An isoperimetric inequality for Laplace eigenvalues on the sphere

arXiv:1706.05713v1 [math.DG] 18 Jun 2017

AN ISOPERIMETRIC INEQUALITY FOR LAPLACE EIGENVALUES ON THE SPHERE MIKHAIL KARPUKHIN, NIKOLAI NADIRASHVILI, ALEXEI V. PENSKOI, AND IOSIF POLTEROVICH Abstract. We prove that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k = 1 (J. Hersch, 1970), k = 2 (N. Nadirashvili, 2002 and R. Petrides, 2014) and k = 3 (N. Nadirashvili and Y. Sire, 2015). Our proof uses the bubbling feature of the extremal metrics for higher eigenvalues exhibited by Nadirashvili-Sire, as well as certain properties of harmonic maps between spheres. In particular, the key new ingredient of the proof is a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.

1. Introduction and main results Let M be a closed surface and g be a Riemannian metric on M. Consider the Laplace-Beltrami operator ∆ : C ∞ (M ) −→ C ∞ (M ) associated with the metric g, p 1 ∂ ij ∂f p ∆f = − , |g|g ∂xj |g| ∂xi and its eigenvalues (1)

0 = λ0 (M, g) < λ1 (M, g) 6 λ2 (M, g) 6 λ3 (M, g) 6 . . . ,

where each eigenvalue is repeated according to its multiplicity. Let Area(M, g) denote the area of M with respect to the Riemannian metric g. Since the eigenvalues possess a rescaling property, ∀t > 0

λi (M, tg) =

λi (M, g) , t

2010 Mathematics Subject Classification. 58J50, 58E11, 53C42. The work of M. K. was supported by Tomlinson Fellowship. The work of A. P. was supported by Russian Science Foundation grant no. 16-11-10260 at Moscow State University. The work of I. P. was supported by NSERC, FRQNT and the Canada Research Chairs program. 1

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M. KARPUKHIN, N. NADIRASHVILI, A. PENSKOI, AND I. POLTEROVICH

it is natural to consider functionals ¯ i (M, g) = λi (M, g) Area(M, g), λ ¯ i (M, g) invariant under the rescaling transformation g 7→ tg. The functional λ is sometimes called an eigenvalue normalized by the area or simply a normalized eigenvalue. Our principal interest is in the following geometric optimization problem for the Laplace-Beltrami eigenvalues: given a compact surface M and a number i ∈ N find the quantity (2)

¯ i (M, g), Λi (M ) = sup λ g

where the supremum is taken over the space of all Riemannian metrics g on M. ¯ i (M, g) is invariant under rescaling of the metric Since the functional λ g 7→ tg, where t ∈ R+ , this problem is equivalent to the question of finding sup λi (M, g), where the supremum is taken over the space of all Riemannian metrics g of area 1 on M. This problem is often referred to as the isoperimetric problem for the Laplace-Beltrami eigenvalues, since for any metric g of unit area on M , one has the inequality λi (M, g) 6 Λi (M ). Definition 1.1. Let M be a closed surface. A metric g0 on M is called ¯ i (M, g) if maximal for the functional λ ¯ i (M, g0 ) Λi (M ) = λ Note that if a maximal metric exists, it is defined up to multiplication by a positive constant due to the rescaling invariance of the functional. The main goal of the present paper is to provide a complete solution of the isoperimetric problem for the Laplace-Beltrami eigenvalues on a sphere. We will say that a metric on a sphere is round if it is isometric to the standard round metric of constant Gaußian curvature. Theorem 1.2. The equality (3)

Λk (S2 ) = 8πk

holds for any k > 1. For k = 1 the supremum in (2) is attained if and only if g is the standard round metric on a sphere. For k > 2 the supremum can not be attained on a smooth metric, and is realized in the limit if and only if the corresponding sequence of metrics degenerates to a union of k touching identical round spheres. Here “touching” means “have exactly one point in common”. The choice of the points where the touching occurs has no significance, as it does not change the spectrum. Moreover, the spectra of touching and disjoint spheres are the same.

