Bounds on Multiplicities of Laplace-Beltrami Operator Eigenvalues on the Real Projective Plane
Aleksandr S. Berdnikov, Nikolai S. Nadirashvili, Alexei V. Penskoi
aa r X i v : . [ m a t h . DG ] D ec BOUNDS ON MULTIPLICITIES OFLAPLACE-BELTRAMI OPERATOR EIGENVALUES ONTHE REAL PROJECTIVE PLANE
ALEKSANDR S. BERDNIKOV, NIKOLAI S. NADIRASHVILI,AND ALEXEI V. PENSKOI
Abstract.
The known upper bounds for the multiplicities of theLaplace-Beltrami operator eigenvalues on the real projective planeare improved for the eigenvalues with even indexes. Upper boundsfor Dirichlet, Neumann and Steklov eigenvalues on the real projec-tive plane with holes are also provided.
Introduction
Consider a closed surface Σ with a Riemannian metric g . Let0 = λ < λ λ . . . be Laplace-Beltrami operator eigenvalues (counting with multiplici-ties). Let m (Σ , g, λ i ) denote the multiplicity of eigenvalue λ i , i.e. thedimension of the eigenspace corresponding to λ i .Finding upper bounds for m (Σ , g, λ i ) is one of classical problems ofSpectral Geometry. These bounds are interesting from several points ofview. For example, they play an important role in the study of metricsextremal for the Laplace-Beltrami eigenvalues, see e.g. [16].In the present paper we consider the case of Σ being the real projec-tive plane RP . The best known bound for the real projective plane(1) m ( R P , g, λ i ) i + 3was proven in the paper [14], see Theorem 5 below. Recently thisbound was improved in the paper [15] for i = 2 ,m ( R P , g, λ ) , see Theorem 7 below.The first goal of the present paper is to improve the bounds (1) forall even i and prove the following theorem. The work of the third author was supported by Russian Science Foundationgrant no. 16-11-10260 at Moscow State University.
Theorem 1.
Let l be a positive integer, then m ( R P , g, λ l ) l + 1 . We also consider the case of a surface Σ with a boundary ∂ Σ. Inthis case, let 0 λ λ . . . denotes the spectrum of the Laplace-Beltrami operator, given eitherDirichlet or Neumann boundary condition on each connected compo-nent of the boundary ∂ Σ.We also consider Steklov eigenvalue problem on (Σ , ∂ Σ) . Let ρ be abounded non-negative function on ∂ Σ and σ a real number. Then afunction u on Σ is called a Steklov eigenfunction with eigenvalue σ if ( ∆ u = 0 in Σ, ∂u∂n = σρu on ∂ Σ,where n denotes the unit outer normal on ∂ Σ.Let us consider the case where ρ is the density of an absolutelycontinuous Radon measure s on ∂ Σ. Then the spectrum of Steklovproblem is non-negative and discrete [1], so we denote it by0 = σ σ . . . Let us denote the multiplicity corresponding to the Steklov eigenvalue σ i by m (Σ , g, s, σ i ).Let now RP h denote RP \ ( ∪ D i ), i.e. the real projective plane witha positive number of holes. Our second result then is the followingtheorem. Theorem 2.
Let l be a positive integer and s be an absolutely contin-uous Radon measure on ∂ R P h with bounded density, then (2) m ( R P h , g, λ l ) l + 2 , (3) m ( R P h , g, s, σ l ) l + 2 . The proofs follow the technique from the paper [14] but uses a morerefined topological argument at the last step.Let us recall some already known results about upper bounds onmultiplicities of eigenvalues.
Theorem 3 (Cheng [7]) . Let Σ be an oriented surface of genus γ .Then for any metric g one has m (Σ , g, λ i ) (2 γ + i + 1)(2 γ + i + 2)2 . OUNDS ON MULTIPLICITIES OF LAPLACE OPERATOR EIGENVALUES 3
Theorem 4 (Besson [4]) . Let Σ be an oriented surface of genus γ .Then for any metric g one has m (Σ , g, λ i ) γ + 2 i + 1 . Let Σ be a non-orientable surface of Euler characteristic χ (Σ) . Thenfor any metric g one has m (Σ , g, λ i ) − χ (Σ)) + 4 i + 2 . Theorem 5 (Nadirashvili [14]) . For any metric g on the sphere S ,the real projective plane RP , the torus T , or the Klein bottle KL thefollowing inequalities hold, m ( S , g, λ i ) i + 1 ,m ( RP , g, λ i ) i + 3 ,m ( T , g, λ i ) i + 4 ,m ( KL , g, λ i ) i + 3 . For any other surface Σ , i.e. for a surface Σ with χ (Σ) < , with anymetric g the following inequality holds, m (Σ , g, λ i ) i − χ (Σ) + 3 . Theorem 6 (M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and Nadi-rashvili [9]) . Let Σ be a closed surface of genus 0. Then for any metric g the inequality m (Σ , g, λ i ) i − holds for i > . Theorem 7 (Nadirashvili and Penskoi [15]) . The following upper boundfor the multiplicity of the second eigenvalue of the Laplace-Beltrami op-erator on the projective plane holds for any metric g,m ( R P , g, λ ) . For a surface Σ with boundary ∂ Σ let us denote by Σ the closed(topological) manifold, obtained by collapsing each connected compo-nent of ∂ Σ into a point.
