On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces
OON LARGENESS AND MULTIPLICITY OF THE FIRSTEIGENVALUE
SUGATA MONDAL
Abstract.
We apply topological methods to study the smallest non-zero number λ in the spectrum of the Laplacian on finite area hyper-bolic surfaces. For closed hyperbolic surfaces of genus two we show thatthe set { S ∈ M : λ ( S ) > } is unbounded and disconnects the modulispace M . Introduction
In this paper we identify hyperbolic surfaces with quotients of the Poincar´eupper halfplane H by discrete torsion free subgroups of PSL(2 , R ) called Fuchsian groups . The
Laplacian on H is the differential operator ∆ whichassociates to a C - function f the function∆ f ( z ) = y ( ∂ f∂x + ∂ f∂y ) . (0.1)For any Fuchsian group Γ, the induced differential operator on S = H / Γ,∆ = ∆ S is called the Laplacian on S . It is a non-positive operator whosespectrum spec(∆) is contained in a smallest interval ( −∞ , − λ ( S )] ⊂ R − ∪{ } with λ ( S ) ≥
0. Points in the discrete spectrum will be referred toas an eigenvalue . In particular this means λ ≥ C -function f ∈ L ( S ), called a λ - eigenfunction , such that∆ f + λf = 0 . When 0 < λ ≤ / λ is called a small eigenvalue and f iscalled a small eigenfunction .We shall restrict ourselves to hyperbolic surfaces with finite area. Anysuch surface S is homeomorphic to a closed Riemann surface S of certaingenus g from which some n many points are removed. In that case S iscalled a finite area hyperbolic surface of type ( g, n ). Each of these n pointsis called a puncture of S .The Laplace spectrum of a closed hyperbolic surface S consists of a dis-crete set: 0 = λ < λ ( S ) ≤ ... ≤ λ n ( S ) ≤ ... ∞ (0.2)such that λ i ( S ) → ∞ as i → ∞ . Each number in the above sequence isrepeated according to its multiplicity as eigenvalue. The number λ i ( S ) iscalled the i -th eigenvalue of S . It is known that the map λ i : M g → R Date : October 30, 2018.1991
Mathematics Subject Classification.
Primary 35P05, 58G20, 43A85, 58G25; Sec-ondary 58J5.
Key words and phrases.
Laplace operator, first eigenvalue, small eigenvalues. a r X i v : . [ m a t h . DG ] O c t SUGATA MONDAL that assigns a surface S ∈ M g to its i -th eigenvalue λ i ( S ) is continuous andbounded [B3]. Hence Λ i ( g ) = sup S ∈M g λ i ( S ) < ∞ . (0.3)For non-compact hyperbolic surfaces of finite area the spectrum of the Lapla-cian is more complicated. It consists of both continuous and discrete com-ponents (see [I] for detail). However, the part of the spectrum lying in [0 , )is discrete. Keeping resemblance to above definition for any hyperbolic sur-face S let us define λ ( S ) to be the smallest positive number in spec(∆). Inparticular, if λ < then it is an eigenvalue. The function λ , so defined,is bounded by because S has a continuous spectrum on [ , ∞ ). As beforewe consider the quantity Λ ( g, n ) = sup S ∈M g,n λ ( S ) . (0.4)In [Se] Atle Selberg proved that for any congruence subgroup Γ of SL(2 , Z ) λ ( H / Γ) ≥ . (0.5)Recall that a congruence subgroup is a discrete subgroup of SL(2 , Z ) thatcontains one of the Γ n whereΓ n = { (cid:18) a bc d (cid:19) ∈ SL(2 , Z ) : a ≡ ≡ d and b ≡ ≡ c (mod n ) } (0.6)is the principal congruence subgroup of level n . Moreover he conjectured Conjecture 0.7.
For any congruence subgroup Γ , λ ( H / Γ) ≥ . M. N. Huxley [Hu] proved this conjecture for Γ n with n ≤
6. Severalattempts have been made to prove it (see [I, Chapter 11] for details) in thegeneral case. The best known bound is due to Kim and Sarnak [K-S].This conjecture motivated, in particular, the question of our interest:
Question 0.8.
Given any genus g ≥ g with λ at least ?A slightly weaker question than the above one would be: Is Λ ( g ) ≥ ?This question is studied in [BBD] by P. Buser, M. Burger and J. Dodziukand in [B-M] by R. Brooks and E. Makover. The ideas in [BBD] and [B-M],in the light of the bound of Kim and Sarnak in [K-S], provide the following. Theorem 0.9.
Given any (cid:15) > , there exists N (cid:15) ∈ N such that for any g ≥ N (cid:15) there exist closed hyperbolic surfaces of genus g with λ ≥ − (cid:15) . The constant in the above theorem can be replaced by if conjecture0.7 is true. Hence it is tempting to conjecture: Conjecture 0.10.
For every g ≥ there exists a closed hyperbolic surfaceof genus g whose λ is at least . Remark 0.11.
Observe that even if Selberg’s conjecture (conjecture 0.7)is true, theorem 0.9 do not provide a positive answer to conjecture 0.10.However it would answer positively the weaker version of our question i.e.it would imply Λ ( g ) ≥ , for large values of g . N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 3
The existence of genus two hyperbolic surfaces with λ > has beenknown in the literature for sometime [Je]. It is known that the Bolza surfacehas λ approximately 3 . B ( ) = { S ∈ M : λ ( S ) > } of the moduli space M . Fromthe continuity of λ it is clear that B ( ) is open. Our first result providesbetter understanding of this set.0.1. Eigenvalue branches.
Recall that the moduli space M g is the quo-tient of T g by the Teichm¨uller modular group M g (see [B3]). We are shiftingfrom the moduli space to the Teichm¨uller space mainly because we wish totalk about analytic paths which involves coordinates and on T g one has theFenchel-Nielsen coordinates (given a pants decomposition) which is easy todescribe.Let γ : [0 , → T be an analytic path. Since, in this case, λ is simple aslong as small, the function λ ( S t ) ( S t = γ ( t )) is also analytic (see theorem0.12) if λ ( S t ) ≤ for all t ∈ [0 , λ may not be simpleeven if small (see § γ : [0 , → T g , λ ( S t ) is continuous but need not be analytic even if λ ( S t ) ≤ for all t ∈ [0 , Theorem 0.12.