AN INEQUALITY FOR LAPLACE EIGENVALUES ON THE SPHERE

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Remark 1.3. In fact, it may happen that the limit of the maximizing sequence of metrics contains additional spheres of zero measure. We ignore these zero measure spheres, as they do not contribute to the spectrum. We refer the reader to Case II in the proof of Theorem 1.2 for further details. Theorem 1.2 settles a conjecture that was proposed in 2002 by the second author [30] (see also [34]). For k = 1, the result is a classical theorem due to J. Hersch [15]. For k = 2, the proof of the equality (3) was outlined in [30], and a different argument was found later in [41]. Recently, (3) has been also established in the case k = 3 [34]. For k > 4, Theorem 1.2 is new. We also note that the uniqueness of the maximal metric was known before for k = 1, but to the best of our knowledge it has not been proved earlier in the cases k = 2, 3. As follows from the previous discussion, Theorem 1.2 could be stated as an isoperimetric inequality λk (S2 , g) 6 8πk for any metric g of unit area on a sphere. Let us also remark that Theorem 1.2 remains valid if we take the supremum over the space of metrics with (possibly) conical singularities, see Section 4 and the proof of Theorem 1.2. Organization of the paper. In the next section we give a brief overview of results on isoperimetric inequalities and extremal metrics for LaplaceBeltrami eigenvalues on surfaces. In Section 3 we recall the relation between extremal metrics and minimal immersions into spheres. In Section 4 we pass from minimal immersions to harmonic immersions, extend our considerations to harmonic immersions with branch points and metrics with conical singularities, and explain why the results from the previous section also hold in this case. In Section 5 we state the Calabi–Barbosa theorem on harmonic immersions S2 −→ Sn with branch points. We also introduce the key new ingredient of our approach: an upper bound due to N. Ejiri [8] on the index k of the eigenvalue λk extremized by a metric induced by a harmonic immersion S2 −→ S2m . Section 6 contains a summary of results obtained in [33] on maximization of Laplace eigenvalues in a conformal class. In particular, we describe the so-called “bubbling phenomenon” exhibited by the extremal metrics for higher eigenvalues. Finally, in Section 7 we prove Theorem 1.2. Acknowledgements. A.P. holds the “Young Russian Mathematics” award as well as Simons-IUM Research Professorship, and would like to thank the sponsors and the jury of these awards. He is also very grateful to the Institut de Mathématiques de Marseille (I2M, UMR 7373) for its hospitality. 2. Isoperimetric inequalities and extremal metrics for eigenvalues on surfaces In this section we present a brief overview of the main developments in the study of the isoperimetric inequalities and extremal metrics for the

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eigenvalues of the Laplace-Beltrami operator on surfaces. The story began almost half a century ago when Hersch [15] obtained a sharp isoperimetric inequality for the first eigenvalue on a sphere which was mentioned in the previous section. Later, Yang and Yau [44] proved that if M is an orientable surface of genus γ then ¯ 1 (M, g) 6 8π(γ + 1). λ Actually, the arguments of Yang and Yau imply a stronger, though in general still non-sharp estimate, γ + 3 ¯ 1 (M, g) 6 8π , λ 2

see [10]. Here [·] denotes the integer part of a number. A similar result in the non-orientable case has been recently obtained in [21]. For a non-orientable surface M of genus γ γ +3 ¯ λ1 (M, g) 6 16π . 2

Here the genus of a non-orientable surface is defined to be the genus of its orientable double covering. For higher eigenvalues, Korevaar [24] proved that there exists a constant C such that for any i > 0 and any compact surface M of genus γ the following upper bound holds, ¯ i (M, g) 6 C(γ + 1)i. λ ¯ i (M, g) are bounded from above, These results imply that the functionals λ and, as a result, Λi (M ) < +∞. A number of important results on the existence and regularity of maximizing metrics has been recently obtained, see [23, 27, 32, 33, 40]. However, the list of known values Λi (M ) for all surfaces other than the sphere is quite short. Li and Yau proved in 1982 in the paper [26] that the standard metric on ¯ 1 (RP2 , g) and the projective plane is the unique maximal metric for λ Λ1 (RP2 ) = 12π.

The second author proved in 1996 in the paper [29] that the standard ¯ 1 (T2 , g) metric on the equilateral torus is the unique maximal metric for λ and 8π 2 Λ1 (T2 ) = √ . 3 ¯ Note that the functional λi (M, g) depends continuously on the metric g. ¯ i (M, g) is a multiple eigenvalue this functional is not in However, when λ general differentiable. If we consider an analytic variation gt of the metric g = g0 , then it was proved by Berger [3], Bando and Urakawa [1], El Soufi ¯ i (M, gt ) and Ilias [13] that the left and right derivatives of the functional λ with respect to t exist. This leads us to the following definition given by the first author in the paper [29] and by El Soufi and Ilias in the papers [12, 13].


