ CP . Then f : S −→ CP is an algebraic curve of degree d. Now we need some results from the theory of higher singularities ofthese holomorphic horizontal lifts, see e.g. the paper [8] by Bolton andWoodward. Let [ f ( z )] = [ f ( z ) , f ( z ) , f ( z ) , f ( z )] be a representativeof f : S −→ CP in the homogeneous coordinates in CP . Let f ( i ) ( z )denotes the i th derivative of f ( z ) . Let Z ( f ) = { z | f ( z ) ∧ f ′ ( z ) ∧ . . . ∧ f (3) ( z ) = 0 } . Remark that Z ( f ) consists of isolated points if f is linearly full, i.e. ifthe image of f is not inside a hyperplane.Let us apply the Gram-Schmidt orthogonalization process to f ( z ) , f ′ ( z ) , f ′′ ( z ) , f ′′′ ( z ) at z Z ( f ) and obtain ˜ f ( z ) = f ( z ) , ˜ f ( z ) , ˜ f ( z ) , ˜ f ( z ) . Then it turns out that the trivial bundle S × C has an orthogonaldecomposition as a sum of holomorphic linear bundles S × C = L ⊕ . . . ⊕ L , such that L i is spanned by ˜ f i for z Z ( f ) . These L i describe the Fr´enetframe for f. The bundle map ∂ i : T , S ⊗ L i −→ L ⊥ i given by ∂ i (cid:18) ∂∂z ⊗ s i (cid:19) = (cid:18) ∂s i ∂z (cid:19) ⊥ , where s i is a local holomorphic section of L i , and (cid:18) ∂s i ∂z (cid:19) ⊥ denotes thecomponent of ∂s i ∂z orthogonal to L i , satisfies ∂ i (cid:18) ∂∂z ⊗ ˜ f i ( z ) (cid:19) = ˜ f i +1 ( z ) . It follows that ∂ i is a holomorphic map and has the image in L i +1 . Definition 7.5.
A (linearly full) holomorphic curve f : S −→ CP has a higher singularity of type ( r ( p ) , r ( p ) , r ( p )) at a point p ∈ Z ( f ) if for i = 0 , , the holomorphic bundle maps ∂ i has a zero of order r i ( p ) at p and r ( p ) + r ( p ) + r ( p ) > . It turns out that for a horizontal curve one has r ( p ) = r ( p ) , i.e. itshigher singularity type at a point p is described by two integers r ( p ) , and r ( p ) . Let us define quantities r = X p r ( p ) , r = X p r ( p ) . The next Proposition relates them to the degree d. Proposition 7.6 (Bolton, Woodward [9]) . For a linearly full holomor-phic horisontal curve in CP the following equation holds, r + r = 2 d − . We need here to recall the definition of an umbilic point.
Definition 7.7.
Let ( M, g ) and ( N, h ) be Riemannian manifolds and ∇ M and ∇ N be the corresponding Levi-Civita connections.Let F : M −→ N be an immersion. Then a) the second fundamentalform II F of F is defined by the formula ∇ NdF ( X ) dF ( Y ) = dF ( ∇ MX Y ) + II F ( X, Y ); b) the vector field ζ = 1dim M tr II F is called a mean curvature normal vector;c) a point p ∈ M is called an umbilic point if there exists a vector v ∈ T F ( p ) N such that at the point p one has (12) II Fp ( X, Y ) = g p ( X, Y ) · v. It follows immediately from Definition 7.7 that if p is an umbilic then II Fp ( X, Y ) = g p ( X, Y ) · ζ ( p ) . As an example it is useful to consider a classical case of an immer-sion F of a two-dimensional surface M to N = R equipped with theeuclidean metric h. Let us consider the induced metric g = F ∗ h on M. Then it is easy to check that II F ( X, Y ) = II ( X, Y ) · ~n, where II ( X, Y )is the classical second fundamental form of the surface M and ~n is aunit normal vector field on M. Let us recall that in the basis consistingof principal directions the metric g has the identity matrix and theclassical second fundamental form II has the diagonal matrix with theprincipal curvatures λ and λ on the diagonal. Then formula (12) isequivalent to the equality λ = λ which is the classical definition ofan umbilic point for a two-dimensional surface in the Euclidean space R . Let z be a conformal parameter on S . It is easy to check that thefollowing Proposition holds.
N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 21
Proposition 7.8.
A point p ∈ S is an umbilic point of a harmonicimmersion F : S −→ S if and only if (13) II Fp ( ∂/∂z, ∂/∂z ) = 0 . Proof.