Let ( S t ) t ∈ I be a real analytic path in T g . Then there existreal analytic functions λ tk : I → R such that for each t ∈ I the sequence ( λ tk ) consist of all eigenvalues of S t (listed with multiplicities, though not inincreasing order). Each function λ tk is called a branch of eigenvalues along S t . More precisely Definition 0.13.
Let α : [0 , → T g be an analytic path. An analyticfunction λ t : [0 , → R is called a branch of an eigenvalue along α if, foreach t , λ t is an eigenvalue of α ( t ). If λ = λ i ( α (0)) then we shall say that λ t is a branch of eigenvalues along α that starts as λ i . If the underlying path α is fixed then we shall skip referring to it.Here, instead of considering λ , we consider branches of eigenvalues thatstart as λ and modify question 0.8 as: Question 0.14.
For any g ≥ T g that start as λ and exceeds eventually ?Fortunately this modified question turns out to be much easier than theoriginal one and we have a positive answer to it (see theorem 1.3).0.2. Multiplicity.
For any eigenvalue λ of S , the dimension of ker(∆ − λ. id)is called the multiplicity of λ . If the multiplicity of λ were one for all closedhyperbolic surfaces of genus g then theorem 1.3 would have showed theexistence of surfaces with λ > implying conjecture 0.10. However this isnot the case and in fact the following is proved in [C-V]: Theorem 0.15.
For every g ≥ and n ≥ there exists a surface S ∈ M g,n such that λ ( S ) is small and has multiplicity equal to the integral part of √ g +12 . SUGATA MONDAL
For g ≥ g = 2 do not work for g ≥
3. In [O] the following upper bound onthe multiplicity of a small eigenvalue is proved
Proposition 0.16.
Let S be a finite area hyperbolic surface of type ( g, n ) .Then the multiplicity of a small eigenvalue of S is at most g − n . Our last result is an improvement of this result for hyperbolic surfaces oftype (0 , n ) (see theorem 1.4). 1. results
As mentioned before, it is known that there are closed hyperbolic surfacesof genus two with λ > (in fact with > . B ( 14 ) = { S ∈ M : λ ( S ) > } . Theorem 1.1. B ( ) is an unbounded set that disconnects M . Sketch of the proof of Theorem 1.1:
We first prove that B ( ) disconnects M . We argue by contradiction andassume that M \ B ( ) is connected. Now for any S ∈ M \ B ( ), λ ( S )is small and hence has multiplicity exactly one by [O]. We shall see that,in fact, the nodal set of the λ ( S )-eigenfunction (see §
3) consists of simpleclosed curves. With the help of this property we shall deduce that the nodalset of the first eigenfunction is constant, up to isotopy, on M \ B ( ).Finally, using an argument involving geodesic pinching we shall show thatthere exist surfaces S and S in M \B ( ) such the nodal sets of the λ ( S )-eigenfunction is not isotopic to the nodal set of the λ ( S )-eigenfunction.This provides the desired contradiction. The rest of the theorem i.e. B ( )is unbounded is deduced from a description of the components of M \B ( ).For finite area hyperbolic surfaces with Euler characteristic two the ideasin the above proof carries over to provide the following. Theorem 1.2.
For any ( g, n ) with g − n = 2 (i.e. ( g, n ) = (2 , , (1 , or (0 , ) the set C g,n ( ) = { S ∈ M g,n : λ ( S ) ≥ } disconnects M g,n .Moreover for ( g, n ) = (2 , and (1 , it is unbounded. Our next result is on the existence of branches of eigenvalues in T g , forany g ≥
3, that start as λ and eventually becomes larger than . Theorem 1.3.
There are branches of eigenvalues in T g that start as λ andtake values strictly bigger than . Recall that T can be embedded in T g as an analytic subset containingsurfaces with certain symmetries (see § T by the above embeddingΠ : T → T g . We shall use a geodesic pinching argument to prove thatamong these branches there are ones that start as λ .Our last result is on the multiplicity of λ of genus zero hyperbolic sur-faces, punctured spheres. N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 5
Theorem 1.4.
Let S be a hyperbolic surface of genus . If λ ( S ) ≤ is aneigenvalue then the multiplicity of λ ( S ) is at most three. Sketch of proof:
Let S be a hyperbolic surface of genus 0 with n punc-tures. Let S denote the closed surface obtained by filling in the punctures of S . Assume that λ ( S ) ≤ is an eigenvalue. Let φ be a λ ( S )-eigenfunctionwith nodal set Z ( φ ) ( §
2) which is a finite graph by [O] (see lemma 2.7).Using
Jordan curve theorem and
Courant’s nodal domain theo-rem we shall deduce the simple description of Z ( φ ) as a simple closed curvein S . In particular, if one of the punctures p of S lies on Z ( φ ) then thenumber of arcs in Z ( φ ) emanating from p is at most two.Let p be one of the punctures of S . It is a standard fact that in any cusparound p any λ ( S )-eigenfunction φ has a Fourier development of the form: φ ( x, y ) = φ y − s + (cid:88) j ≥ (cid:114) jyπ K s − ( jy )( φ ej cos( j.x ) + φ oj sin( j.x )) (1.5)where λ ( S ) = s (1 − s ) with s ∈ ( ,
1] and K is the modified Bessel functionof exponential decay (see § λ ( S )-eigenfunctions by E and consider the map π : E → R given by π ( φ ) =( φ , φ e , φ o ). This is a linear map and so if dim E > π is non-empty. Let ψ ∈ ker π i.e. ψ = ψ e = ψ o = 0. Then by the result [Ju]of Judge, the number of arcs in Z ( ψ ) emanating from p is at least four, acontradiction to the above description of Z ( φ ) at p .2. Preliminaries
In this section we recall some definitions and results that will be necessaryin later sections. Let S be a finite area hyperbolic surface. Then S ishomeomorphic to a closed surface with finitely many points removed. Eachof these point, called punctures, has special neighborhoods in S called cusp s.2.1. Cusps.