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Definition 2.1. A Riemannian metric g on a closed surface M is called an ¯ i (M, g) if for any analytic deformation extremal metric for the functional λ gt such that g0 = g one has d¯ d¯ λi (M, gt ) 606 λ (M, g ) . i t dt dt t=0+ t=0−

It was proved in [17] that the metric on the Klein bottle realized as the ¯ 1 (KL, g). It was conso-called bipolar Lawson surface τ˜3,1 , is extremal for λ jectured in this paper that this metric is the unique extremal metric and, moreover, the maximal one. It was later shown by El Soufi, Giacomini and Jazar [9] that this metric on τ˜3,1 is indeed the unique extremal metric for ¯ 1 (KL, g). the functional λ In fact, it follows from the results of [27] that there exists a smooth (up to at most a finite number of conical points) metric gK on the Klein bottle ¯ 1 (KL, g) is attained on gK . It could be then shown (a detailed such that sup λ exposition of this argument will appear in [6]) that the metric on τ˜3,1 is the maximal one and, hence, √ ! 2 2 ¯ 1 (KL, gτ˜ ) = 12πE , Λ1 (KL) = λ 3,1 3 where E is the complete elliptic integral of the second kind and gτ˜3,1 is the metric on τ˜3,1 . More results on extremal metrics on tori and Klein bottles could be found in the papers [12, 18, 19, 20, 25, 36, 37, 39]. A review of these results is given by the third author in the paper [38]. It was shown in [16] using a combination of analytic and numerical tools that the maximal metric for the first eigenvalue on the surface of genus two is the metric on the Bolza surface P induced from the canonical metric on the sphere using the standard covering P −→ S2 . The authors stated this result as a conjecture, because the argument is partly based on a numerical calculation. The proof of this conjecture was outlined in a recent preprint [35]. For surfaces of genus γ > 3, no maximal (and even extremal) metrics for Laplace eigenvalues are known. The only surface other than the sphere for which a sharp isoperimetric inequality was established for a higher eigenvalue is the projective plane. It was shown in [31] that Λ2 (RP2 ) = 20π. It turns out that there is no maximal metric, but the supremum could be obtained as a limit on a sequence of metrics converging to a union of a projective plane and a sphere touching at a point, each endowed with the standard metric, such that the ratio of the areas of the projective plane and the sphere is 3 : 2. Inspired by this result, as well as by Theorem 1.2, we would like to conclude this Section with the following conjecture:

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Conjecture 2.2. The equality Λk (RP2 ) = 4π(2k + 1) holds for any k > 1. For k > 2 the supremum can not be attained on a smooth metric, and is realized in the limit if and only if the corresponding sequence of metrics degenerates to a union of k − 1 identical round spheres and a standard projective plane touching each other, such that the ratio of the areas of the projective plane and the spheres is 3 : 2. 3. Extremal metrics and minimal immersions to spheres In this section we recall the relation between extremal metrics and minimal immersions into spheres. Let us recall the definition of a minimal map, see e.g. [5, 7]. Let (M, g) be a Riemannian manifold of dimension m. Let α be a symmetric bilinear 2-form on T M. Let σk be the k-th elementary symmetric function. Let σk (α) = σk (λ1 , . . . , λm ), where λi are eigenvalues of α related to g, i.e. roots of the polynomial det(αij − λgij ) = 0. Definition 3.1. Let (M, g) and (N, h) be Riemannian manifolds. A smooth map f : M −→ N is called minimal if f is extremal for the volume functional Z p |σm (f ∗ h)| dV olg , V [f ] = M

where m = dim M.

It is well-known that a surface M # R3 is minimal if and only if the coordinate functions xi are harmonic with respect to the Laplace-Beltrami operator on M. A similar result holds for a minimal submanifold in Rn . Since harmonic functions are eigenfunctions with eigenvalue 0, it is natural to ask what is an analogue of this statement for a non-zero eigenvalue. The answer was given by Takahashi in 1966. Theorem 3.2 (Takahashi [43]). If an isometric immersion f : M # Rn+1 ,

f = (f 1 , . . . , f n+1 ),

is defined by eigenfunctions f i of the Laplace-Beltrami operator ∆ with a common eigenvalue λ, ∆f i = λf i , then (i) the image f (M ) lies on the sphere SnR of radius R with the center at the origin such that dim M , (4) λ= R2 (ii) the immersion f : M # SnR is minimal. If f : M # SnR ⊂ Rn+1 , f = (f 1 , . . . , f n+1 ),


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is a minimal isometric immersion of a manifold M into the sphere SnR of radius R, then f i are eigenfunctions of the Laplace-Beltrami operator ∆, ∆f i = λf i , with the same eigenvalue λ given by formula (4). Let us introduce the eigenvalue counting function N (λ) = #{λi |λi < λ}. Theorem 3.2 implies that if M is isometrically minimally immersed in the sphere SnR , then among the eigenvalues of M there are at least n + 1 eigenM . It is easy to see that λN dim M is the first eigenvalue values equal to dim R2 R2

equal to

dim M . R2

It is important due to the following theorem.