Theorem 8 (Karpukhin, Kokarev, and Polterovich [12]) . Let ( M, g ) be a compact Riemannian surface with a non-empty boundary, and µ be an absolutely continuous Radon measure on ∂M whose density isbounded. Then the multiplicity m ( M, g, µ, σ k ) of a Steklov eigenvalue σ k ( g, µ ) satisfies the inequalities (4) m ( M, g, µ, σ k ) − χ ( M )) + 2 k + 1 , (5) m ( M, g, µ, σ k ) − χ ( M )) + k, A. S. BERDNIKOV, N. S. NADIRASHVILI, AND A. V. PENSKOI for all k = 1 , . . . .Besides, the latter inequality is strict for an even k . Theorem 9 (Jammes [11]) . Let Σ be a compact surface with boundary,then for k > the following inequality holds, (6) m (Σ , g, µ, σ k ) k − χ (Σ) + 3 . Theorem 10 (T. Hoffmann-Ostenhof, Michor, and Nadirashvili [10]) . Let k > . Then the multiplicity of the k -th eigenvalue λ k for theDirichlet problem on a planar simply-connected domain D satisfies theinequality m ( D, λ k ) k − . Theorem 11 (Berdnikov [2]) . Let M be the surface with χ ( M ) < .Then m ( M, g, λ k ) k − χ ( M ) + 3 ,m ( M, g, vol ∂M , σ k ) k − χ ( M ) + 3 . Let us also recall results concerning the relation between bounds onmultiplicities of eigenvalues and the chromatic number.
Definition 1.
A chromatic number chr(Σ) of a surface Σ is the max-imal n such that one can embed in Σ the complete graph K n on n vertices. Let us consider a Schr¨odinger operator H = ∆+ V. Let ¯ m (Σ) denotethe supremum over all H of the multiplicitiy of the eigenvalue λ on Σ . Theorem 12 (Colin de Verdi`ere [5]) . For any surface Σ , one has ¯ m (Σ) > chr(Σ) − . For the four surfaces Σ with χ ( S ) > , namely S , RP , T , KL , onehas ¯ m (Σ) = chr(Σ) − . This theorem leads to a natural conjecture that for any surface Σone has ¯ m (Σ) = chr(Σ) − . This conjecture was proved by S´evennecfor some surfaces Σ using the following upper bound for ¯ m (Σ) . Theorem 13 (S´evennec [17]) . If χ (Σ) < then ¯ m (Σ) − χ (Σ) . It follows that the above conjecture holds for all surfaces Σ with χ (Σ) > − , the four new cases being T T and n RP , n = 3 , , . Since m (Σ , g, λ ) ¯ m (Σ) , these results provide interesting upperbounds on m (Σ , g, λ ) . Let us now compare Theorem 2 with the above mentioned results.The bound given by formula (2) seems to be a first bound of this kindfor the projective plane with holes.
OUNDS ON MULTIPLICITIES OF LAPLACE OPERATOR EIGENVALUES 5
The comparison of the bound given in formula (3) with bounds (4),(5) and (6) gives a complicated answer. Bounds (4), (5) are linear in k, bound (4) has a better constant, bound (5) has a better asymp-totic. Bound (3) improves bound (4) for RP by 1 . Bound (6) improvesbound (5) by 1 . The third author is very grateful for the the Institut de Math´ema-tiques de Marseille (I2M, UMR 7373) for the hospitality. The thirdauthor is a Young Russian Mathematics award winner and would liketo thank its sponsors and jury. The third author is a Simons research-professorship award winner and would like to express his deep gratitudeto the jury and to the Simons foundation. The third author is veryindebted to Pavel Winternitz for fruitful discussions.1.