Denote by ι the parabolic isometry ι : z → z + 2 π . For a choiceof t >
0, a cusp P t is the half-infinite cylinder { z = x + iy : y > πt } / < ι > .The boundary curve { y = πt } is a horocycle of length t . The hyperbolicmetric on P t has the form: ds = dx + dy y . (2.1)Any function f ∈ L ( P t ) has a Fourier development in the x variable of theform f ( z ) = (cid:88) n ∈ Z ∗ f n ( y ) cos( nx + θ n ) . (2.2)If f satisfy the equation ∆ f = s (1 − s ) f then the above expression can besimplified as f ( z ) = f ( y ) + (cid:88) j ≥ f j (cid:114) jyπ K s − ( jy ) cos( j.x − θ j )= f ( y ) + (cid:88) j ≥ (cid:114) jyπ K s − ( jy )( f ej cos( j.x ) + f oj sin( j.x )) (2.3) SUGATA MONDAL where K s is the modified Bessel function (see [Ju]) and f ( y ) = f , y s + f , y − s if s (cid:54) = 12 and f ( y ) = f , y + f , y log y if s = 12 . (2.4)The function f is called cuspidal if f ( y ) ≡ Nodal sets.
For any function f : S → R , the set { x ∈ S : f ( x ) = 0 } iscalled the nodal set Z ( f ) of f . Each component of S \ Z ( f ) is called a nodaldomain of f . In a neighborhood of a regular point p ∈ Z ( f ) ( ∇ p f (cid:54) = 0)the implicit function theorem implies that Z ( f ) is a smooth curve. In aneighborhood of a critical point p ∈ Z ( f ) ( ∇ p f = 0), it is not so simple todescribe Z ( f ). When f is an eigenfunction of the Laplacian we have thefollowing description due to S. Y. Cheng [Che]: Theorem 2.5.
Let S be a surface with a C ∞ metric. Then, for any solutionof the equation (∆ + h ) φ = 0 , h ∈ C ∞ ( S ) , one has: ( i ) Critical points on the nodal set Z ( φ ) are isolated. ( ii ) Any critical point in Z ( φ ) has a neighborhood N in S which is diffeo-morphic to the disc { z ∈ C : | z | < } by a C -diffeomorphism that sends Z ( φ ) ∩ N to an equiangular system of rays. Remark 2.6.
In particular, if p ∈ Z ( φ ) is a critical point of φ then thedegree of the graph Z ( φ ) at p is at least 4. Hence if a component of Z ( φ ) isa simple closed loop then it is automatically smooth.When S is closed theorem 2.5 implies that Z ( φ ) is a finite graph. When S is non-compact with finite area it implies local finiteness of Z ( φ ) but not global . In this particular case we have the following lemma due to Jean-Pierre Otal [O, Lemma 6] (the second part is [O, Lemma 1]) Lemma 2.7.
Let S be a hyperbolic surface with finite area and let φ : S → R be a λ -eigenfunction with λ ≤ . Then the closure of Z ( φ ) in S is a finitegraph. Moreover, each nodal domain of φ has negative Euler characteristic. In particular, Z ( φ ) is a union (not necessarily disjoint) of finitely many cycles in S that may contain some of the punctures of S . Next we recallCourant’s nodal domain theorem Theorem 2.8.
Let S be a closed hyperbolic surface. Then the number ofnodal domains of a λ i ( S ) -eigenfunction can be at most i + 1 . The proof (see [Cha] or [Che]) of this theorem works also for finite areahyperbolic surfaces if λ i < . In particular, for a hyperbolic surface S withfinite area if λ ( S ) < then the number of nodal domains of a λ ( S )-eigenfunction is at most two. Since any λ -eigenfunction φ has mean zero, Z ( φ ) must disconnect S . Hence any λ -eigenfunction has exactly two nodaldomains. N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 7 Genus two: Proof of Theorem 1.1
We begin by proving that B ( ) disconnects M . We argue by contradic-tion and assume that M \B ( ) is connected. Now, for any S ∈ M \B ( ): λ ( S ) ≤ and so λ ( S ) is simple by [O]. Hence to a surface S ∈ M \ B ( )one can assign its first non-constant eigenfunction φ S without any ambiguity.We assume that φ S is normalized i.e. (cid:90) S φ S dµ S = 1 . (3.1)Let Z ( φ S ) denote the nodal set of φ S . Since φ S is the first eigenfunction, byCourant’s nodal domain theorem, S \ Z ( φ S ) has exactly two components.Denote by S + ( φ S ) (resp. S − ( φ S )) the component of S \ Z ( φ S ) where φ S is positive (resp. negative). By Euler-Poicar´e formula applied to the celldecomposition of S consisting of nodal domains of φ S as the two skeletonand the nodal set Z ( φ S ) as the one skeleton we have the following equality: χ ( S ) = χ ( S + ( φ S )) + χ ( S − ( φ S )) + χ ( Z ( φ S )) . (3.2)Since χ ( S ) = − χ ( S + ( φ S )) and χ ( S − ( φ S )) are negative by lemma2.7, we conclude from (3.9) that χ ( Z ( φ S )) = 0. This means that Z ( φ S )consists of simple closed curve(s) that divide S into exactly two components.Moreover, since no nodal domain of φ S is a disc or an annulus by lemma2.7, each curve in Z ( φ S ) is essential (homotopically non-trivial in S ) and notwo curves in Z ( φ S ) are homotopic. In particular, Claim 3.3.
For any S ∈ M \ B ( ) , the nodal set Z ( φ S ) of φ S consistseither of tree smooth simple closed curves that divide S into two pair ofpants (the first picture below) or of a unique smooth simple closed curvethat divides S into two tori with one hole (the second picture below). Decomposition
Now we have the following:
Claim 3.4.