Theorem 3.3 (Nadirashvili [29], El Soufi, Ilias [13]). Let M # SnR be an immersed minimal compact submanifold. Then the metric induced on M by ¯ dim M (M, g). this immersion is extremal for the functional λ N

R2

If a metric on a compact manifold M is extremal then there exists an isometric minimal immersion to the sphere M # SnR by eigenfunctions with M of the Laplace-Beltrami operator corresponding to this meteigenvalue dim R2 ric. We later use Theorem 3.3 for M = S2 . In this case dim M = 2. Using rescaling one can consider only the case R = 1. Hence, the value dim S2 N = N (2) R2 is particularly important for our further considerations. 4. Harmonic maps with branch points and metrics with conical singularities Let us recall the definition of a harmonic map, see e.g. the review [5]. Definition 4.1. Let (M, g) and (N, h) be Riemannian manifolds. A smooth map f : M −→ N is called harmonic if f is an extremal for the energy functional Z |df (x)|2 dV olg . (5) E[f ] = M

The following theorem (see, e.g. the paper [7]) explains the relation between minimal and harmonic maps in the class of isometric immersions. Theorem 4.2. Let M, N be Riemannian manifolds. If f : M # N is an isometric immersion, then f is harmonic if and only if f is minimal.

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It is well-known that if dim M = 2, then the property of f to be harmonic depends only on the conformal class of the metric g, see e.g. [5]. A harmonic map f : M −→ N is called conformal if the metric induced by f belongs to the same conformal class on M for which the map f is extremal for the energy functional. Proposition 4.3. Let M be a compact surface. Let a metric g on M be ¯ k (M, ·). Then there exists a conformal harmonic extremal for a functional λ n immersion f : M # S from M (endowed with the conformal class of the metric g) to Sn (endowed with the canonical metric gSn of radius 1), such that g = f ∗ gSn , i.e. g is induced by f. Conversely, let M be a compact surface with a fixed conformal class and f : M # Sn be a conformal harmonic immersion from M to Sn endowed with the canonical metric gSn of radius 1. Then the metric g = f ∗ gSn induced by f is extremal for the functional λ¯k (M, ·) for k = N (2). ¯ k . Using the Proof. Let a metric g on M be extremal for a functional λ ¯ invariance of λk under dilations g 7→ tg, we may assume that λk = 2. Then by Theorem 3.3 there exists a minimal isometric immersion of M to the unit sphere, f : M # Sn ⊂ Rn+1 , and by Theorem 3.2 the components f i of f are eigenfunctions with the eigenvalue λk = 2. Proposition 4.2 then implies that f is a harmonic isometric immersion of M to the sphere Sn , and hence, f is a conformal harmonic map from M (endowed with the conformal class defined by g) to Sn . Consider now M with a fixed conformal class and a conformal harmonic immersion f : M # Sn . Then the induced metric f ∗ gSn defines the same conformal class on M as the inital one and the map f is an isometric harmonic immersion. It follows from Proposition 4.2 that f is a minimal isometric immersion. By Theorem 3.2, the components f i of f : M # Sn ⊂ Rn+1 are eigenfunctions corresponding to the eigenvalue λk = 2, where k = N (2). It follows then from Theorem 3.3 that the induced metric f ∗ gSn is extremal ¯ k (M, ·). for λ It is a well-known fact that there is only one conformal class of Riemannian metrics on S2 . Hence, for M = S2 any harmonic immersion is conformal. Corollary 4.4. The extremal metrics for the eigenvalues of the LaplaceBeltrami operator on the sphere S2 are exactly the metrics induced on S2 by harmonic immersions f : S2 # Sn . It turns out that it is useful to consider a wider class of harmonic immersions with branch points. Definition 4.5 (see e.g. [14]). Let M be a manifold of dimension 2. A smooth map f : M −→ N has a branch point of order k at point p if there exist local coordinates u1 , u2 centered at p and x1 , . . . , xn centered at f (p) such


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that in these coordinates f is written as x1 + ix2 = wk+1 + σ(w), xk = χk (w),

k = 3, . . . , n,

σ(w), χk (w) = o(|w|k+1 ), ∂σ ∂χk (w), (w) = o(|w|k ), ∂uj ∂uj where w = u1 + iu2 .

j = 1, 2,

k = 3, . . . , n,

If M is compact, a map f : M −→ N may have at most a finite number of branch points. Note that the presence of branch points leads to a minor technical issue: if (N, g) is a Riemannian manifold and f : M # N is an immersion with branch points, then the induced metric f ∗ g is not a smooth Riemannian metric. Definition 4.6 (see e.g. [22]). A point p on a surface M is called a conical singularity of order α > −1 or angle 2π(α + 1) of a metric g, if in an appropriate local complex coordinate z centered at p the metric has the form g(z) = |z|2α ρ(z)|dz|2 in a neighborhood of p, where ρ(0) > 0. Then we immediately obtain the following Proposition. Proposition 4.7. If M is a compact surface, (N, h) is a Riemannian manifold and f : M # N is an immersion with branch points, then g = f ∗ h is a smooth Riemannian metric except at a finite number of branch points of the map f. At these points the metric g has conical singularities. The order of the conical singularity at a point p is equal to the order of p as a branch point. Thus, we switch to a geometric setting slightly larger than the initial one: consider not only smooth Riemannian metrics and harmonic immersions, but also Riemannian metrics with a finite number of conical singularities and harmonic immersions with branch points. However, it remains to verify that all the key results from the previous section still hold in this setting. It is well-known that the eigenvalues of the Laplace-Beltrami operator could be defined using a variational approach, (6)