If a point p is umbilic then II Fp ( ∂/∂z, ∂/∂z ) = g p ( ∂/∂z, ∂/∂z ) · ζ ( p ) = 0 , since z is a conformal coordinate and g p = 2Φ | dz | for some Φ . Let equality (13) holds. Since F is real, this implies that II Fp ( ∂/∂ ¯ z, ∂/∂ ¯ z ) = 0 . If follows that formula (12) holds for v = II Fp ( ∂/∂z, ∂/∂ ¯ z ) g p ( ∂/∂z, ∂/∂ ¯ z ) , and p is umbilic. (cid:3) The higher singularities of a holomorphic horizontal lift f of a har-monic immersion with branched points F : S −→ S are related to thebranch points and the umbilics of F. Proposition 7.9 (Bolton, Woodward [8, 9]) . A point p is a branchpoint of F if and only if r ( p ) > . Moreover, r ( p ) is equal to theorder of zero of dF ( ∂/∂z ) at p. If r ( p ) = 0 then p is an umbilic if and only if r ( p ) > . Moreover, r ( p ) is equal to the order of zero of II F ( ∂/∂z, ∂/∂z ) at p. The higher singularities of f occur exactly at the branch points andumbilics of F. Combining Propositions 7.6 and 7.9, we obtain the following Propo-sition.
Proposition 7.10.
Let F : S −→ S be a harmonic immersion withbranch points of harmonic degree d. Then (i) if d = 3 then F does not have either branch points or umbilics, (ii) if d > then F has at least one branch point or an umbilic. Existence of maximal metrics
What can we say about the existence of the maximal metric for agiven eigenvalue on a given surface? The situation in the case of thefirst eigenvalue is the following.
Theorem 8.1 (Matthiesen, Siffert [39]) . For any closed surface M, there is a metric g on M, smooth away from finitely many conicalsingularities, achieving Λ ( M ) , i.e. Λ ( M ) = ¯ λ i ( M, g ) = λ ( M, g ) Area(
M, g ) . However, as we observed in the Introduction, the situation is morecomplicated for higher eigenvalues. In particular, on the sphere thereis no maximal metrics for ¯ λ k if k > , see the papers [42, 51] for k = 2 , [45] for k = 3 and [33] for arbitrary k > . It turns out that extremal metrics for higher eigenvalues on thesphere exhibit the so-called “bubbling phenomenon”. This phenom-enon was studied in details by the first author and Sire in the pa-pers [43, 44] and also by Petrides [52] in the context of maximizationof eigenvalues in a given conformal class. More precisely, they inves-tigated the question of existence of Riemannian metrics with conicalsingularities for which the quantityΛ k ( M, [ g ]) = sup h ∈ [ g ] ¯ λ k ( M, h )is attained, where [ g ] denotes the class of metrics conformally equivalentto g. The equality Λ k ( S ) = 8 πk. proven in the recent paper [33] combined with [52, Theorem 2] impliesthe following result. Proposition 8.2 ([33]) . Let ( M, g ) be a closed Riemannian surfaceand k > . If Λ k ( M, [ g ]) > Λ k − ( M, [ g ]) + 8 π, then there exists a maximal metric ˜ g ∈ [ g ] , smooth except possibly at afinite set of conical singularities, such that Λ k ( M, [ g ]) = ¯ λ k ( M, ˜ g ) . Since there is only one conformal structure on RP , see e.g. the book[53], Λ ( RP , [ g ]) = Λ ( RP ) and we have the following Proposition. Proposition 8.3. If Λ ( RP ) > Λ ( RP ) + 8 π = 20 π, then there exists a maximal metric ˜ g , smooth except possibly at a finiteset of conical singularities, such that Λ ( RP ) = ¯ λ ( RP , ˜ g ) . N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 23 Proof of Theorem 1.3
Let us consider a sequence { g n } of metrics of area one on the pro-jective plane such that the limiting metric is a singular metric realizedas a union of the projective plane and the sphere touching at a point,with standard metrics and the ratio of the areas 3 : 2 . In this casethe limit spectrum is the union of spectra of the projective plane withstandard metric g ′ of area and of the sphere with standard metric g ′′ of area , see e.g. [14, Section 2] and [1] for more details about thelimit spectrum. Thenlim n →∞ λ ( RP , g n ) = λ ( RP , g ′ ) = λ ( S , g ′′ ) = 20 π. Hence, lim n →∞ ¯ λ ( RP , g n ) = 20 π. If Λ ( RP ) = 20 π, then the proof is finished. If Λ ( RP ) > π, thenby Proposition 8.3 there exists a maximal metric ˜ g , smooth exceptpossibly at a finite set of conical singularities, such