Preliminaries
The standard relation between the original problem and the topologyof the surface is well-known. Recall that for an eigenfunction f of theLaplacian, the connected components of f − ( R \ { } ) are called nodaldomains of f and the set f − (0) is called a nodal graph N ( f ) of f .We will denote the number of nodal domains of a (nodal) graph N by µ ( N ). We will also use notation N ( f ) for N ( f ) ∩ Int (Σ) ⊂ Σ. Theterm “graph” is justified by the following theorem.
Theorem 14 (Bers theorem, [3]) . For a Laplacain eigenfunction f and x ∈ M there exists an integer n > and polar coordinates ( r, θ ) centered at x such that the following formula holds f ( x ) = r n (sin( nθ )) + O ( r n +1 ) . The Bers theorem implies that the nodal graph is, in fact, an embed-ded graph. It also implies that each eigenfunction has a well-definedorder ord ( f, x ) := n of vanishing at any point x ∈ M . Note that thequotient of eigenfunctions f such that ord ( f, x ) = n by those g with ord ( g, x ) = n +1 is contained in the span of r n (sin( nθ )) and r n (cos( nθ ))and hence at most 2-dimensional (if n = 0 then sin( nθ ) = cos( nθ )and the quotient is at most 1-dimensional). That implies that high-dimensional eigenspaces contain functions with high vanishing order ata given point. Proposition 1.
Let U be an eigenspace of Laplacian with dim( U ) > n and x be a point in M . Then the dimension of the linear subspace offunctions f ∈ U such that ord( f, x ) > k > (or, equivalently, x is avertex of N ( f ) of deg > k ) is at least n − k . The following theorem is crucial for the considered approach, it con-tains an upper bound for the number of nodal domains.
A. S. BERDNIKOV, N. S. NADIRASHVILI, AND A. V. PENSKOI
Theorem 15 (Courant nodal domain theorem, [8], [6], [13]) . Let Σ bea smooth manifold with smooth boundary ∂ Σ (possibly empty) and s be an absolutely continuous Radon measure on ∂M with bounded den-sity. Consider either the Laplacian eigenvalue problem on M with theDirichlet or Neumann boundary condition on each connected compo-nent of ∂ Σ , or, if ∂ Σ = ∅ , the Steklov eigenvalue problem for themeasure s . Then for each non-zero function f in the i -th eigenspace U i of the considered spectral problem the number µ ( N ( f )) of its nodaldomains is not greater than i + 1 . The Theorems 1 and 2 follow immediately from the Courant nodaldomain theorem and the following theorem, which we prove later inthis paper.
Theorem 16.
Let λ be a real number and U be a linear space of func-tions on the closed surface Σ = R P or a surface with the boundary Σ = R P h . Suppose that every f ∈ U is a Laplacian eigenfunction ∆ f = λf . Suppose sup f ∈ U ( µ ( N ( f ))) is not greater than an odd num-ber n = 2 l + 1 . Let i = n − . Then dim( U ) n = 2 i + 2 .Moreover, if Σ = R P , i. e. ∂ Σ = ∅ , then this inequality is strict,i.e. dim( U ) < n = 2 i + 2 . We will use a notion of a star fibration, which we explain now. Berstheorem tells us that the nodal graph of the eigenfunction f is dif-feomorphic near x to 2 n rays in R emitting from 0 at equal anglesbetween the adjacent lines. That makes it natural to consider a star fi-bration E M (2 n ) over Int ( M ) introduced in paper [2]. It can be definedas follows. Recall that the spherisation S ( E ) of a vector bundle E isthe fiber bundle ( E \ / R > . In the case when E carries a positive-definite metric (i.e. tangent bundle of a Riemannian manifold), thespherisation S ( E ) is isomorphic to the fiber bundle of the unit spheresin E . Now, consider the subset E ′ M (2 n ) ⊂ Q n S ( T M ) such that thepoint (( x , v ) , . . . , ( x n , v n )) belongs to E ′ M (2 n ) iff all x i are equal and v i are representing equidistant rays in T x M . Define E M (2 n ) as a quo-tient of E ′ M (2 n ) by the natural action of the permutation group S n .The fiber F x (2 n ) at a point x of the fibration E M (2 n ) consists then ofall 2 n -stars in the tangent space T x M , i.e. configurations of 2 n rays(or equally n lines) in T x M with equal angles between adjacent lines.In these terms the Bers theorem states that if ord ( f, x ) = n thenthe nodal graph N ( f ) defines an element of F x (2 n ) which we denoteby s ( N ( f ) , x ). OUNDS ON MULTIPLICITIES OF LAPLACE OPERATOR EIGENVALUES 7 First bound
We start by proving Lemma 1 from paper [2] for our particular caseΣ = RP or Σ = RP h . Lemma 1.