λk =

min max R[v],

V ⊂H 1 (M ) u∈V

dim V =k u⊥1

where

R |∇u|2 dV ol R[v] = RM 2 M |u| dV ol is the Rayleigh quotient. This formula holds also in the case of metrics with conical singularities, see e.g. [22].

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Branch points do not affect convergence neither of the energy functional (5) nor the integrals in the proof of Theorem 3.3. Hence Proposition 4.3 holds also in the setting of metrics with conical singularities and harmonic maps with branch points. Let us also remark that for any manifold equipped with a metric with isolated conical singularities it is possible to construct a sequence of smooth Riemannian manifolds such that their area as well as their eigenvalues converge to the area and eigenvalues of the initial manifold, see e.g. [42]. 5. Harmonic maps into spheres and bounds on the harmonic degree We now focus on harmonic immersions S2 # Sn with branched points. Let us introduce the following useful definition. Definition 5.1. A harmonic map f : Σ −→ Sn ⊂ Rn+1 of a surface Σ to the standard unit sphere Sn ⊂ Rn+1 is called linearly full if the image f (Σ) does not lie in a hyperplane of Rn+1 . The following theorem was proved by Calabi in 1967 and later refined by Barbosa in 1975. Let gSn denote the standard metric on Sn of radius 1. Theorem 5.2 (Calabi [4], Barbosa [2]). Let f : S2 −→ Sn be a linearly full harmonic immersion with branch points. Then (i) the area of S2 with respect to the induced metric Area(S2 , f ∗ gSn ) is an integer multiple of 4π; (ii) n is even, n = 2m, and Area(S2 , f ∗ gSn ) > 2πm(m + 1). Definition 5.3. If Area(S2 , f ∗ gSn ) = 4πd, then we say that f is of harmonic degree d. We obtain immediately a lower bound for the harmonic degree. Proposition 5.4. Let f : S2 −→ S2m be a linearly full harmonic immersion . with branch points. Then d > m(m+1) 2 Theorem 5.2 implies the following proposition. Proposition 5.5. Extremal metrics (possibly with conical singularities) on S2 are induced by linearly full harmonic immersions with branch points f : S2 −→ S2m . The harmonic degree d of such an immersion satisfies the inequality d > m(m+1) . 2 Indeed, consider a metric g induced by a harmonic immersion f : S2 −→ S2m ⊂ R2m+1

that is not linearly full. Among subspaces of R2m+1 containing the image f (S2 ), we can find a linear subspace V ⊂ R2m+1 of minimal possible dimension 2m′ + 1. Take an orthogonal map A ∈ SO(2m + 1) such that A(V ) is


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generated by the first 2m′ + 1 standard basis vectors in R2m+1 . Then g is induced by the linearly full harmonic immersion ′

′

A ◦ f : S2 −→ S2m ⊂ R2m +1 . As we know from Theorem 3.3, a metric induced by a linearly full harmonic immersion with branch points f : S2 −→ S2m is extremal for the ¯ N (2) (S2 , g). eigenvalue λ A priori, it is not clear how to find N (2) for a metric on a surface induced by a harmonic immersion into a sphere. The following upper bound on N (2) due to Ejiri is a particular case of [8, Theorem C] and is crucial for our considerations. Theorem 5.6 (Ejiri [8]). Let f : S2 −→ S2m be a linearly full harmonic map of harmonic degree d > 1 of S2 to the standard unitary sphere S2m . Then N (2) > d + 1. We note that while the condition d > 1 is not explicitly stated in the formulation of Theorem C in [8], it is assumed and used in its proof. 6. Maximization of higher eigenvalues in a conformal class and the bubbling phenomenon The last ingredient of the proof of Theorem 1.2 is the theory of maximization of higher eigenvalues in a conformal class developed by the second author and Sire in the papers [32, 33]. We summarize these results below. Consider a Riemannian metric g on a connected compact closed surface M. Let us denote by [g] the following class of metrics conformally equivalent to g, [g] = {˜ g |˜ g = µg},

where µ : M → R+ is an L1 function on M with mass 1, i.e. a probability density. Usually [g] is called the conformal class of g. Remark, however, that [g] contains only metrics of area 1 conformally equivalent to g. Let us consider the supremum of an eigenvalue over metrics of area 1 conformally equivalent to g, ˜ k (M, [g]) = sup λk (˜ Λ g ). g˜∈[g]

Let ground denote the standard metric on the sphere S2 of radius 1. Theorem 6.1 (Nadirashvili, Sire [33]). Let (M, g) be a smooth connected compact Riemannian surface without boundary. For any k > 1 and a sequence of metrics {gi′ }i>1 ∈ [g] of the form gi′ = µ′i g such that ˜ k (M, [g]) lim λk (gi′ ) = Λ

i→∞

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there exists a subsequence of metrics {gn }n>1 = {gi′n }n>1 ∈ [g], where gn = µn g, such that ˜ k (M, [g]) lim λk (gn ) = Λ n→∞

and a probability measure µ such that µn ⇀∗ µ weakly in measure as n → +∞.