thatΛ ( RP ) = ¯ λ ( RP , ˜ g )As we already know from Proposition 5.3, the metric ˜ g is induced on RP by a harmonic immersion with branched points RP −→ S n . The upper bound m ( RP , g, λ ) λ are among immersions RP S . Let p : S −→ RP be the standard projection. We can lift a har-monic immersion with branch points f : RP S to a harmonicimmersion with branch points F = f ◦ p : S −→ S . Calabi-Barbosa’s Theorem 6.1 implies that it is sufficient to considera harmonic immersion with branch points F : S −→ S of harmonicdegree d > F : S −→ S , see Proposition 6.4. However,Proposition 6.5 says that we can exclude harmonic maps RP −→ S since they are constant. As a result, we should consider only a harmonicimmersion with branch points RP −→ S . Consider a harmonic immersion with branch points f : RP −→ S corresponding to λ and its lift F = f ◦ p : S −→ S . As we knowfrom Proposition 7.10, there are two different cases depending on itsharmonic degree d. Consider the case d = 3 . Let g S n denote the standard metric on S n . Since d = 3 , one has Area( S , F ∗ g S n ) = 12 π due to Calabi-BarbosaTheorem 6.1. Then Area( RP , f ∗ g S n ) = 6 π because p : S −→ RP is a two-sheeted covering. Since the radius of S n is 1 , Takahashi Theo-rem 2.2 implies that λ = 2 . As a result, ¯ λ ( RP , f ∗ g S n ) = 12 π < π and the induced metric is not maximal.Consider the case d > . In this case Proposition 7.6 implies that F = f ◦ p : S −→ S and hence f : RP −→ S have at least one branchpoint or umbilic. Let us prove that an immersion by eigenfunctionscorresponding to λ cannot have either branch points or umblilics.Let us suppose that f = ( f , . . . , f ) and p ∈ RP is a branch point.It follows that f i are linearly independent eigenfunctions with eigen-value λ = 2 such that df i ( p ) = 0 . One can then construct at least 4 lin-early independent eigenfunctions ˜ f i , i = 1 , . . . , , such that ˜ f i ( p ) = 0 ,d ˜ f i ( p ) = 0 . This means that all ˜ f i have zero of order 2 at p. Using BersTheorem 3.1 one can then construct at least 2 linearly independenteigenfunctions with eigenvalue λ = 2 with zero of order 3 at p, butthis contradicts Lemma 4.5.Let us suppose that f = ( f , . . . , f ) and p ∈ RP is an umbilic.Let z be a local conformal parameter on RP in a neighborhood of thepoint p. Let ds = 2Φ | dz | be the induced metric. It is well-known that f z ¯ z = − Φ f, see e.g. [3, 11], this is in fact a harmonic map equation inthis particular setting. Since p is an umbilic, II fp ( ∂/∂z, ∂/∂z ) = 0 . Bydefinition of the second fundamental form, this means that f zz ( p ) is atangent vector and hence f zz ( p ) is a linear combination of f z ( p ) and f ¯ z ( p ) . It follows that there exist α, β ∈ C such that for any i = 1 , . . . , f iz ¯ z ( p ) = − Φ( p ) f i ( p ) , (14) f izz ( p ) = αf iz ( p ) + βf i ¯ z ( p ) , (15) f i ¯ z ¯ z ( p ) = ¯ βf iz ( p ) + ¯ αf i ¯ z ( p ) . (16)Remark that these equations are linear. This implies that they holdfor any linear combination of f i . Now one can construct two linear combinations ϕ = X i =1 A i f i , ψ = X i =1 B i f i with real coefficients A i , B i such that ϕ and ψ have zero of order 2at p. It follows that ϕ ( p ) = ϕ z ( p ) = ϕ ¯ z ( p ) = 0 , ψ ( p ) = ψ z ( p ) = ψ ¯ z ( p ) = 0 . As it was remarked before, the equations (14) , (15) , (16) hold for ϕ and ψ. It follows that they are eigenfunctions with eigenvalue λ = 2with zero of order 3 at the point p. This contradicts Lemma 4.5.
N ISOPERIMETRIC INEQUALITY ON THE PROJECTIVE PLANE 25
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E-mail address : [email protected] (Alexei V. Penskoi) Department of Higher Geometry and Topology,Faculty of Mathematics and Mechanics, Moscow State University,Leninskie Gory, GSP-1, 119991, Moscow, Russia and
Faculty of Mathematics, National Research University Higher Schoolof Economics, 6 Usacheva Str., 119048, Moscow, Russia and
Interdisciplinary Scientific Center J.-V. Poncelet (ISCP, UMI 2615),Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia
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