Let λ ∈ R be a real number, U be a linear space of functionson Σ ∈ { RP , RP h } satisfying ∆( f ) = λf and let n = sup f ∈ U µ ( N ( f )) .Suppose that dim( U ) > n . For each x ∈ Int (Σ) consider the set U n ( x ) ⊂ U consisting of eigenfunctions f x whose nodal graph N f con-tains a vertex x of degree at least n .Then dim( U n ( x )) > . Moreover, • any f x has a nodal graph with a unique vertex in Σ ; • ord ( f x , x ) = n ; • faces of N ( f x ) are homeomorphic to D ; • dim( U n ( x )) . Proof . The first inequality is a consequence of Proposition 1.Now, consider the nodal graph N ( f ) in Σ of a function f ∈ U n ( x )with highest ord ( f, x ). Add new edges cutting non-simply-connectedfaces into discs (and hence making the graph connected). Contractsome edges to merge all the vertices with x . We have obtained a newgraph N ′ . If some of propositions of Lemma 1 failed, namely, therewere other vertices except x , or there were non-simply-connected facesof N , or ord ( f x , x ) > n , or dim( U n ( x )) >
2, then the degree of vertex x in the new graph satisfies deg N ′ ( x ) > n + 2 and hence N ′ has atleast n + 1 edges. The number of faces of N ′ is the same as for N , nomore than n . Now Euler characteristic of Σ ∼ = RP can be estimatedas 1 = { vertices of N ′ } − { edges of N ′ } + { faces of N ′ } − ( n + 1) + n = 0 , but it is a contradiction.The inequality dim( U ( x )) f x ∈ U n +1 ( x ), such that ord ( f x , x ) > n which is already ruled out.Hence all the ways Lemma 1 could fail, lead to the contradiction andLemma 1 is proven. (cid:3) Lemma 1 implies that if dim( U ) > n then for each point x ∈ RP there is either unique (up to R ∗ ) eigenfunction f x whose nodal graph N f x has the vertex x of degree 2 n , or a 2-dimensional space U ( x ) ofsuch functions. We prove now that the case dim( U n ( x )) = 2 describedabove is impossible in the case of odd n . A. S. BERDNIKOV, N. S. NADIRASHVILI, AND A. V. PENSKOI
Lemma 2.
In the setting of Lemma 1 if n is an odd number then wehave got dim( U n ( x )) = 1 . Before we get to the proof of Lemma 2, let us quickly mention atechnical property of nodal graphs in the surface Σ with the collapsedboundary.
Proposition 2.
Suppose f ( p ) is a continuous family of functions de-pending on a parameter p , each f ( p ) satisfy ∆ f ( p ) = λf ( p ) for somereal number λ ∈ R and the nodal graph N ( p ) = N ( f ( p )) has only onevertex x ( p ) in Σ and deg N ( p ) ( x ( p )) = 2 n . Then the loops provided byedges of N ( p ) do not change their homotopy class in the local system π (Σ , x ( p )) while p moves along Int (Σ) . The idea of the proof is that since x ( p ) is the only vertex of thegraph in Σ, there is no more than two rays of the nodal graph in Σapproaching each connected component of the boundary. Hence forall p close to p these rays can connect only with each other near thecollapsed boundary component in Σ. For details, see [2, Proposition 3]. Proof of Lemma 2 . Suppose that dim( U n ( x )) = 2. Then, accord-ing to Bers theorem, for some polar coordinates ( r, θ ) near x and polarcoordinates ( R, φ ) in U ( x ) we have the following approximation f ( R,φ ) ( r, θ ) = Rr n cos(( n ) θ + φ ) + O ( r n ) . Hence, taking the star of the function σ ( • , x ) : P U → F x is a diffeo-morphism of P U ∼ = S ∼ = F x , and the generator loop of P U rotateseach ray of σ ( f, x ) to the subsequent one.Consider the universal cover π : R × [ − , ( • / Z ) × id −−−−−→ S × [ − , ∼ = S \ {± x } → • / {± } −−−−→ R P \ { x } ∼ = S ⋉ [ − , R P \ { x } acts by integer shifts on R -factor. Then the lift of the edge of N ( f ) joins ( y, −
1) and ( y + jn , e )for some j ∈ Z and e ∈ {− , } . Applying Proposition 2 to an isotopyof a graph induced by a generator loop of P U , we conclude that all thepoints ( y + in , −
1) are joined with ( y + i + jn , e ) by the lift of some edge of N ( f ). Since this identification is a part of an involution prescribed bythe lifts of the edges, e has to be equal to 1. We conclude that ( y + in , y + i − jn , − S \ {± x } replaces j defined in this way by − j . But the graph, lifted along thequotient by {± } is invariant under the antipodal map, therefore j = 0. OUNDS ON MULTIPLICITIES OF LAPLACE OPERATOR EIGENVALUES 9