Moreover, the following decomposition holds, µ = µr + µs where µr is a nonnegative not trivial, by the formula

C∞

function and µs is the singular part given, if µs =

K X

ci δxi

i=1

for some K > 1, ci > 0 and some “bubbling points” xi ∈ M . Furthermore, the number K satisfies the bound K 6 k − 1.

Moreover, there exist mj such that 1 6 mj 6 k and cj =

˜ m (S2 , [ground ]) Λ j . ˜ k (M, [g]) Λ

The regular part of the limit density µ, i.e. µr , is either identically zero or µr is absolutely continuous with respect to the Riemannian measure with a smooth positive density vanishing at most at a finite number of points on M. Furthermore, if we denote by Ar the volume of the regular part µr , i.e. Ar = Area(M, µr g), then either Ar = 0 or there exists m0 such that 1 6 m0 6 k and ˜ m (M, [g]) Λ 0 Ar = . ˜ Λk (M, [g]) Finally, if we denote by U the eigenspace of the Laplace-Beltrami opera˜ k (M, [g]), then there exists a tor on (M, µr g) associated to the eigenvalue Λ family of eigenvectors {u1 , . . . , ul } ⊂ U such that the map ϕ = (u1 , . . . , ul ) : M → Rl

is a harmonic isometric immersion into the sphere Sl−1 . Theorem 6.1 could be interpreted in the following way: the supremum of λk is attained as a limit on a sequence of metrics converging to a singular metric on M with K < k spheres touching M at the points x1 , . . . , xK . The restriction of this limit singular metric on M is µr g and on the sphere ¯ m on the sphere of area ci . touching M at xi is the metric maximizing λ j That’s why this phenomenon is called “bubbling”, these spheres bubble up out of the surface M. The metric µr g has area Ar .


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Remark 6.2. The following identity holds, K X

(7)

ci + Ar = 1,

i=1

since the area of µg is equal to 1. Remark 6.3. The metric µr g on M is maximal on M for the functional ¯ m (M, ·). λ 0 This fact is not mentioned in the statement of Theorem 6.1, however, it follows from the proof of this theorem in the paper [33]. ˜k Theorem 6.1 explains that the bubbling phenomenon could occur for Λ with k > 2. Note that in the case k = 1 no bubbling phenomenon occurs, i.e. µs is always identically zero. For k = 2 and k = 3 the bubbling phenomenon was observed on the example of the sphere in the papers [30, 34] and on the example of the real projective plane in the paper [31]. Let us explain the mechanism of bubbling in more detail. If in Theorem 6.1 we have µs 6= 0, then we say that the sequence of metrics gn is bubbling at points xi and ci is a part of total area which is concentrated at the point xi . One can choose a sequence of disks Dk on (M, g) centered at xi with radii tending to zero and a subsequence gnk such that disks (Dk , gnk ) as Riemannian manifolds are tending to (D, g), where g = gr + gs , with a regular metric gr and possibly a nonzero singular part gs . Lemma 6.4 below shows that the spectrum of (D, g) is a subset of the limit of spectra of (M, gn ). This lemma was in fact proved in the paper [33] although it was not stated there in the form presented here. Let us denote by Dr (x) the disk centered at x ∈ M and of radius r with respect to the metric g. The following lemma describes the decomposition of the spectrum in the case on a singular extremal metric (i.e. µs is not identically zero), see [34]. ′ Lemma 6.4. There exists a subsequence n o µn g of the sequence µi g of metrics

from Theorem 6.1 with eigenvalues λnj

j>0

such that the following property

holds. Consider a smooth cut-off function ψr on M such that 0 6 ψr 6 1, ψr = 0 on Dr/2 (˜ x) and ψr = 1 on M \ Dr (˜ x) where x ˜ is a bubbling point in Theorem 6.1. Define the sequence of metrics hn = ψ2−n µn g on M and ρn on S2 such that (S2+ , ρn ) is isometric to (D2−n , µn g − hn ). Let us extend ρn by 0 on S2− . Denote by {αni }i>0 and {βin }i≥0 the sequences of eigenvalues of the Laplace-Beltrami operator on (M, hn ) and (S2 , ρn ) respectively. Fix a natural number N > 1. Suppose that the following limits exist, lim λni = λi ,

n→∞

i = 0, . . . , N.

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Then there exists a subsequence nm and natural numbers N1 , N2 > 1 such that the following limits hold, lim αni m = αi

m→∞

and lim βinm = βi

m→∞

and, furthermore, {λ0 , . . . , λN } = {α0 , . . . , αN1 } ∪ {β0 , . . . , βN2 } ,

where the union of sets is taken considering the multiplicity of the eigenvalues. Remark 6.5. Note that metrics ρn are vanishing on open subsets of S2 . For nonnegative metrics, the eigenvalues of the Laplace-Beltrami operator are defined via Rayleigh’s variational formula (6). 7. Proof of Theorem 1.2 It is already known that Λk (S2 ) = 8πk for k = 1 (see the paper [15]), k = 2 (see the paper [30]) and k = 3 (see the paper [34]). It is also well known that in the case k = 1 the round metric on S2 is the unique maximal metric. This could be proved e.g. in the following way. It follows from the paper [26, Theorem 1] that the maximal metric is smooth. Then this metric is induced by a minimal immersion to a sphere by first eigenfunctions as we know from Theorem 3.3. It was proved in [28] (see also [11]) that there exists at most one metric in a conformal class admitting an isometric immersion into a sphere by the first eigenfunctions. The uniqueness of a maximal metric for the first eigenvalue on S2 then follows from the fact that there is only one conformal structure on S2 . We prove Theorem 1.2 by induction in k. Indeed, the result holds for k = 1 as we have just discussed. Let us suppose that it holds for l = 1, ..., k − 1. Let {gn } be a maximizing sequence of metrics of area 1 for the functional ¯ k (S2 , g), i.e. λ ¯ k (S2 , gn ) = Λk (S2 ). lim λ n→∞

Since there is only one conformal structure on S2 , the metrics gn can be written as gn = µn ground , where ground is the round metric on S2 of Gaußian curvature 1. It follows that the sequence {gn } is a maximizing sequence for ˜ k (S2 , [ground ]). Choosing a subsequence if needed, we can assume that {gn } Λ is the sequence from Theorem 6.1. For convenience, we use the notation of Theorem 6.1 in the rest of the proof. Let µr and µs be the regular part and the singular part from Theorem 6.1. Consider three possible cases.


15

Case I. Let µs 6= 0 and µr 6= 0. As we know, the quantity K of “bubbling points” satisfies 1 6 K 6 k − 1. If m0 = k, then Ar = 1 and hence K X

ci = 0.

i=1

This implies that c1 = . . . = cK = 0. This contradicts the assumption µs 6= 0. It follows that 1 6 m0 6 k − 1. As we know from Remark 6.3, the metric ¯ 2 , g). Since m0 6 k − 1, by inductive µr ground is a maximal metric for λ(S hypothesis the regular metric µr ground is a metric on m0 touching round spheres. This is possible if and only if m0 = 1 and µr = 1. This implies that ˜ 1 (S2 , [ground ]) 8π Λ1 (S2 ) Λ = = (8) Ar = 2) 2 2 ˜ Λ (S Λ Λk (S , [ground ]) k k (S ) by the inductive hypothesis. Then formula (7) implies that K X Λmj (S2 ) j=1

or, equivalently,

K X

(9)

Λk

(S2 )

+

8π =1 Λk (S2 )

Λmj (S2 ) + 8π = Λk (S2 ).

j=1

As we know, mj 6 k. If at least one mj = k then formula (9) cannot hold. It follows that for all j we have mj 6 k − 1. Then the metrics on the ¯ m on the sphere with mj < k and by the bubbling spheres are maximal for λ j inductive hypothesis these metrics are the singular metrics on mj touching spheres with standard metric of the same Gaußian curvature. Let us now use Lemma 6.4 or, more precisely, the similar statement for K bubbling points, for N = k. Then the set of the first k + 1 limit eigenvalues {λ0 , . . . , λk },

where

λj = lim λj (S2 , µn ground ), n→∞

coincides with the first k + 1 elements of the ordered union of spectra of K K P P mj + 1 spheres. This implies that if mj + 1 > k, then we have λk = 0

j=1

j=1

and the considered metric could not be maximal. Hence,   K K X X mj + 1 6 8πk. Λmj (S2 ) + 8π = 8π  Λk (S2 ) = j=1

j=1

16


The value 8πk is then attained and, moreover, attained exactly as a limit on a sequence of metrics on the sphere converging to a singular metric obtained on a connected space consisting of k touching (in arbitrary way) spheres with round metrics of the same Gaußian curvature. Different ways of representing K P k as a sum k = mj + 1, 1 6 K 6 k − 1, give different arrangements of j=1

these spheres. Case II. Let µs 6= 0 and µr = 0. As in Case I, the number K of “bubbling points” satisfies 1 6 K 6 k − 1, but in this case Ar = 0. This implies that instead of (9) we obtain K X

Λmj (S2 ) = Λk (S2 ).

j=1

The analysis of Case II is then similar to Case I. We obtain 2

Λk (S ) =

K X j=1

2

Λmj (S ) = 8π

K X

mj 6 8πk.

j=1

The value 8πk is then attained and, moreover, attained exactly as a limit on a sequence of metrics on the sphere converging to a singular metric obtained on a connected space consisting of k touching (in arbitrary way) spheres with round metrics of the same Gaußian curvature. Different ways of representing K P k as a sum k = mj , 1 6 K 6 k − 1, give different arrangements of these j=1

spheres. Let us note that in this case µr = 0, and this means that in the limit the initial sphere has measure zero and does not affect the spectrum (see Remark 1.3). It is only the “bubbles” that have non-zero measure and contribute to the spectrum. Case III. Let µs = 0. Let us consider Riemannian metrics on S2 (possibly ¯ k (S2 , g). As we alwith conical singularities) extremal for the functional λ ready know from Corollary 4.4, these metrics are induced on S2 by harmonic immersions (possibly with branched points) S2 −→ Sn . Theorem 5.2 and its corollary Proposition 5.5 imply that it is sufficient to consider linearly full harmonic immersions with branch points f : S2 −→ S2m . of harmonic degree d > m(m+1) 2 Consider the case d > 1. Let f : S2 −→ S2m be a linearly full harmonic immersion of harmonic degree d. Theorem 5.6 implies that for the metric f ∗ ground on S2 induced by f we have (10)

N (2) > d + 1.

On the other hand, by Definition 5.3 we have Area(S2 , f ∗ ground ) = 4πd.


17

Since the value of λN (2) = 2 by Theorem 3.2, we have ¯ N (2) (S2 , f ∗ ground ) = 2 Area(S2 , f ∗ ground ) = 8πd. λ If we denote N (2) = k, inequality (10) implies ¯ k (S2 , f ∗ ground ) = 8πd < 8πk, λ ¯k i.e. any smooth metric (with possibly conical singularities) extremal for λ ¯ k strictly less then limit of λ ¯ k on a sequence converging has the value of λ to k touching spheres. Hence, any smooth extremal metric induced by a harmonic immersion of harmonic degree d > 1 is not a maximal metric. Finally, consider the case d = 1. Then Area(S2 , f ∗ gcan ) = 4π and ¯ k (S2 , f ∗ ground ) = 4π. λ However, in this case the induced metric f ∗ ground could be maximal only for k = 1, but we have assumed that k > 1. This completes the proof of the theorem. References [1] S. Bando, H. Urakawa, Generic properties of the eigenvalue of Laplacian for compact Riemannian manifolds. Tˆ ohoku Math. J., 35:2 (1983), 155–172. [2] J. Barbosa, On minimal immersions of S 2 into S 2m , Trans. Amer. Math. Soc., 210 (1975), 75–106. [3] M. Berger, Sur les premières valeurs propres des variétés Riemanniennes. Compositio Math., 26 (1973), 129-149. [4] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry, 1 (1967), 111–125. [5] J. Eells, L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10:1 (1978), 1–68. [6] D. Cianci, V. Medvedev, in preparation. [7] J. Eells, J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. [8] N. Ejiri, The boundary of the space of full harmonic maps of S2 into S2m (1) and extra eigenfunctions, Japan. J. Math, 24:1 (1998), 83–121. [9] A. El Soufi, H. Giacomini, M. Jazar, A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle, Duke Math. J., 135:1 (2006), 181-202. Preprint arXiv:math/0701773. [10] A. El Soufi, S. Ilias, Le volume conforme et ses applications d’après Li et Yau, Sém. Théorie Spectrale et Géométrie, Institut Fourier, 1983–1984, No.VII, (1984). [11] A. El Soufi, S. Ilias, Immersions minimales, première valeur propre du laplacien et volume conforme, Math. Ann., 275:2 (1986), 257–267. [12] A. El Soufi, S. Ilias, Riemannian manifolds admitting isometric immersions by their first eigenfunctions. Pacific J. Math., 195:1 (2000), 91-99. [13] A. El Soufi, S. Ilias, Laplacian eigenvalues functionals and metric deformations on compact manifolds. J. Geom. Phys., 58:1 (2008), 89-104. Preprint arXiv:math/0701777. [14] R. D. Gulliver II, R. Osserman, H. L. Royden, A Theory of Branched Immersions of Surfaces, Amer. J. Math., 95:4 (1973), 750–812. [15] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér A-B, 270 (1970), A1645-A1648.

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