Let S ∈ M such that λ ( S ) is simple and the nodal set Z ( φ S ) of the λ ( S ) -eigenfunction φ S is also simple. Then S has a neighborhood SUGATA MONDAL N ( S ) in M such that for any S (cid:48) ∈ N ( S ) the nodal set Z ( φ S (cid:48) ) is isotopicto Z ( φ S ) .Proof. First observe that λ ( S ) being simple we have a neighborhood N (cid:48) ( S )in M such that for any S (cid:48) ∈ N ( S ) λ ( S (cid:48) ) is simple. Hence φ (cid:48) S is welldefined too.Now φ S is the λ ( S )-eigenfunction, so S \Z ( φ S ) has exactly two connectedcomponents S + and S − such that φ S has positive sign on S + . So necessarily φ S has negative sign on S − . Now consider a tubular neighborhood T S of Z ( φ S ). By [M, Theorem 3.36](see also [H], [Ji]) we have a neighborhood N ( S ) ⊂ N (cid:48) ( S ) of S such that for any S (cid:48) ∈ N ( S ), φ S (cid:48) has positive sign on S + \ T S and negative sign on S − \ T S . In particular, Z ( φ S (cid:48) ) ⊂ T S . Henceby the description of Z ( φ S (cid:48) ) as in claim 3.3 the proof follows. (cid:3) Therefore, there exists S ∈ M \ B ( ) such that Z ( φ S ) consists of onlyone curve if and only if for all S (cid:48) ∈ M \ B ( ), Z ( φ S (cid:48) ) consists of only onecurve. This is a contradiction to proposition 3.6. Definition 3.5.
The systole s ( S ) of a surface S is the minimum of thelengths of closed geodesics on S . The injectivity radius of S at a point p isthe maximum of the radius of the geodesic discs with center p that embedin S . For any (cid:15) > S with injectivity radius at least (cid:15) is denoted by S [ (cid:15), ∞ ) . Each point in the complement S (0 ,(cid:15) ) = S \ S [ (cid:15), ∞ ) hasinjectivity radius at most (cid:15) . S [ (cid:15), ∞ ) and S (0 ,(cid:15) ) are respectively called (cid:15) -thickpart and (cid:15) -thin part of S . Proposition 3.6.
Let S be a finite area hyperbolic surface of type ( g, n ) .Let G = ( γ i ) ki =1 be a collection of smooth, mutually non-intersection simpleclosed curves on S that separates S in exactly two components. Assume that G is minimal in the sense that no proper subset of G can separate S . Thengiven any (cid:15), δ > there exists a finite area hyperbolic surface S G of type ( g, n ) with s ( S G ) < (cid:15) such that λ ( S G ) < δ is simple and the nodal set ofthe λ ( S G ) -eigenfunction is isotopic to G . Remark 3.7.
It is not very difficult to construct two collections of curveson S , as in the above lemma, that are not isotopic. In particular for ( g, n ) =(2 ,
0) claim 3.3 provides two such collections. Therefore the above lemmaindeed provide two surfaces S and S in M such that S , S ∈ M \ B ( )and Z ( φ S ) is not isotopic to Z ( φ S ).Proof of Proposition 3.6 uses the behavior of sequences of small eigenpairsover degenerating sequences of hyperbolic surfaces. For precise definitionsof these concepts we refer the reader to [M]. Proof.
Without loss of generality we may assume that each curve in G is a ge-odesic. Extend G to a pants decomposition P = ( γ i ) g − ni =1 of S . Let ( l i , θ i )denote the Fenchel-Nielsen coordinates on T g,n with respect to ( γ i ) g − ni =1 .Here l i denotes the length parameter and θ i denotes the twist parameteralong γ i .Now consider the sequence of surfaces ( S m ) in T g,n such that l i ( S m ) = m for i ≤ k , l j = c > j > k and θ j = c > ≤ j ≤ g − n . Then,up to extracting a subsequence, ( S m ) converges to a finite area hyperbolic N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 9 surface S ∞ ∈ ∂ M g,n . Let us denote the extracted subsequence by ( S m ) itself. Observe that S ∞ is obtained from S by pinching the geodesics in G .Namely, for each i = 1 , ..., k there is a geodesic γ mi in S m , in the homotopyclass of γ i , whose length tends to zero as m → ∞ .The number of components of S ∞ ∈ M g,n is exactly two. Hence by[C-C], λ ( S m ) → S m stay away from zero. Inparticular λ ( S m ) is simple for m sufficiently large. Let φ S m be the λ ( S m )-eigenfunction with L -norm 1. Recall that we want to prove that for any (cid:15), δ > S G with s ( S G ) < (cid:15) such that λ ( S G ) < δ is simple andthe nodal set of the λ ( S G )-eigenfunction is isotopic to G . Since s ( S m ) → λ ( S m ) → Z ( φ S m )is isotopic to G for sufficiently large m .Now we apply [M, Theorem 3.34] to extract a subsequence of φ S m thatconverges uniformly over compacta to a 0-eigenfunction φ ∞ of S ∞ with L -norm 1. Let us denote the extracted subsequence by ( S m ) itself. Since0-eigenfunctions are constant functions, φ ∞ is constant on each componentsof S ∞ . Claim 3.8.
The two constant values of φ ∞ on the two components of S ∞ are non-zero and have opposite sign.Proof. For (cid:15) > L -norm of φ S m restricted to S (0 ,(cid:15) ) m by (cid:107) φ S m (cid:107) S (0 ,(cid:15) ) m . By the uniform convergence of φ S m to φ ∞ over compacta wehave (cid:90) S [ (cid:15), ∞ ) ∞ φ ∞ = lim m →∞ (cid:90) S [ (cid:15), ∞ ) m φ S m = 1 − lim m →∞ (cid:107) φ S m (cid:107) S (0 ,(cid:15) ) m . Since (cid:82) S ∞ φ ∞ = lim (cid:15) → (cid:82) S [ (cid:15), ∞ ) ∞ φ ∞ = 1 we obtain that for any δ > (cid:15) > m →∞ (cid:107) φ S m (cid:107) S (0 ,(cid:15) ) m ≤ δ . Now | (cid:90) S [ (cid:15), ∞ ) ∞ φ ∞ | = lim m →∞ | (cid:90) S [ (cid:15), ∞ ) m φ S m | = | − lim m →∞ (cid:90) S (0 ,(cid:15) ) m φ S m |≤ lim m →∞ (cid:113) | S (0 ,(cid:15) ) m |(cid:107) φ S m (cid:107) S (0 ,(cid:15) ) m ( by Holder inequality) ≤ δ lim m →∞ (cid:113) | S (0 ,(cid:15) ) m | . Here | S (0 ,(cid:15) ) m | denotes the area of S (0 ,(cid:15) ) m . Recall that, for any m ∈ N ∪ ∞ ,lim (cid:15) → | S (0 ,(cid:15) ) m | = 0. So for m ≥ (cid:15) sufficiently small: | (cid:90) S [ (cid:15), ∞ ) ∞ φ ∞ | < δ and | S (0 ,(cid:15) ) m | < δ. Finally, taking (cid:15) to be sufficiently small, we calculate: | (cid:90) S ∞ φ ∞ | ≤ | (cid:90) S [ (cid:15), ∞ ) ∞ φ ∞ | + | (cid:90) S (0 ,(cid:15) ) ∞ φ ∞ | ≤ δ + (cid:113) | S (0 ,(cid:15) ) ∞ |(cid:107) φ S ∞ (cid:107) S (0 ,(cid:15) ) ∞ ≤ δ since (cid:107) φ S ∞ (cid:107) S (0 ,(cid:15) ) ∞ < (cid:107) φ S ∞ (cid:107) = 1 . Since δ is arbitrary we conclude that (cid:82) S ∞ φ ∞ = 0. Hence φ ∞ has L -norm 1 and mean zero.Since φ ∞ has L -norm 1 at least one of the two constant values of φ ∞ on the two components of S ∞ is non-zero. Since φ ∞ has mean zero both ofthese values are non-zero have opposite sign. (cid:3) As the length of γ mi tends to zero, we may assume that the collar neigh-borhood C mi of γ mi with two boundary components of length 1 embeds in S m and ( C mi ) ki =1 are mutually disjoint. At this point we recall that G isminimal in the sense that no proper subset of G can separate S . Hencenot only S m \ ∪ ki =1 ( C mi ) separates S in exactly two components but also noproper sub-collection of ( C mi ) ki =1 can separate S m . In particular, for each i , the limits of the two components of ∂C mi belong to two different compo-nents of S ∞ . Using claim 3.8 let us denote the limits of these two boundarysets by B ∞ i (+) and B ∞ i ( − ) such that φ ∞ | B ∞ i (+) > φ ∞ | B ∞ i ( − ) < ∂C mi by B mi (+) and B mi ( − )such that B ∞ i ( ± ) is the limit of B mi ( ± ) respectively. By the uniform con-vergence of φ S m to φ ∞ over compacta we conclude that, for sufficiently large m , φ S m | B mi (+) > φ S m | B mi ( − ) <
0. Hence, for m sufficiently large, atleast one component of Z ( φ S m ) is contained in C mi . Let Z i denote the unionof the components of Z ( φ S m ) that are contained in C mi .Let α be a simple closed loop in Z i . Since π ( C mi ) is Z there are only twopossibilities for α . Either it bounds a disc in C mi or it is homotopic to γ mi .Since λ ( S m ) is small, each component of S m \ Z ( φ S m ) has negative Eulercharacteristic by lemma 2.7. This discards the possibility that α bounds adisc in C mi . Hence α is homotopic to γ mi . Let β be another simple closed loopin Z i . Then β is also homotopic to γ mi implying that one of the componentsof S m \ Z ( φ S m ) has non-negative Euler characteristic. This leaves us withthe observation that each C mi contains exactly one loop α mi from Z ( φ S m ).By remark 2.6 α mi is in fact smooth. Therefore we have an isotopy of S that sends α mi to γ mi . Combining these isotopies we obtain that Z ( φ S m ) isisotopic to ( γ mi ) ki =1 . (cid:3) It remains to show that B ( ) is unbounded. We argue by contradictionand assume that B ( ) is bounded. Then we have (cid:15) > B ( )is contained in the compact set I (cid:15) = { S ∈ M : s ( S ) ≥ (cid:15) } [B]. Nowapplying lemma 3.6 obtain S and S in M such that s ( S i ) < (cid:15) , λ ( S i ) < is simple and the nodal set of the λ ( S )-eigenfunction is not isotopic to λ ( S )-eigenfunction. Since M \ I (cid:15) is path connected (see lemma A.1)we may have a path β in M \ I (cid:15) that joins S and S . Then lemma 3.4implies that the nodal set of the λ ( S )-eigenfunction is isotopic to λ ( S )-eigenfunction. This is a contradiction to our choice of S and S .3.1. Proof of Theorem 1.2.
The case ( g, n ) = (2 ,
0) follows from theabove theorem. It remains to show theorem 1.2 for ( g, n ) = (1 ,
2) and (0 , For the rest of the proof we refer to the pair ( g, n ) for only thesetwo cases. We argue by contradiction and assume that M g,n \ C g,n ( ) isconnected. By definition λ ( S ) < for any S ∈ M g,n \C g,n ( ). Hence λ ( S )is an eigenvalue and by [O-R] it is the only non-zero small eigenvalue of S .So we can consider the first non-constant eigenfunction φ S of S . As beforelet Z ( φ S ) be the nodal set of φ S . Denote by S the surface obtained from S by filling in its punctures and by Z ( φ S ) the closure of Z ( φ S ) in S . Bylemma 2.7 Z ( φ S ) is a finite graph. Now apply Euler-Poincar´e formula to thecell decomposition of S defined as follows: the punctures on S that do notlie on Z ( φ S ) is the zero skeleton, Z ( φ S ) is the one skeleton and S \ Z ( φ S ) N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 11 is the two skeleton. If k is the number of punctures of S that do not lie on Z ( φ S ) then χ ( S ) − k = χ ( S \ Z ( φ S )) + χ ( Z ( φ S )) . (3.9)By lemma 2.7 each component of S \Z ( φ S ) has negative Euler characteristicand so χ ( S \ Z ( φ S )) ≤ −
2. For ( g, n ) = (1 , χ ( S ) = 0 and so we havethe only possibility k = 2 and χ ( Z ( φ S )) = 0. For ( g, n ) = (0 , χ ( S ) = 2leaving us with the only possibility k = 4 and χ ( Z ( φ S )) = 0. Hence none ofthe punctures of S lie on the closure of the nodal set Z ( φ S ) i.e. Z ( φ S ) = Z ( φ S ) is a compact subset of S . Since χ ( Z ( φ S )) = 0 we conclude that Z ( φ S ) is a union of simple closed curves. Also by lemma 2.7 we know thatno loop in Z ( φ S ) can bound a disc and no two components of Z ( φ S ) can behomotopic. Summarizing these observations we get: Claim 3.10.
Let S ∈ M g,n \ C g,n ( ) . ( i ) If ( g, n ) = (1 , then Z ( φ S ) consists of either exactly one simple closedcurve or two simple closed curves. In the first case Z ( φ S ) divides S into twocomponents one of which is a surface of genus one with a copy of Z ( φ S ) as itsboundary and the other one is a twice punctured sphere with a copy of Z ( φ S ) as its boundary. In the last case Z ( φ S ) divides S into two components eachof which is a once punctured sphere with two boundary components comingfrom Z ( φ S ) . ( ii ) If ( g, n ) = (0 , then Z ( φ S ) consists of exactly one simple closed curve(there are two possibilities for this up to isotopy) that separates S into twocomponents each of which is a twice punctured sphere with one boundarycomponent coming from Z ( φ S ) . Next we have the following modified version of claim 3.4. Let S ∈ M g,n \C g,n ( ) with φ S the λ ( S )-eigenfunction. Claim 3.11.
There exists a neighborhood N ( S ) of S in M g,n such that forany S (cid:48) ∈ N ( S ) : λ ( S (cid:48) ) is simple and the nodal set Z ( φ S (cid:48) ) of the λ ( S (cid:48) ) -eigenfunction φ S (cid:48) is isotopic to Z ( φ S ) .Proof. Since λ ( S ) is < , λ is a continuous function in a neighborhood of S by [H](see also [C-C], [M]) and so we have a neighborhood N (cid:48) ( S ) of S which is contained in M g,n \ C g,n ( ). In particular, φ S (cid:48) is well-defined for S (cid:48) ∈ N (cid:48) ( S ) and Z ( φ S (cid:48) ) has the description in claim 3.10. Now consider atubular neighborhood T S of Z ( φ S ) in S such that ∂ T S has two components ∂ T + S and ∂ T − S each of which is a simple closed curve with φ S | ∂ T + S > φ S | ∂ T − S < λ < and simple on N (cid:48) ( S ). Hence by [H] for any compact subset K of S the map: Φ : K × N (cid:48) ( S ) → R given by Φ( x, S (cid:48) ) = φ S (cid:48) ( x ) is continuous.Considering K = ∂ T S we obtain N ( S ) ⊂ N (cid:48) ( S ) such that for any S (cid:48) ∈ N ( S ): φ S (cid:48) | ∂ T + S > φ S (cid:48) | ∂ T − S <
0. In particular, for any S (cid:48) ∈ N ( S ): Z ( φ S (cid:48) ) hasa component inside T S . Hence by the description of Z ( φ S (cid:48) ) in claim 3.10 weobtain the claim. (cid:3) Since by our assumption M g,n \ C g,n ( ) is connected the above claimimplies that only one of the two possibilities in claim 3.10 can actuallyoccur. This is a contradiction to proposition 3.6. Now we show that C , ( ) is unbounded. We argue by contradiction andassume that C , ( ) is bounded. Then we have (cid:15) > C , ( ) iscontained in the compact set I (cid:15) = { S ∈ M , : s ( S ) ≥ (cid:15) } [B]. Applyinglemma 3.6 we obtain S and S in M , such that s ( S i ) < (cid:15) , λ ( S i ) < is simple and the nodal set of the λ ( S )-eigenfunction is not isotopic to λ ( S )-eigenfunction. Since M , \ I (cid:15) is path connected (see lemma A.1)we may have a path β in M , \ I (cid:15) that joins S and S . Then lemma3.11 implies that the nodal set of the λ ( S )-eigenfunction is isotopic to λ ( S )-eigenfunction. This is a contradiction to our choice of S and S . ¯ ® ° ± Branches of eigenvalues
In this section we consider branches of eigenvalues along paths in T g . Mainpurpose of doing so is that the multiplicity of λ i , in particular λ is not onein general. Therefore along ’nice’ paths in T g the functions λ i may not be’nice’ enough (see introduction). However, theorem 0.12 shows that up tocertain choice at points of multiplicity λ i ’s are in fact ’nice’. This ’nice’choice makes λ i into a branch of eigenvalues. Theorem 1.3 says that if werestrict ourselves to branches of eigenvalues then we have a positive answerto conjecture 0.10, namely there are branches of eigenvalues that start as λ and becomes more than . Proof of Theorem 1.3.
We begin by explaining the the embedding Π : T →T g (see the next figure). Let S be the closed hyperbolic surface of genus twoand α, β, γ , δ are four geodesics on S as in the following picture. Now cut S along δ to obtain a hyperbolic surface S ∗ with genus one and two geodesicboundaries (each a copy of δ ). Consider g − S ∗ and glue N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 13 them along their consecutive boundaries after arranging them along a circleas in the picture below. Let Π( S ) denote the resulting hyperbolic surface.Now take a geodesic pants decomposition ( ξ i ) i =1 , , of S involving δ = ξ and consider the Fenchel-Nielsen coordinates ( l i , θ i ) i =1 , , on T with respectto this pants decomposition. Here l i = l ( ξ i ) is the length of the closedgeodesic ξ i and θ i is the twist parameter at ξ i . The images of ( ξ i ) i =1 , , in Π( S ), ( ξ ji ) i =1 , , j =1 , ,...,g − is a geodesic pants decomposition of Π( S ).Consider the the Fenchel-Nielsen coordinates ( l ji , θ ji ) i =1 , , j =1 , ,...,g − on T g with respect to this pants decomposition. As before, l ji = l ( ξ ji ) is the lengthof the closed geodesic ξ ji and θ ji is the twist parameter at ξ ji . With respectto these pants decompositions Π is expressed as( l , l , l , θ , θ , θ ) → ( l , l , l , θ , θ , θ (cid:124) (cid:123)(cid:122) (cid:125) , ..., l , l , l , θ , θ , θ (cid:124) (cid:123)(cid:122) (cid:125) g − ) . (4.1)This is an analytic map and the image Π( S ) of any S ∈ T has an isometry τ of order ( g −
1) that sends one 6-tuple ( l , l , l , θ , θ , θ ) to the next one.Also Π( S ) /τ is isometric to S i.e. Π( S ) is a ( g −
1) sheeted covering of S .Hence each eigenvalue of S is also an eigenvalue of Π( S ). In particular, abranch λ t of eigenvalues in T along η ( t ) is a a branch of eigenvalues in T g along Π( η ( t )).To finish the proof we need only to find S ∈ T such that λ ( S ) = λ (Π( S )). Once we find such a S , we can consider any analytic path η in T such that η ( o ) = S and λ ( η (1)) > . Then the branch of eigenvalues λ t = λ ( η ( t )) along Π( η ( t )) would be a branch that we seek.To show this we employ the technique in claim 3.6. Let S n be a sequence ofsurfaces of genus two on which the lengths of the geodesics α, β and γ tendsto zero. In particular, S n → S ∞ ∈ M , ∪ M , implying λ ( S n ) → λ ( S n ) (cid:57)
0. The sequence Π( S n ) converges to a surface in M ,g +1 ∪ M ,g +1 and so λ (Π( S n )) → λ (Π( S n )) (cid:57)
0. So for large n , λ ( S n ) <λ (Π( S n )) implying λ ( S n ) = λ (Π( S n )). (cid:3) Punctured spheres
We begin this section by recapitulating the ideas in [BBD]. By purelynumber theoretic methods Atle Selberg showed that for any congruencesubgroup Γ of SL(2 , Z ), λ ( H / Γ) ≥ . The purpose in [BBD] was to con-struct explicit closed hyperbolic surfaces with λ close to . To achieve thisgoal the authors of [BBD] considered principal congruence subgroups Γ n (see introduction) and corresponding finite area hyperbolic surfaces H / Γ n .Then they replaced the cusps in H / Γ n , which is even in number, by closedgeodesics of small length t and glued them in pairs (see [BBD] for details).The surface S t obtained in this way is closed, their genus g is independent of t and as t → S t → H / Γ n in the compactification of the moduli space M g .Rest of the proof showed that λ is lower semi-continuous over the family S t .This approach together with the result of Kim and Sarnak provides theorem0.9.Limiting properties of eigenvalues over degenerating family of hyperbolicmetrics have been studied well in the literature (to name a few Denis Hejhal[H], Gilles Courtois-Bruno Colbois [C-C], Lizhen Ji [Ji], Scott Wolpert [Wo], Chris Judge [ ? ]) (see also [M, Theorem 2]). These limiting results can besummarized as: Theorem 5.1.
Let ( S m ) be a sequence of hyperbolic surfaces in M g,n thatconverges to a finite area hyperbolic surface S ∈ ∂ M g,n . Let ( λ m , φ m ) bean eigenpair of S m such that λ m → λ < ∞ . Then, up to extracting a sub-sequence and up to rescalling, the sequence ( φ m ) converges to a generalizedeigenfunction over compacta if one of the following is true ( i ) n = 0 ( [Ji] ) ( ii ) n (cid:54) = 0 and λ < ( [H] , [C-C] ) ( iii ) n (cid:54) = 0 and λ > ( [Wo] ) ( iii ) n (cid:54) = 0 , λ m ≤ and φ m is cuspidal ( [M] ). Recall that there is a copy of M , g + n in the compactification M g,n of M g,n . The ideas in [BBD] along with above limiting results imply Lemma 5.2.
For any pair ( g, n ) , Λ ( g, n ) ≥ Λ (0 , g + n ) . Motivated by this we focus on Λ (0 , n ). Although we would not be ableto prove conjecture 0.10 we have theorem 1.4 on the multiplicity of λ whichwe prove now.5.1. Proof of Theorem 1.4.
Let S be a hyperbolic surface of genus 0 andassume that λ ( S ) ≤ is an eigenvalue. Let φ be a λ ( S )-eigenfunction.Then the closure Z ( φ ) of the nodal set Z ( φ ) of φ is a finite graph in S bytheorem 0.16. In particular, Z ( φ ) is a union of closed loops in S . Observealso that the number of components of S \ Z ( φ ) is same as that of S \ Z ( φ ).Now let Z ( φ ) consists of more than one closed loop. Then by Jordancurve theorem the number of components of S \ Z ( φ ) is at least three.This is a contradiction to Courant’s nodal domain theorem 2.8 which saysthat a λ ( S )-eigenfunction can have at most two nodal domains. Hence weconclude that Z ( φ ) is a simple closed curve in S . In particular, we have thefollowing description of Z ( φ ) at any puncture. Claim 5.3.
If one of the punctures p of S is a vertex of Z ( φ ) then thenumber of arcs in Z ( φ ) emanating from p is at most two. Let λ ( S ) = s (1 − s ) with s ∈ ( , p be one of the punctures of S . Let P t be a cusp around p (see § S being a puncturedsphere, does not have any cuspidal eigenvalue [Hu], [O]. Thus any λ ( S )-eigenfunction φ is a linear combination of residues of Eisenstein series (see[I]). It follows from [I, Thorem 6.9] that the y s term can not occur in theFourier development (see (2.1)) of these residues in P t . Hence φ has aFourier development in P t of the form (see § φ ( x, y ) = φ y − s + (cid:88) j ≥ (cid:114) jyπ K s − ( jy )( φ ej cos( j.x ) + φ oj sin( j.x )) . (5.4)Now we consider the space E generated by λ ( S )-eigenfunctions. The map π : E → R given by π ( φ ) = ( φ , φ e , φ o ) is linear and so if dim E > π is non-empty.Let ψ ∈ ker π i.e. ψ = ψ e = ψ o = 0. Then by the result [Ju] of Judge,the number of arcs in Z ( ψ ) emanating from p is at least four, a contradictionto claim 5.3. N LARGENESS AND MULTIPLICITY OF THE FIRST EIGENVALUE 15
Acknowledgement
I would like to thank my advisor Jean-Pierre Otal for all his help startingfrom suggesting the problem to me. I am thankful to Peter Buser andWerner Ballmann for the discussions that I had with them on this problem.I would like to thank the Max Planck Institute for Mathematics in Bonn forits support and hospitality.
Appendix
A.For the convenience of the reader we give a proof of the fact that, for( g, n ) (cid:54) = (0 , , (1 , M g,n \ I (cid:15) of the compact set I (cid:15) = { S ∈ M g,n : s ( S ) ≥ (cid:15) } [B] is path connected. Lemma A.1.
For any ( g, n ) (cid:54) = (0 , , (1 , with g − n > and any (cid:15) > the set M g,n \ I (cid:15) is path connected.Proof. Let S and S be two surfaces in M g,n such that s ( S i ) < (cid:15) . So wehave simple closed geodesics γ on S and γ on S such that the length l γ i of γ i is < (cid:15) . Recall that it has always been our practise to treat M g,n as a subset of all possible metrics on a fixed surface S and the geodesicsare understood to be parametric curves on S that satisfy certain differentialequations provided by the metric.With this understanding let us first assume that γ does not intersect γ .So we may consider a pants decomposition P of S containing both γ and γ .Let the Fenchel-Nielsen coordinates of S i be given by ( l j ( S i ) , θ j ( S i )) g − nj =1 .Here l , l are the length parameters along γ , γ and θ , θ are twist param-eters along γ , γ . Then consider the path β : [0 , → T given by: l ( β ( t )) = (cid:40) l ( S ) if t ∈ [0 , ] , − t ) l ( S ) + (2 t − l ( S ) if t ∈ [ , l ( β ( t )) = (cid:40) (1 − t ) l ( S ) + 2 tl ( S ) if t ∈ [0 , ] ,l ( S ) if t ∈ [ , l ( β ( t )) = (1 − t ) l ( S ) + tl ( S ) and θ j ( β ( t )) = (1 − t ) θ j ( S ) + tθ j ( S ).Since l ( β ( t )) < (cid:15) for t ∈ [0 , ] and l ( β ( t )) < (cid:15) for t ∈ [ ,
1] we observe that s ( β ( t )) < (cid:15) for all t . The image of β under the quotient map T g,n → M g,n produces the required path joining S and S .Now let us assume that γ intersects γ . Let γ be a simple closed geo-desic that does not intersect γ and γ . By our assumption i.e. ( g, n ) (cid:54) =(0 , , (1 ,
1) such a geodesic exists. Then by the procedure described aboveboth S and S can be joined by a path in M g,n \ I (cid:15) to a surface on which γ has length < (cid:15) . This finishes the proof. (cid:3) References [B] Bers, L.; A remark on Mumford’s compactness theorem, Israel J. Math. 12 (1972),400-407.[B1] Buser, Peter; Cubic graphs and the first eigenvalue of a Riemann surface, Math. Z.162 (1978), 87-99.[B2] Buser, Peter; On the bipartition of graphs, Discrete Applied Mathematics, 9 (1984),105-109. [B3] Buser, Peter; Geometry and spectra of compact Riemann surfaces. Progress in Math-ematics, 106. Birkh¨auser Boston, Inc., Boston, MA, 1992.[BBD] Burger, M., Buser, P., Dodziuk, J.; Riemann surfaces of large genus and large λ .Geometry and Analysis on Manifolds (T. Sunada, ed.), Lecture Notes in Math. 1339,Springer-Verlag, Berlin, 1988, 54-63.[B-M] Brooks, R., Makover E., Riemann surfaces with large first eigenvalue. J. Anal.Math. 83 (2001), 243 - 258.[Cha] Chavel, Isaac; Eigenvalues in Riemannian geometry. Pure and Applied Mathemat-ics, 115. Academic Press, 1984.[Che] Cheng, S. Y.; Eigenfunctions and nodal sets, Comment. Math. Helvetici 51 (1976),43-55.[C-C] Colbois, B., Courtois, G., Les valeurs propres inf´erieures ´a 1/4 des surfaces deRiemann de petit rayon d’injectivit´e. Comment. Math. Helv. 64 (1989), no. 3, 349- 362.[C-V] Colbois, B.; Colin de Verdire, Y.; Sur la multiplicit de la premire valeur propre d’unesurface de Riemann courbure constante. (French) [Multiplicity of the first eigenvalue ofa Riemann surface with constant curvature] Comment. Math. Helv. 63 (1988), no. 2,194208.[H] Hejhal, D. ; Regular b-groups, degenerating Riemann surfaces and spectral theory,Memoires of Amer. Math. Soc. 88, No. 437, 1990.[Hu] Huxley, M. N.; Cheeger’s inequality with a boundary term, Commentarii Mathe-matici Helvetici 58 (1983).[I] Iwaniec, H., Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev.Mat. Iberoamericana, Revista Matem´atica Iberoamericana, Madrid, 1995.[Je] Jenni, F.; Uber den ersten Eigenwert des Laplace-Operators auf ausgewhlten Beispie-len kompakter Riemannscher Flchen. (German) [On the first eigenvalue of the Laplaceoperator on selected examples of compact Riemann surfaces] Comment. Math. Helv. 59(1984), no. 2, 193-203.[Ji] Ji, Lizhen; Spectral degeneration of hyperbolic Riemann surfaces. J. DifferentialGeom. 38 (1993), no. 2, 263 - 313.[Ju] Judge, Chris; The nodal set of a finite sum of Maass cusp forms is a graph. Proceed-ings of Symposia in Pure Mathematics, Volume 84, 2012[K-S] Kim, Henry H.; Functoriality for the exterior square of GL and symmetric fourthof GL . J. Amer. Math. Soc. 16 (2003), no. 1, 139 - 183.[M] Mondal, Sugata; Topological bounds on the number of cuspidal eigenvalues of finitearea hyperbolic surfaces (preprint)[O] Otal, Jean-Pierre; Three topological properties of small eigenfunctions on hyperbolicsurfaces. Geometry and Dynamics of Groups and Spaces, Progr. Math. 265, Birkh¨auser,Bassel, 2008.[O-R] Otal, Jean-Pierre; Rosas, Eulalio; Pour toute surface hyperbolique de genre g, λ g − > /
4. Duke Math. J. 150 (2009), no. 1, 101 - 115.[Se] Selberg, A.; On the estimation of Fourier coefficients of modular forms. Proc. Symp.Pure Math. VII, Amer. Math. Soc. (1965), 1-15.[S-U] Strohmaier, A., Uski, V.; An algorithm for the computation of eigenvalues, spectralzeta functions and zeta-determinants on hyperbolic surfaces. Comm. Math. Phys. 317,(2013), no. 3, 827–869.[Wo] Wolpert, S. A.; Spectral limits for hyperbolic surface, I, Invent. Math. 108 (1992),67 - 89.
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn.
E-mail address ::