But this couldn’t be either, as in this case for sufficiently small ε weget sgn ( f ( π ( y + 12 n , − (1 − ε )))) = sgn ( f ( π ( y + 12 n , − ε )) == − sgn ( f ( π ( y + n + 12 n , − ε ))) = − sgn ( f ( π ( y + 12 n , − (1 − ε ))))which is a contradiction. Here for the first equality we use j = 0, forthe second equality we use the fact that n is odd and hence we have sgn (sin( π )) = − sgn (sin( nπ + sin( π )) and for the third equality we usethe invariance of f under the action of π ( RP \ { x } ). (cid:3) Finally, for an odd n we are left with only possibility that for eachpoint x ∈ R P there is unique (up to R ∗ ) eigenfunction f x such that ord ( f x , x ) = n and no functions with ord ( f x , x ) > n + 1. Note thatit completes the proof of the first bound in the Theorem 16 requiredfor the Theorem 2. Indeed, if in the conditions of the Theorem 16 wehave dim( U ) > n + 1, then by Proposition 1 we have the inequalitydim( U n ( x )) > n + 1) − n = 2, which contradicts Lemma 2 forodd n .Therefore Theorem 2 is now proven. (cid:3) Second bound: closed surface case
We switch to the case of Σ = RP . Suppose that there is a coun-terexample to the Theorem 1, so that there is an eigenspace U l withdim( U l ) > l + 2. Then due to Courant nodal domain theorem theassumption sup f ∈ U ( µ ( N ( f ))) l + 1 = n of the Lemmas 1 and 2 holdsand for each point x ∈ R P we get a function f x , defined up to aconstant, such that ord ( f x , x ) = n . The stars of the functions f x formthen a smooth section σ of the star fibration E RP (2 n ) according to thefollowing technical proposition. Proposition 3.
Let U be a finite-dimensional eigenspace of the Lapla-cian. For every x ∈ Int (Σ) consider the subspace [ f x ] ⊂ U of functions f x with ord ( f x , x ) > n . Suppose that for any x ∈ Int (Σ) the subspace [ f x ] is of dimension 1 (i. e. nonzero f x are defined up to R ∗ ), and that ord ( f x , x ) = n . Define a section σ ( x ) = s ( N ( f x ) , x ) . Then this section σ ∈ Γ( E Int (Σ) (2 n )) is smooth. The proof follows directly from inverse function theorem, see thepaper [2, Proposition 1].Now in order to achieve the final contradiction let us pull σ backto the star fibration E S (2 n ) via the universal cover S → RP and observe that the Euler class e ( E S (2 n )) = 2 ne ( T S ) = 4 n is non-zeroand hence E S (2 n ) can not have a continuous section. Hence, the initialassumption that the multiplicity of λ l is at least 4 l + 2 is false andthus the multiplicity is at most 4 k + 1.Unfortunately, we did not find any evident way to make the analo-gous final consideration with the Euler class in the case ∂ Σ = ∅ , sincethe surface with boundary has no such class. However, it would bepossible to use the relative Euler class if we could put some restrictionson the behavior of the star section σ near the boundary. In the caseof a surface M of positive genus such restrictions come from the factthat some loops of nodal graph represent a (constant) non-zero class in H ( ˜ M ) for orienting cover ˜ M . This consideration, based on the methodfrom the paper [14], is the crucial argument in the paper [2]. But in ourcase of RP the orienting cover S has zero homology in dimension 1and there is no apparent topological obstructions for the nodal graphsin the consideration to exist. References [1] Bandle, C.: Isoperimetric inequalities and applications,
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Department of Mathematics, Massachusetts Insti-tute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139,USA and
Independent University of Moscow, Bolshoy Vlasyevskiy per. 11, 119002,Moscow, Russia and
National Research University Higher School of Economics, VavilovaStr. 7, 117312, Moscow, Russia
E-mail address : [email protected] (N. S. Nadirashvili) Aix Marseille Universit´e, CNRS, I2M UMR 7353 —Centre de Math´ematiques et Informatique, 13453, Marseille, France
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
Laboratoire J.-V.Poncelet (UMI 2615), Bolshoy Vlasyevskiy Pereulok11, 119002, Moscow, Russia
E-mail address , corresponding author:, corresponding author: