Toric aspects of the first eigenvalue
aa r X i v : . [ m a t h . DG ] F e b TORIC ASPECTS OF THE FIRST EIGENVALUE
EVELINE LEGENDRE AND ROSA SENA-DIAS
Abstract.
In this paper we study the smallest non-zero eigenvalue λ of theLaplacian on toric K¨ahler manifolds. We find an explicit upper bound for λ interms of moment polytope data. We show that this bound can only be attainedfor CP n endowed with the Fubini-Study metric and therefore CP n endowedwith the Fubini-Study metric is spectrally determined among all toric K¨ahlermetrics. We also study the equivariant counterpart of λ which we denote by λ T . It is the the smallest non-zero eigenvalue of the Laplacian restricted totorus-invariant functions. We prove that λ T is not bounded among toric K¨ahlermetrics thus generalizing a result of Abreu-Freitas on S . In particular, λ T and λ do not coincide in general. Introduction
Toric K¨ahler manifolds are very symmetric K¨ahler manifolds for which thereis a concrete parametrization of the space of K¨ahler metrics. More concretely,they are symplectic manifolds admitting a Hamiltonian action from a maximaltorus and endowed with a compatible, torus invariant Riemannian metric givingrise to an integrable complex structure. The underlying symplectic manifold iscompletely characterized by a combinatorial object which is a convex polytopecalled moment polytope arising as the image of the moment map for the torusaction. Toric K¨ahler metrics are parametrized by convex functions on that momentpolytope satisfying certain properties as we will discuss in section 2. Toric K¨ahlermanifolds have played a crucial role in studying important questions in geometry.In [10], Donaldson was able to fully characterize those toric K¨ahler surfaces whichadmit constant scalar curvature thus settling an important conjecture in K¨ahlergeometry in the toric context for real dimension 4. There has been a lot of interestin studying toric spectral geometry and, in particular, inverse spectral questionsin this toric context as well (see [3], [13]).Given a Riemannian manifold (
M, g ), the Riemannian metric determines aBeltrami-Laplace operator whose smallest non-zero eigenvalue, which we also referto as the first eigenvalue , and is denoted by λ ( g ), carries a surprising amount ofgeometric information. There has been great deal of effort put into finding sharpbounds for λ with geometric meaning (see [6]). Bourguignon–Li–Yau found anupper bound for λ for K¨ahler manifolds endowed with a full holomorphic em-bedding into projective space. Polterovich (see [21]) looked at boundedness of λ in the context of symplectic manifolds. He showed, in particular, that there Date : February 9, 2016.RSD was partially supported by FCT/Portugal through projects PEst-OE/EEI/LAOO9/2013, EXCL/MAT-GEO/0222/2012 and PTDC/MAT/117762/2010. Wewould also like to thank CAST for a travel grant that allowed EL to visit Lisbon. are symplectic manifolds admitting compatible Riemannian metrics whose λ isarbitrarily large. One of the questions we want to address here is: “are theregeometric bounds on λ ( M, g ) where M is a toric manifold and g is a toric K¨ahlermetric on it?”. As it turns out, one can always use Bourguignon–Li–Yau’s resultin the toric context, and we use it to give an explicit bound for λ in terms ofmoment polytope data. More precisely, we prove the following theorem. Theorem 1.1.
Let ( M n , ω ) be a toric symplectic manifold endowed with a toricK¨ahler structure whose Riemannian metric we denote by g . Let P ⊂ R n be itsmoment polytope. There is an integer, k ( P ) ≥ such that for any k ≥ k ( P ) λ ( g ) ≤ nk ( N k + 1) N k (cid:18) Cn R M c ∧ ω n − k R M ω n (cid:19) . where C can be taken to be the supremum of the norms of the primitive normalvectors to facets of P and N k + 1 = ♯ ( P ∩ Z n /k ) . If P is integral (i.e its verticeslies in Z n ), then we have a finer bound given by λ ( g ) ≤ n ( N + 1) N , where N + 1 = ♯ ( P ∩ Z n ) is the number of integer points in P . We will make k ( P ) explicit ahead. The Fubini-Study metric realizes the boundin the above theorem. In fact we show that this is the only toric K¨ahler metricthat does saturate this bound in the integral case. Theorem 1.2.
Let ( M n , ω ) be an integral toric symplectic manifold endowed witha toric K¨ahler structure whose Riemannian metric we denote by g . Let N + 1 bethe number of integer points in the moment polytope of M . If λ ( g ) = 2 n ( N + 1) N , then M is equivariantly symplectomorphic to CP n and this symplectomorphismtakes g into the Fubini-Study metric on CP n . It was previously known (see [6]) that the Fubini-Study metric on CP n is de-termined by the spectrum among all K¨ahler metrics on CP n compatible with thestandard complex structure. It was also proved by Tanno (see [22]) that, if aK¨ahler manifold of real dimension less than 12 has the same spectrum as CP n with the Fubini-Study metric, then it is holomorphically isometric to it. A simpleconsequence of the above theorem is that the spectrum of the Laplacian of a toricK¨ahler metric on an integral toric manifold determines if the manifold is CP n endowed with the Fubini-Study metric. Corollary 1.3.
An integral toric K¨ahler manifolds which has the same spectrumas ( CP n , ω F S , J ) is holomorphically isometric to it. Another interesting question is that of spectrally characterizing either constantscalar-curvature, extremal or K¨ahler-Einstein toric K¨ahler metrics. In [13] theauthors prove that the equivariant spectrum determines if a toric K¨ahler metric
ORIC ASPECTS OF THE FIRST EIGENVALUE 3 has constant scalar curvature. A variation of the argument there would show thatthe equivariant spectrum also determines if a metric is extremal.Going back to the first eigenvalue, there are various bounds that one can writedown for toric K¨ahler manifolds using Bourguignon–Li–Yau’s bound, see § λ –extremal , where λ –extremal meansextremal for the first eigenvalue with respect to local variations in the K¨ahlermetrics space. Hence, in general, we cannot expect a toric K¨ahler–Einstein metricto saturate fine bounds. Another natural candidate to consider is a balancedmetric when it exists, see discussion § § λ . Proposition 1.4. Let ( M, ω, T ) be a compact symplectic toric orbifold withmoment map x : M → t ∗ . Then ( M, g, J, ω, T ) is a K¨ahler–Einstein toric orbifoldwith Einstein constant λ if and only if, up to an additive constant, the momentmap satisfies (1) 2 λ h x, b i = ∆ g h x, b i ∀ b ∈ t , and λ is the smallest non-vanishing eigenvalue for the K¨ahler–Einstein orbifoldtoric metric. Matsushima’s theorem implies that a necessary condition for a toric K¨ahler met-ric to be K¨ahler-Einstein is that its λ be a multiple eigenvalue with multiplicityat least equal to half the dimension of the manifold. What’s more, it follows fromthe above proposition that one can see if a metric is K¨ahler- Einstein by simplychecking if its moment map coordinates are eigenfunctions for 2 λ .On a toric manifold endowed with a torus invariant metric one can considera toric version of λ namely λ T defined to be the smallest non-zero invarianteigenvalue of the Laplacian i.e. the smallest eigenvalue of the Laplacian restrictedto torus invariant functions. We clearly have λ ≤ λ T . In [3], Abreu–Freitasstudied λ T for the simplest toric manifold, namely S with the usual S action byrotations around an axis. They proved it was unbounded (both above and below)among S -invariant metrics. In this paper we generalize their results, by usingan original approach for the upper bound, on all toric manifolds. We are able toprove the following. It is possible that this result was previously known but the authors did not find a referencefor it in the literature and thus state it and prove it.
EVELINE LEGENDRE AND ROSA SENA-DIAS
Theorem 1.5.
Let ( M, ω, T ) be a compact symplectic toric orbifold, let K Tω be theset of all toric K¨ahler metrics on ( M, ω, T ) . inf { λ ( g ) | g ∈ K Tω } = 0 . and sup { λ T ( g ) | g ∈ K Tω } = + ∞ . Combining Theorem 1.1 and 1.5, we see that there are toric K¨ahler manifolds forwhich λ does not coincide with λ T . For toric K¨ahler–Einstein metrics, it followsfrom Matsushima Theorem [19] that λ = λ T as there are invariant eigenfunctionsfor λ . It would be interesting to characterize those toric K¨ahler manifolds forwhich this occurs. Given a weight vector m ∈ Z n , one could also define m -equivariant λ which we denote by λ m as the lower non-vanishing eigenvalue ofthe Laplacian restricted to the set of m -equivariant functions { f ∈ C ∞ ( M, C ) : f ( e i θ p ) = e i θ · m f ( p ) , ∀ p ∈ M, θ ∈ R n } . One could prove a similar result in this setting and again it would be interestingto understand which metrics have λ = λ m and how this depends on w . Notethat λ T = λ . Recently, in [18], Hall-Murphy proved that on any toric manifold λ T restricted to the class of toric K¨ahler metrics whose scalar curvature in non-negative is bounded and this generalizes another result in [3].The paper is organized as follows. In section 2 we quickly review some basic factsabout toric manifolds and their toric K¨ahler metrics. The reader is encouragedto consult the references for more details and proofs. We also give a proof ofProposition 1.4. In section 3 we study λ T and generalize Abreu-Freitas’ result toprove Theorem 1.5. Section 4 deals with λ and there we prove Theorems 1.1 and1.2. Acknowledgements.
The authors would like to thank Emily Dryden and JulienKeller for interesting conversations concerning the topic of this paper and alsoStuart Hall and Tommy Murphy for sharing their preprint.2.
Background
Toric K¨ahler geometry.
This section does not contain all the ingredientsof symplectic toric geometry needed in subsequent sections, we only lay down thenotation and refer to the classical references for this theory (in particular for proofsof what is claimed in this section) like [1, 2, 8, 11, 16, 17, 20].Let ( M n , ω, T n ) be a compact toric symplectic orbifold. It admits a momentmap x : M → P ⊂ t ∗ where t = Lie T is the Lie algebra of T and t ∗ is its dualsuch that for all a ∈ t − d h x, a i = ω ( X a , · )where X a is the vector field on M induced by the 1–parameter subgroup associatedto a . The image of x , that we denote P , is called the moment polytope. It is aconvex simple (i.e. its vertex are the intersection of exactly n –facets) polytopein t ∗ . Let ν = { ν , . . . , ν d } be a set of vectors in t which are normal to thefacets of P and inward pointing. Let Λ be a lattice in t such that T = t / Λ. If
ORIC ASPECTS OF THE FIRST EIGENVALUE 5 ν ⊂ Λ, the triple (
P, ν,
Λ) is rational and if, each subset of vectors in ν normal tofacets meeting at any given vertex, forms a basis of Λ, then we say that ( P, ν,
Λ)is Delzant. The Delzant–Lerman–Tolman correspondence states that compacttoric symplectic orbifolds are in one to one correspondence with rational labelledpolytopes and are smooth if and only if the rational labelled polytopes is Delzant.In this text, we often identify t with R n and Λ with Z n . Definition 2.1.
Let (
P, ν ) be a labelled polytope. The functions L , . . . , L d ∈ Aff( t ∗ , R ) are said to be the defining functions of ( P, ν ) if P = { x ∈ t ∗ | L k ( x ) ≥ } and dL k = ν k .On the pre-image of the interior of the polytope ˚ M = x − ( ˚ P ), the action of T is free. The action–angle coordinates ( x, θ ) = ( x , . . . , x n , θ . . . , θ n ) are localcoordinates on ˚ M used to (locally) identify ˚ M with ˚ P × T where the first projectioncoincides with the moment map and(2) ω = n X i =1 dx i ∧ dθ i . The space of compatible T –invariant K¨ahler metrics on ( M, ω, T ) is parametrizedby the set of symplectic potentials which is denoted by S ( P, ν ) (up to the additionof an affine linear function). The set S ( P, ν ) is defined as the subset of functions u ∈ C ∞ ( ˚ P , R ) ∩ C ( P, R ), where ˚ P denotes the interior of P , such that(i) u − P dk =1 L k log L k ∈ C ∞ ( P, R );(ii) the restriction of u to ˚ P is strictly convex;(iii) for each face F of P , the restriction of u to ˚ F (the relative interior of F )is strictly convex.Given u ∈ S ( P, ν ), the metric defined by(3) g u = n X i,j =1 u ij dx i ⊗ dx j + u ij dθ i ⊗ dθ j where u ij = ∂ u∂x i ∂x j and ( u ij ) = ( u ij ) − , is a t –invariant K¨ahler metric on ˚ P × T ≃ ˚ M compatible with ω . Conditions ( i ) , ( ii ) , ( iii ) ensure that g u is the restrictionof a smooth metric on M . For convenience, we denote H uij = u ij , G uij = u ij , H u = ( H uij ) and G u = ( G uij ). One can prove that any toric K¨ahler structure on( M, ω ) can be written using a symplectic potential in S ( P, ν ) as above.Abreu [1] computed the curvature of a compatible K¨ahler toric metric, g u , interms of its symplectic potential u . The scalar curvature of g u is the pull-back by x of the function(4) scal u = − n X i,j =1 ∂ H uij ∂x i ∂x j . To recover the original convention introduced by Lerman and Tolman in the rational case,take m k ∈ Z such that m k ν k is primitive in Λ so ( P, m , . . . m d , Λ) is a rational labelled polytope.
EVELINE LEGENDRE AND ROSA SENA-DIAS
Moreover, the Ricci curvature is(5) ρ g u = − X i,l,k H uli,ik dx k ∧ dθ l . K¨ahler–Einstein metrics and moment map as eigenfunctions of theLaplacian.
Let (
M, g u , J, ω, T ) be a compact K¨ahler toric manifold with momentmap x and denote by ∆ u , the Laplacian with respect to the Riemannian metric g u .Recall that ( M, g u , J, ω ) is K¨ahler–Einstein if there exists λ such that λω = ρ g u where ρ g u is the Ricci form of the Chern and Levi connection. We say that λ isthe Einstein constant. In the compact toric setting, λ > Proposition 2.2.
Let ( M, ω, T ) be a compact symplectic toric orbifold with mo-ment map x : M → t ∗ . Then ( M, g u , J, ω, T ) is a K¨ahler–Einstein toric orbifoldwith Einstein constant λ if and only if, up to an additive constant, the momentmap satisfies (6) 2 λ h x, b i = ∆ g u h x, b i ∀ b ∈ t . Matsushima Theorem [19] already states that on a given a K¨ahler–Einsteinmanifolds, of Einstein constant λ , the eigenvalues of the Laplacian are boundedbelow by 2 λ . Proposition 2.2 is a finer converse of this: for a toric K¨ahler metricif the moment map coordinates are eigenfunctions for the same eigenvalue thenthe metric is K¨ahler–Einstein (and then, by Matsushima’s theorem, this commoneigenvalue is the smallest one). Proof.
We have(7) ∆ u = − n X i,j =1 G ij ∂ ∂θ i ∂θ j + ∂∂x i (cid:18) H ij ∂∂x j (cid:19) , so that(8) d ∆ u h x, b i = − n X i,j,k =1 H ij,ik b j dx k = − ρ g u ( X b , · ) ∀ b ∈ t using (5). From (8), we see that ∆ g u x is a moment map for 2 ρ g u and, in theK¨ahler–Einstein case λω = ρ g u , this implies that 2 λx − ∆ g u x = α ∈ t ∗ is constant.Thus x − α λ satisfies (1).The converse is also a simple computation. Indeed, assuming (1), we have∆ g u x i = − n X j =1 ∂H ij ∂x j = 2 λx i ORIC ASPECTS OF THE FIRST EIGENVALUE 7 for i = 1 , . . . n . Inserting this in (5), we get ρ g u ( · , · ) = − n X i,l,k =1 H li,ik dx k ∧ dθ l = 12 n X l,k =1 ∂∂x k (2 λx l ) dx k ∧ dθ l = λ n X k =1 dx k ∧ dθ k = λω, (9)as in (2). (cid:3) The first invariant eigenvalue λ T Minimizing λ T . The goal of this subsection is to show the first part ofTheorem 1.5. With the notation introduced in Section 2, Theorem 1.5 wouldfollow from(10) inf u ∈S ( P,ν ) { λ ( g u ) } = 0 . An easy computation shows that for any T –invariant function Z M g u ( ∇ g u f, ∇ g u f ) dv g u = Z T n dθ ∧ · · · ∧ dθ n Z P H u ( df, df ) dx ∧ · · · ∧ dx n . Here df denotes the differential of f seen as a function on P . We fix coordinateson t ∗ and, by translating if necessary, we assume that R P x i d̟ = 0 where we haveset d̟ = dx ∧ · · · ∧ dx n . The Rayleigh characterization of the first eigenvaluetells us that for any i = 1 , . . . , n (11) λ ( g u ) ≤ R P H u ( dx i , dx i ) d̟ R P x i d̟ = R P u ii d̟ R P x i d̟ with equality if and only if x i is an eigenfunction of the Laplacian ∆ g u . Sincethe denominator does not depend on u , to show (10), it is sufficient to show thatwe can find u ∈ S ( P, ν ) with arbitrarily small u ii , as Abreu and Freitas did for S –invariant metrics on S in [3].Take any u o ∈ S ( P, ν ) and for any positive real number c > u c = u o + cx i . First, we will show that u iic decreases when c increases. We have Hess u c =Hess u o + cE i where E i = ( δ li δ ki ) ≤ l,k ≤ n and δ li being the Kronecker symbol. Inparticular,(12) det Hess u c = det Hess u o + c det M ii where M lk denotes the ( l, k )-minor matrix of Hess u o . Note that M ii is positivedefinite on each point of the interior ˚ P since it corresponds to the restriction ofthe metric g u o (as a metric on ˚ P ) to the orthogonal space to ∂∂x i . In particular for EVELINE LEGENDRE AND ROSA SENA-DIAS any c >
0, formula (12) gives det Hess u c >
0. Now, since the ( i, i )-minor matricesof Hess u o and Hess u c are the same we have(13) u iic = det M ii det Hess u o + c det M ii Thus, u iic → c → + ∞ .Now, we will show that u c ∈ S ( P, ν ) for all c > i ) , ( ii ) and ( iii ) of the definition, see § u c − P dk =1 L k log L k = cx i + ( u o − P dk =1 L k log L k ) is smooth since u o ∈ S ( P, ν );(ii) let x ∈ ˚ P , (Hess u c ) x is positive definite if all its eigenvalues are positive.The smallest eigenvalues of (Hess u c ) x ismin = v ∈ T x ˚ P ((Hess u c ) x v, v )( v, v ) = min = v ∈ T x ˚ P (cid:26) ((Hess u o ) x v, v )( v, v ) + c v i ( v, v ) (cid:27) ≥ min = v ∈ T x ˚ P (cid:26) ((Hess u o ) x v, v )( v, v ) (cid:27) > F be a face of P and x ∈ ˚ F (we may assume that F is not a vertex sincein this case there is nothing to check). Again, the smallest eigenvalues of(Hess ( u c ) | ˚ F ) x ismin = v ∈ T x ˚ F ((Hess u c ) x v, v )( v, v ) = min = v ∈ T x ˚ F (cid:26) ((Hess u o ) x v, v )( v, v ) + c v i ( v, v ) (cid:27) ≥ min = v ∈ T x ˚ F (cid:26) ((Hess u o ) x v, v )( v, v ) (cid:27) > . (15)Hence u c ∈ S ( P, ν ) for all c > λ ( g u c ) −→ c → + ∞ . Thisproves (10).3.2. Maximizing λ T . The goal of this subsection is to show the second partof Theorem 1.5. Let (
P, ν ) be the labelled moment polytope of a symplectictoric orbifold (
M, ω, T ). Without loss of generality, we assume, in this section,that 0 ∈ ˚ P . In particular, the defining functions L k ( x ) = h x, ν k i + c k satisfy L k (0) = c k >
0. Let u o ∈ S ( P, ν ) be the Guillemin potential, that is, u o = 12 d X i =1 L i log L i − L i and G o = Hess u o and H o = (Hess u o ) − . Choosing coordinates and an innerproduct ( · , · ), we see G o and H o as matrices. For s >
1, we denote u so , theGuillemin potential of sP which is the dilation of P by an s -factor. The definingaffine-linear functions of sP are L sk = h x, ν k i + sc k . Consider the following familyof functions on P u s = u o − u so s . ORIC ASPECTS OF THE FIRST EIGENVALUE 9
We will show that u s ∈ S ( P, ν ). Since u so is smooth on P when s >
1, to showthat u s ∈ S ( P, ν ) it is sufficient to show that G s = Hess u s is positive definite on˚ P . This is clear since L sk ( x ) > L k ( x ) on P and G s = 12 d X k =1 (cid:18) L k − sL sk (cid:19) ν k ⊗ ν k . In [3], the authors show that for the 2–sphere, λ T ( g u s ) ր + ∞ when s goes to 1.We will use another approach to show that the same holds in higher dimension.The rough idea is that, since u s → P , the eigenvalues of the inverseof its Hessian should, in some way, tend to infinity and thus the Rayleigh quotientof any function should go to infinity. We write • G so = Hess u so , and H so = ( G so ) − • G s = Hess u s and H s = ( G s ) − .We start by proving the following simple lemma. Lemma 3.1.
For any f ∈ C ( P )(16) Z P H s ( df, df ) d̟ ≥ Z P H o ( df, df ) d̟. In particular, the variation λ T ( s ) := λ T ( g u s ) is bounded below by λ ( g u o ) Proof.
First note that the eigenvalues of H o G s are, strictly smaller than 1 on ˚ P .In fact, if λ is an eigenvalue for H o G s and u is the corresponding eigenvector, G s u = λG o u , so that d X k =1 (cid:18) − λL k ( x ) − sL sk ( x ) (cid:19) h ν k , u i = 0 , which is not possible if λ ≥
1. Since G s = G o − s G so , we have H s = Id n + ∞ X k =1 (cid:18) s H o G so (cid:19) k ! H o , on the interior of P . From this expression, we get that for any f ∈ C ( P ) , (17) Z P H s ( df, df ) d̟ ≥ Z P H o ( df, df ) d̟. (cid:3) We are now in a position to prove that for the family of metrics determined by u s , λ T is unbounded. Proposition 3.2. sup { λ T ( s ) | s > } = + ∞ . Proof.
Assume that λ T ( s ) is also bounded above by a constant, say κ >
0, then,we can find a sequence s k → + such that λ T ( s k ) converges to some λ > f s k ∈ C ∞ ( M ) T = C ∞ ( P ) of eigenfunctions∆ g usk f s k = λ T ( s k ) f s k normalized such that k f s k k L = R P ( f s k ) d̟ = 1. Note that the inequality (16)implies that the Sobolev norms of { f s k } in H ( M, g u o ) are bounded above by κ +1.Indeed, combining the hypothesis and (16), we have κ > λ T ( s k ) = Z P H s ( df s k , df s k ) d̟ ≥ k∇ g uo f s k k g uo . Consequently, there exists a subsequence, that we still index by s for simplicity,of eigenfunctions f s ∈ C ∞ ( M ) T converging in the L ( M, g u o ) topology to somefunction f ∈ L ( M ). We have k f k L = 1, R P f ( x ) d̟ = 0.A straightforward calculation yields G s ( x ) = s − s d X k =1 (cid:18) L k ( x ) + sc k L k ( x ) L sk ( x ) (cid:19) ν k ⊗ ν k , and thus, for any x ∈ ˚ P , B x := lim s → + G s ( x ) s − d X k =1 (cid:18) L k ( x ) + c k L k ( x ) (cid:19) ν k ⊗ ν k is positive definite and depends smoothly on x ∈ ˚ P . For x ∈ ˚ P , let A x = lim s → + ( s − H s ( x )be the inverse of B x .Let K be a compact subset of ˚ P . The integral Z K H s ( df s , df s ) d̟ can be written as(18) Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟. Now for any ǫ > (cid:12)(cid:12)(cid:12)(cid:12)Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup K (( s − H s − A x ) Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≤ ǫ Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ORIC ASPECTS OF THE FIRST EIGENVALUE 11 when s is sufficiently close to 1. On K , the symmetric bilinear form A is positivedefinite and its smallest eigenvalue µ is strictly positive. Hence, on K , the normof A x is equivalent to the Euclidean norm i.e. Z K A (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≥ Γ K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ is bounded. The inequality Z K H s ( df s , df s ) d̟ ≤ κ, implies Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≤ κ, but Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≥ (Γ K − ǫ ) Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ and we conclude that Γ K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≤ κ. Using the Poincar´e inequality, there exists C K , a constant depending only on K , such that C K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≥ s − Z K (cid:18) f s − R K f s R K d̟ (cid:19) d̟. However, since f s → f in the L –topology on P and on K , Z K (cid:18) f s − R K f s d̟ R K d̟ (cid:19) d̟ −→ Z K (cid:18) f − R K f d̟ R K d̟ (cid:19) d̟. But 0 ≤ Z K (cid:18) f s − R K f s d̟ R K d̟ (cid:19) d̟ ≤ s − C K κ Γ K −→ , and so it must tend to zero. This implies that Z K (cid:18) f − R K f d̟ R K d̟ (cid:19) d̟ = 0 , and f is a constant on K . Since K is arbitrary, f is constant on P . But R P f = 0so that f must be identically zero which contradicts R P f = 1. (cid:3) This proposition proves the second part of Theorem 1.5. We have thus provedTheorem 1.5. Bounds on λ for toric manifolds The Bourguignon–Li–Yau bound of an integral polytope.
Given acomplex projective manifold (
M, J, L ) where (
M, J ) is complex manifold and L → M is a complex line bundle giving a fixed embedding Φ : M ֒ → CP N ≃ P ( H ( M, L )) ∗ , in [7], Bourguignon–Li–Yau gave a bound on the first eigenvalueof any K¨ahler metric ω , compatible with J . The bound depends only on thedimension of M , the K¨ahler class [ ω ] ∈ H ( M, R ) and the embedding class[Φ ∗ ω F S ] = 2 πc ( L ). The aim of this subsection is to discuss and review theresult, as well as apply it to integral toric manifolds. We start by stating the mainresult of [7]. Theorem 4.1 (Bourguignon–Li–Yau) . Let M n be a compact complex manifoldand let Φ : M → CP N be a holomorphic immersion such that Φ( M ) is not con-tained in any hyperplane in CP N . Then for any K¨ahler metric ω on M , compatiblewith the given complex structure (19) λ ( M, ω ) ≤ n ( N + 1) R M Φ ∗ ω F S ∧ ω n − N R M ω n , where ω F S = i ∂ ¯ ∂ log( | Z | + · · · + | Z N | ) is the Fubini-Study form on CP N . Given a complex projective manifold (
M, J, L ), we say that an immersion Φ : M → CP N is full if its image is not contained in any hyperplane of CP N . Givena full holomorphic immersion, we set B ([Φ] , [ ω ]) = 2 n ( N + 1) R M Φ ∗ ω F S ∧ ω n − N R M ω n . It is clear that B ([Φ] , [ ω ]) only depends on the H , ( M, R )–cohomology classes [ ω ]and [Φ ∗ ω F S ].Arezzo–Ghigi–Loi generalized Theorem 4.1 to provide a bound on the first eigen-value of K¨ahler manifolds admitting a Gieseker stable bundle, see [5]. In [21],Polterovich used the Bourguignon–Li–Yau Theorem to give a bound on λ for allrational symplectic manifold whose symplectic class π [ ω ] is in H ( M, Q ). In thetoric case, this theorem can be applied directly to provide (various) bounds onthe first eigenvalue of compact toric K¨ahler manifolds. Indeed, given a toric com-pact K¨ahler manifold ( M, ω, J, T ) it is known, see for e.g. [16], that H dR ( M ) = H , ∂ ( M ). Hence, one can pick a symplectic form ˜ ω , compatible with J and lyingin an integral and very ample class. Using Kodaira’s embedding Theorem, weknow that there exists a full embedding Φ : M → CP N such that [Φ ∗ ω F S ] = [˜ ω ].Hence, by Theorem 4.1 λ ( ω, J ) ≤ B (Φ , [ ω ]). The point of Theorem 1.1 is to givea bound that depends only (and explicitly) on the polytope. Remark . Using Bourguignon–Li–Yau Theorem, we can get a finer upper boundfor λ ( M, ω ), namelyinf (cid:8) B ([Φ] , [ ω ]) | Φ : M → CP N full holomorphic immersion (cid:9) . ORIC ASPECTS OF THE FIRST EIGENVALUE 13
Many natural questions arise: given Ω = [ ω ], is this infimum reached for someimmersion ? If so, is this immersion minimal or balanced ? Note that in the Rie-mannian case, there is an Embedding Theorem due to Colin-de-Verdi`ere and ElSoufi–Ilias (see [14]) concerning λ –extremal metrics. These Riemannian metricsare essentially defined as critical points of the map g λ ( g ) on the space ofRiemannian metrics with fixed total volume. In that case, the aforementioned au-thors showed that an orthonormal basis of the first eigenspace provides a minimal embedding into a sphere S N such that the standard round metric on S N pulls-backto the extremal one. Definition 4.3.
Given a compact symplectic toric manifold (
M, ω, T ) with mo-ment polytope P we say that P ⊂ t ∗ is integral if its vertices lie in the dual of thelattice Λ ⊂ t of circle subgroups of the torus T .It is well known that integral polytopes correspond to symplectic toric manifoldswhose cohomology class is integral.Any symplectic toric manifold admits compatible complex structures (see [8,16]) and two such compatible complex structures are biholomorphic. A symplectictoric manifold whose cohomology class is integral caries a holomorphic line bundle L whose first Chern class is the class of the symplectic form. It is known that thisholomorphic line bundle defines a full holomorphic embedding Φ u : M ֒ → CP N where N is the number of lattice points in P . The embedding is associated to abasis for H ( L ) namely { e mz , m ∈ Λ ∩ P } where is a reference holomorphicsection of L . The embedding is defined byΦ( z ) = [ e m z : · · · : e m N z ] , where m , · · · , m N are the lattice points in P . One can express such an embeddingin action-angle coordinates.(20) Φ u ( x, θ ) = [Φ u,m ( z ) : · · · : Φ u,m N ( z )] , Φ m ( z ) = e m · z and z = ∂u∂x + i θ is the complex coordinate on M . If we take theGuillemin potential, u o = P dk =1 L k log L k thenΦ u ( x, θ ) = [Π di =1 L k ( x ) m · νk e i m · θ : · · · : Π di =1 L k ( x ) mN · νk e i m N · θ ] . It is known, see [16] that [Φ ∗ u o ω F S ] = [ ω ]. Remark . It is known, see for example [9], that for two distinct symplecticpotentials u, u o ∈ S ( P, ν ) the map γ u,u o : P × T −→ P × T defined by γ u,u o ( x, e i θ ) = (( du o ) − ( d x u ) , e i θ ) extends as an equivariant diffeomor-phism on M sending J u to J u o and γ ∗ u,u o ω = ω + dd c h where h ∈ C ∞ ( M ) T . Wehave γ ∗ u,u o Φ u o = Φ u and, since the space of symplectic potentials is convex themap γ u,u o preserves cohomology classes and [Φ ∗ u ω F S ] = [ ω ]. Remark . Together with Φ u comes an embedding φ : T ֒ → T N +1 induced bythe map linear map φ ∗ : t → R N +1 , taking θ ∈ t to( θ · α , . . . , θ · α N ) ∈ R N +1 so that the maps Φ u are φ –equivariant embeddings.Applying the Bourguignon–Li–Yau theorem to ( M n , ω, g u , J, T ), we get that(21) λ ( ω, g u ) ≤ n ( N + 1) N Z M Φ ∗ u H ∧ ω n − ω n = 2 n ( N + 1) N where N + 1 is the number of lattice points in P . Remark . Observe that taking kP for k ∈ N ∗ and k ≥ n . However, the left hand side decreases quicklyas well since λ ( kω, kg u ) = k λ ( ω, g u ). Hence, in each rays of K¨ahler cone in H , ( M, Z ) there is an optimal class, the primitive class, on which we may applythe bound B (Φ u , [ ω P ]). Remark . The Bourguignon–Li–Yau bound is an integer if and only if N = 2, N = n or N = 2 N . The two first cases imply M is a projective space and the lastone gives λ ( ω, g u ) ≤ n + 1. Note that, in this last case, the first eigenvalue of aK¨ahler–Einstein metric, which is 2 λ by Proposition 1.4 where λ = 2 πc ( M ) / [ ω ],cannot reach this bound whenever [ ω ] is integral.4.2. A bound on λ for toric manifolds. Let ( M n , ω, g, J ) be a compact toricK¨ahler manifold. The cohomology class of ω π is integral if and only if P is integral.In this subsection we will not assume that P is integral. We start by defining aninteger k ( P ) associated to P . Let k be a fixed integer. Consider the lattice Z n /k ∩ P . Let P k be the convex hull of Z n /k ∩ P . It is a convex polytope in R n contained in P . As k tends to infinity the lattice Z n /k ∩ P becomes finer andeventually P k will look combinatorially like P i.e. it will have the same number offacets and of vertices as P . Definition 4.8.
Let P be a Delzant polytope. Set L min i = min { L i ( m ) , m ∈ P ∩ Z n /k } . We define k ( P ) to be smallest integer k ≥ P k , the convex hull of Z n /k ∩ P can be written as P k = { x ∈ P : L i ≥ L min i , i = 1 , · · · d } . For any integer k , we set N k = ♯ ( Z n /k ∩ P ) − k ≥ k ( P ), P k has the same number of facetsas P . Note that if P is integral, k ( P ) = 1. Theorem 4.9.
Let ( M n , ω ) be a toric symplectic manifold endowed with a toricK¨ahler structure whose Riemannian metric we denote by g . Let P be its momentpolytope. For any k ≥ k ( P ) , λ ( g ) ≤ nk ( N k + 1) N k (cid:18) Cn R M c ∧ ω n − k R M ω n (cid:19) . where C can be taken to be the supremum of the norms of the primitive vectorsnormals to facets of P . If P is integral then we have a finer bound given by λ ( g ) ≤ n ( N + 1) N ,
ORIC ASPECTS OF THE FIRST EIGENVALUE 15 where N + 1 is the number of integer points in P . The theorem is essentially a consequence of the Bourguignon–Li–Yau boundon the first eigenvalue of projective K¨ahler manifolds. By Kodaira’s embeddingtheorem, a multiple of a rational symplectic form on a K¨ahler manifold gives riseto a holomorphic embedding into projective space and Polterovich’s theorem from[21] follows from this. In fact, any K¨ahler toric manifold admits an embeddinginto projective space even in the case where its symplectic form is not rational.This can be seen as a consequence of the fact the complex structure of a toricsymplectic manifold does not depend on the full moment polytope but only onthe fan it determines. Two moment polytopes that are close determine the samefan and therefore an irrational moment polytope can be perturbed into a rationalone without changing the complex structure. We will make this precise in the proofof the theorem so as to have effective bounds to use when applying Bourguignon–Li–Yau’s result.
Proof.
Let k ≥ k ( P ). Consider the set P ∩ Z n /k . We set L min i = min { L i ( m ) , m ∈ P ∩ Z n /k } . The polytope P k is given by P k = { x ∈ P : L i ≥ L min i , i = 1 , · · · d } = { x ∈ P : h x, ν i i + c i − L min i ≥ , i = 1 , · · · d } .P k is a rational polytope and kP k is an integral polytope. This uses the fact that P is a Delzant polytope. A Delzant polytope is integral if and only if its verticesare in Z n as this is equivalent to the c i ’s being integral.We will build an explicit embedding of M into CP N k . On an open dense subsetof M we have symplectic coordinates which we denote by ( x, θ ) ∈ P × T n . In thesecoordinates our metric can be described by a symplectic potential u ∈ S ( P, ν ).The potential u is given by u = u o + v where v is smooth in a neighborhood of P and u o = P di =1 L i log L i − L i is the symplectic potential of the Guillemin metricon M . We also have complex coordinates in the same open dense subset of M and the two are related by Legendre transform. More specifically the complexcoordinate z = y + i θ is given by y = ∂u∂x = 12 d X i =1 ν i log L i + ∂v∂x . Set(22) Φ u,k ( x, θ ) = h(cid:16) Π di =1 L c i − L min i i (cid:17) e km · z i m ∈ P ∩ Z n /k , where we think of z = u x + i θ as in (20) and (cid:16) Π di =1 L c i − L min i i (cid:17) e km · z are the homoge-neous coordinates of Φ u,k ( x, θ ). The first thing to do is check that this embeddingis well defined. We have e km · z = e i km · θ e km · u x = e i km · θ e km · v x Π di =1 L km · ν i i , therefore Φ u,k ( x, θ ) = h(cid:16) Π di =1 L L i ( m ) − L min i i (cid:17) e km · v x e i km · θ i m ∈ P ∩ Z n /k , It is clear that the homogeneous coordinates of Φ u,k do not vanish over the interiorof P . In fact, the function Φ u,k extends to ∂P even though θ does not. To seewhy this is so, let x ∈ ∂P . Without loss of generality assume that L ( x ) = · · · = L r ( x ) = 0 and suppose that x is in the interior of the face F defined by L ( x ) = · · · = L r ( x ) = 0. Then, we should check thatΦ u,k ( x, θ ) = Φ u,k x, θ + r X l =1 α l ν l ! , because the Lie algebra of the subgroup of T n that fixes the points in x − ( F ) isspanned by ν , · · · ν r . The only homogeneous coordinates that do not vanish at x are those corresponding to m ’s such that L i ( m ) = L min i for i = 1 , · · · , r . Let m a be such an m . Then e i ( θ + P rl =1 α l ν l ) · m a = e i θ · m a e i P rl =1 α l ( − c l + L min l ) because m a · ν l = − c l + L min l is independent of m a , that is each component ofΦ u,k ( x, θ + P rl =1 α l ν l ) is (cid:16) e i P rl =1 α l ( − c l + L min l ) (cid:17) times the corresponding componentof Φ u,k ( x, θ ) and thus Φ u,k ( x, θ + P rl =1 α l ν l ) = Φ u,k ( x, θ ). One also needs to checkthat, on the boundary of P , not all homogeneous coordinates vanish simultane-ously. Let x ∈ ∂P say L ( x ) = · · · = L r ( x ) = 0 then, since P k has the samefaces as P , we can choose a vertex of P k which we will denote by m, such that L ( m ) = L min1 , · · · , L r ( m ) = L min r . Because kP k is integral, km ∈ Z n . The m -thhomogeneous coordinates of Φ u,k is non-vanishing at x because the L i ’s whichvanish at x appear raised to power 0 in the expression for the m -th homogeneouscoordinate.Φ u,k is holomorphic because on the interior of P it coincides with[ e km · z ] m ∈ P ∩ Z n /k , which is expressed in terms of complex coordinates as a holomorphic function.It is also clear that Φ u,k is injective and that, because { e k m · z } is a linearlyindependent set of functions, the image of Φ u,k is not contained in any hyperplane.At this point we can apply Bourguignon–Li–Yau’s theorem 4.1 to conclude that(23) λ ( M, ω ) ≤ n ( N k + 1) R P Φ ∗ u,k ω F S ∧ ω n − N k R P ω n . We proceed to calculate the cohomology class [Φ ∗ u,k H ]. We have(24) Φ ∗ u,k ω F S = ∂ ¯ ∂ log X m ∈ P ∩ Z n /k | e km · z | As in the integral case, see Remark 4.4, [Φ ∗ u o ,k ω F S ] = [Φ ∗ u,k ω F S ] for any pair ofsymplectic potentials u an u o . So it is enough to consider the Guillemin potential ORIC ASPECTS OF THE FIRST EIGENVALUE 17 u o = P di =1 ( L i log L i − L i ) and this gives y = Re( z ) = P di =1 ν i log L i . Replacingin equation (24) Φ ∗ u o ,k ω F S = ∂ ¯ ∂ log X m ∈ P ∩ Z n /k e km · y = ∂ ¯ ∂ log X m ∈ P ∩ Z n /k Π di =1 L km · ν i i = ∂ ¯ ∂ log X m ∈ P ∩ Z n /k Π di =1 L k ( L i ( m ) − L min i − c i + L min i ) i = ∂ ¯ ∂ log X m ∈ P ∩ Z n /k Π di =1 L k ( L i ( m ) − L min i ) i + k d X i =1 ( − c i + L min i ) ∂ ¯ ∂ log L i As we have seen before X m ∈ P ∩ Z n /k Π di =1 L k ( L i ( m ) − L min i ) i is nowhere vanishing in ¯ P so that the cohomology class of Φ ∗ u o ,k ω F S is that of k d X i =1 ( − c i + L min i ) ∂ ¯ ∂ log L i . Let E i be the cohomology class that is Poincare dual to the divisor in M whoseimage under the moment map is the facet i . Then[ ∂ ¯ ∂ log L i ] = E i and we conclude that [Φ ∗ u o ,k ω F S ] = k P di =1 ( − c i + L min i ) E i . It is well know (see[16]) that [ ω ] = − P di =1 c i E i therefore[Φ ∗ u o ,k ω F S ] = k [ ω ] + d X i =1 L min i E i ! . Replacing in equation (23) we see that λ ( M, ω ) ≤ n ( N k + 1) N k k + k R P P di =1 L min i E i ∧ ω n − R P ω n ! Because Z n /k gives a grid in P of side length 1 /k we have L min i = L i ( m ) = L i ( m ) − L i ( x ) , x ∈ L − i (0) , hence L min i = | ( m − x ) · ν i | ≤ | ν i | n/k so that Z P d X i =1 L min i E i ∧ ω n − ≤ nCk Z P d X i =1 E i ∧ ω n − , where C is a constant. Because c ( M ) = P di =1 E i it follows that λ ( M, ω ) ≤ nk ( N k + 1) N k (cid:18) nC R P c ( M ) ∧ ω n − k R P ω n (cid:19) . It is clear from the above argument that if P is integral one can take k = 1 andthe bound becomes λ ( M, ω ) ≤ n ( N + 1) N . (cid:3)
The equality case in Bourguignon–Li–Yau’s bound.
The goal of thissubsection is to study K¨ahler toric metrics which saturate the bound in Theorem4.9. We give a quick overview of Bourguignon, Li and Yau’s proof in [7] as thisproof will be important to us.
Proof of Theorem 4.1, see [7] . Recall that the first eigenspace of ( CP N , ω F S ) hasa basis given by the functions [ Z ] Ψ ij ( Z ) − δ ij N +1 where for i, j ∈ { , , . . . , N } ,Ψ ij ( Z ) = Z i Z j P Nk =0 | Z k | is one component of the SU ( N + 1) moment map Ψ : CP N ֒ → su ∗ N +1 . The mainstep of the proof in [7], is to show that, given a full embedding Φ : M ֒ → CP N ,there exists a unique B ∈ SL ( N + 1 , C ) such that B ∗ = B > R M ω n Z M (Ψ ij ◦ B ◦ Φ)( p ) ω n = δ ij N + 1 . Said differently, ( B ◦ Φ) ∗ ω F S is ( ω n /n !)–balanced, (see [12]). To simplify thenotation we write ω –balanced instead of ( ω n /n !)–balanced.Denote f Bij = Ψ ij ◦ B ◦ Φ − δ ij N +1 ∈ C ∞ ( M ). The Rayleigh principle implies that(26) λ ( M, ω ) Z M ( f Bij ) ω n n ! ≤ Z M |∇ ω f Bij | ω n n !with equality if and only f Bij is an eigenfunction of ∆ for the eigenvalue λ ( M, ω ).Taking the sum over i, j = 0 , . . . , N the left hand side of (26) gives NN + 1 Z M ω n n !thanks to (25). Noticing that g ( ∇ f, ∇ f ) ω n = n Re( df ∧ d c f ∧ ω n − ), that the form P i,j df Bij ∧ d c f Bij is real and that, on CP N ,(27) X i,j d Ψ ij ∧ d c Ψ ij = 2 ω F S , ORIC ASPECTS OF THE FIRST EIGENVALUE 19 the right hand side of (26) gives X i,j Z M |∇ ω f Bij | ω n n ! = X i,j Z M df Bij ∧ d c f Bij ∧ ω n − ( n − n − Z M ( B ◦ Φ) ∗ ω F S ∧ ω n − . Finally, B ∗ ω F S and ω F S are in the same cohomology class on CP N , this concludesthe proof. (cid:3) Remark . The equality case in (19) implies the equality case in each inequality(26) and then that each function f Bij is an eigenfunction of ∆ for the first eigenvalue.In the toric context and for the embedding Φ u we get the following refinementof Bourguignon–Li–Yau’s result on the existence of balanced metrics. Lemma 4.11.
Let ( M, ω, g u , J u , T ) be a toric K¨ahler manifold with integral poly-tope P ⊂ t ∗ and corresponding embedding Φ u : M ֒ → CP N . There exists a diagonalmatrix B = diag( α , . . . , α N ) ∈ GL ( N + 1 , R ) with α i > and tr B = 1 satisfyingthe condition (25) .Proof. First, observe that when i = j , the function Ψ ij ◦ Φ u integrates to 0 on M since it does on each orbit of T . Hence to prove the lemma, we only have to provethat there exists α = ( α , . . . , α N ) ∈ R N +1 > such that, for i = 0 , . . . , N , ψ u,i ( α ) := α i Z M | Z i | P Nj =0 α j | Z j | ω n n ! = 1 N + 1where Z i = Φ u,m i ( x, θ ) see (20). Let Σ be the simplex defined byΣ := ( X ∈ R N +1 | N X i =0 X i = 1 , X i > ) . Since P Ni =1 ψ u,i ( α ) = vol( M ) < + ∞ and each component ψ u,i ( α ) ≥ α ∈ R N +1 , by the dominated convergence lemma, the map ψ u can be extendedcontinuously to Σ. Hence, we see ψ u = ( ψ u, , . . . , ψ u,N ) as a continuous map ψ u : Σ −→ Σfrom the closed simplex Σ to itself. It is obvious that ψ u maps ∂ Σ to ∂ Σ.To prove the lemma we need to prove that ψ u is surjective which will follow ifwe prove that the restriction ψ u : ∂ Σ → ∂ Σ has non-trivial degree.Like in [7], instead of integrating on M , we integrate on CP N with the measure dµ u defined to be the pushforward of the measure on M defined by the metric.This is possible because Φ u is a full embedding. Hence ψ u,i ( α ) = α i Z CP N | Z i | P Nj =0 α j | Z j | dµ u . Now if we consider the volume form dµ o induced by the Fubini-Study metric on CP N , the corresponding map ψ o : ∂ Σ → ∂ Σ with components ψ o,i ( α ) = α i Z CP N | Z i | P Nj =0 α j | Z j | dµ o and its restriction ψ o : ∂ Σ → ∂ Σ are bijections. Now ψ t = tψ u + (1 − t ) ψ o is afamily of continuous maps from Σ to itself preserving the boundary. Then thedegree does not depend on t and is non trivial for t = 0. (cid:3) Remark . A straightforward corollary of this lemma is that the ω –balancedmetric is toric as soon as ω is toric.Theorem 1.2 is a consequence of two propositions we state below and whichwe prove using the following observation. Assume that the first eigenvalue of( M, ω, g u , J u , T ) reaches Bourguignon, Li and Yau’s bound, i.e λ ( g u ) = n ( N +1) N .By Remark 4.10 and Lemma 4.11, there exists a set of N + 1 real positive numbers { α k } k ∈ P ∩ Z n such that for each m, k ∈ P ∩ Z n , the function Ψ mk − δ mk / ( n + 1) isan eigenfunction of eigenvalue n ( N +1) N whereΨ mk = α k α m Z m Z k P j | α j Z j | is seen as a function of ( x, θ ). Here we write Z m = e ( u x + i θ ) · m , where u x = ∂u∂x . We assume, without loss of generality, that 0 ∈ P ∩ Z n . Wenormalize the α ’s so as to have α = 1 instead of P k ∈ P ∩ Z n α k = 1 as in theprevious lemma. We recall that(28) Z M Ψ mk ω n n ! = δ mk Z M ω n n ! . Observe that for each pair m, k ∈ P ∩ Z n (29) ∆Ψ mk = ∆( α m Z m Ψ k ) = α m Z m ∆Ψ k − α m h dZ m , d Ψ k i since ∆( Z m ) = 0 where h· , ·i denotes the inner product induced by g u on thecotangent bundle of M .When k = 0 = m , identity (29) becomes2 n ( N + 1) N Ψ m = 2 n ( N + 1) N α m Z m (Ψ − /n + 1) − α m h dZ m , d Ψ i = 2 n ( N + 1) N Ψ m − n ( N + 1) α m Z m N ( n + 1) − α m h dZ m , d Ψ i , and we get 2 n ( N + 1) α m Z m N ( n + 1) = − α m h dZ m , d Ψ i . ORIC ASPECTS OF THE FIRST EIGENVALUE 21
Developing the right hand side in action angle coordinates, using (3), we have2 n ( N + 1) α m Z m N ( n + 1) = − α m h dZ m , d Ψ i = − α m n X i,j =1 H ij ∂ x i Z m ∂ x j Ψ = 4 α m Z m P k ∈ W P ns,t =1 u ts m s k t | α k Z k | (cid:0)P k ∈ W | α k Z k | (cid:1) where W = P ∩ Z n . Dividing both sides by 2 α m Z m , we end up with(30) n ( N + 1) N ( n + 1) = 2 P k ∈ W P ns,t =1 u ts m s k t | α k Z k | (cid:0)P k ∈ W | α k Z k | (cid:1) = − ( d Ψ )( m ) Proposition 4.13.
Let ( ω, g u , J u , T ) be a toric K¨ahler structure on CP n . Assumethat λ = 2( n + 1) i.e. assume that the first eigenvalue of the Laplacian reaches theBourguignon–Li–Yau bound. Then, the toric K¨ahler metric g u is the Fubini-Studymetric on CP n .Proof. The moment polytope P of CP n is a simplex and if ( ω, g u , J u , T ) saturatesthe bound then P is primitive as explained in Remark 4.6. So one can suitablynormalize it so that it has integer vertices 0 , e , · · · e n where e i is the vector in R n whose i -th component is 1 and all others are zero. In the notation above, N = n and we identify W = P ∩ Z n with { , , . . . , n } . Equation (30) implies that, forall m = 1 , . . . , n ,(31) 1 = 2 P nm,k =1 u mk | α k Z k | (1 + P nk =1 | α k Z k | ) = − ∂ Ψ ∂x m where Z k = e ( u k + i θ k ) for k = 1 , . . . , n . HenceΨ = K − n X k =1 x k for some constant K . The additive constant is fixed K = 1 by the integrationconstraint (28).In the case m = k = 0, identity (29) gives2( n + 1)(Ψ mm − /n + 1) = α m Z m n + 1)Ψ m − α m h dZ m , d Ψ m i that is − − α m h dZ m , d Ψ m i . Developing the right hand side using (31), we get − − α m h dZ m , d Ψ m i = 2 | α m Z m | (cid:18) u mm P i | α i Z i | − (cid:19) = − ∂ Ψ mm ∂x m . Therefore, for each m > ∂ Ψ mm ∂x m = 1 In the case 0 = m = k = 0, identity (29) gives 2( n + 1)Ψ mk = α m Z m n +1)Ψ k − α m h dZ m , d Ψ k i , that is0 = − α m h dZ m , d Ψ k i . Developing the right hand side using (31), we get0 = − α m h dZ m , d Ψ k i = 2 α m α k Z m Z k (cid:18) u mk P i | α i Z i | − (cid:19) = − α m α k Z m Z k | α m Z m | (cid:18) ∂ Ψ mm ∂x k (cid:19) . Therefore, for each m, k > k = m , we have ∂ Ψ mm ∂x k = 0 . Together with (32) it gives(33) Ψ mm = x m where again the additive constant is fixed by the integration constraint (28) onΨ mm . We conclude from Remark (4.10) that the components of the moment mapare eigenfunction for the same eigenvalue and thus, by Proposition 2.2 ( g u , J u )is K¨ahler-Einstein. By uniqueness of extremal toric metric [15], ( g u , J u ) is theFubini-Study metric. (cid:3) We can also prove that if a toric manifold admits a toric K¨ahler metric for whichthe embedding given by the integral points of the moment polytope saturates theBourguignon–Li–Yau bound, then that manifold must be CP n and it follows fromthe above proposition that the metric must be the Fubini-Study metric. Proposition 4.14.
Let ( M, ω, g u , J u ) be a toric K¨ahler manifold with integralpolytope P . Assume that λ ( g u ) = n ( N +1) N with N = ♯ ( P ∩ Z n ) − . Then P is thestandard simplex and M is (equivariantly symplectomorphic to) CP n .Proof. Again we assume that the origin lies in P ∩ Z n , more precisely, up to anintegral invertible affine transformation, we may assume that P is standard at theorigin i.e. the facets that meet at 0 have normals e , · · · , e n . In particular, thevertices of P adjacent to the origin, say m , · · · m n , are each an integral multipleof an element of a dual basis of e , · · · , e n respectively.Under the hypothesis of the proposition and with respect to the notation above,Equations (29) and (30) hold for points in P ∩ Z n . Suppose there is m ∈ P ∩ Z n a vertex distinct from the origin and from m , · · · m n . Then, there exist a , · · · a n such that m = P nl =1 a l m l .Equation (30) holds for m as well as for m , · · · m n . So we must have n ( N + 1) N ( n + 1) = − ( d Ψ )( m ) = − n X l =1 a l ( d Ψ )( m l ) = ( n X l =1 a l ) n ( N + 1) N ( n + 1)which implies P nl =1 a l = 1. So m lies on a facet of the simplex of vertices0 , m , · · · m n which contradicts convexity unless P is that simplex. (cid:3) To sum up we proved Theorem 1.2.
ORIC ASPECTS OF THE FIRST EIGENVALUE 23
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Some numerical results in complex differential geometry , Pure Appl.Math. Q. , no. 2, Special issue: In honor of Friedrich Hirzebruch, 571–618.[13] E. Dryden, V. Guillemin, R. Sena-Dias
Equivariant spectrum on toric orbifolds ,Adv. Math. (2012), 3-4, 1271–1290.[14]
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E-mail address : [email protected] Rosa Sena-Dias, Centro de An´alise Matem´atica Geometria e Sistemas Dinˆamicos,Departamento de Matem´atica, Instituto Superior T´ecnico, Av. Rovisco Pais,1049-001 Lisboa, Portugal
E-mail address : [email protected] r X i v : . [ m a t h . DG ] F e b TORIC ASPECTS OF THE FIRST EIGENVALUE
EVELINE LEGENDRE AND ROSA SENA-DIAS
Abstract.
In this paper we study the smallest non-zero eigenvalue λ of theLaplacian on toric K¨ahler manifolds. We find an explicit upper bound for λ interms of moment polytope data. We show that this bound can only be attainedfor CP n endowed with the Fubini-Study metric and therefore CP n endowed withthe Fubini-Study metric is spectrally determined among all toric K¨ahler metrics.We also study the equivariant counterpart of λ which we denote by λ T . It isthe smallest non-zero eigenvalue of the Laplacian restricted to torus-invariantfunctions. We prove that λ T is not bounded among toric K¨ahler metrics thusgeneralizing a result of Abreu-Freitas on S . In particular, λ T and λ do notcoincide in general. Introduction
Toric K¨ahler manifolds are very symmetric K¨ahler manifolds for which thereis a concrete parametrization of the space of K¨ahler metrics. More concretely,they are symplectic manifolds admitting an effective Hamiltonian action froma maximal torus and endowed with a compatible, torus invariant Riemannianmetric giving rise to an integrable complex structure. The underlying symplecticmanifold is completely characterized by a combinatorial object which is a convexpolytope called moment polytope arising as the image of the moment map forthe torus action. Toric K¨ahler metrics are parametrized by convex functions onthat moment polytope satisfying certain properties as we will discuss in section 2.Toric K¨ahler manifolds have played a crucial role in studying important questionsin geometry. Mabuchi was one the first to study their K¨ahler Geometry in [24].In [13], Donaldson was able to fully characterize those toric K¨ahler surfaces whichadmit constant scalar curvature thus settling an important conjecture in K¨ahlergeometry in the toric context for real dimension 4. There has been a lot of interestin studying toric spectral geometry and, in particular, inverse spectral questionsin this toric context as well (see [4], [16]).Given a Riemannian manifold (
M, g ), the Riemannian metric determines aBeltrami-Laplace operator whose smallest non-zero eigenvalue, which we also re-fer to as the first eigenvalue , and is denoted by λ ( g ), carries a surprising amountof geometric information. There has been great deal of effort put into findingsharp bounds for λ with geometric meaning (see [7]). In [23], Hersch discoveredan upper bound for λ for metrics on S . Bourguignon–Li–Yau found an upper Date : February 9, 2016.RSD was partially supported by FCT/Portugal through projects PEst-OE/EEI/LAOO9/2013, EXCL/MAT-GEO/0222/2012 and PTDC/MAT/117762/2010 andEL is partially supported by the ANR French grant EMARKS. We would also like to thankCAST for a travel grant that allowed EL to visit Lisbon. bound for λ for K¨ahler manifolds endowed with a full holomorphic embeddinginto projective space [9]. This result has been extended in some ways to K¨ahlermanifolds carrying Gieseker stable bundle [6] by allowing maps to Hermitian sym-metric spaces [8]. Polterovich (see [28]) looked at boundedness of λ in the contextof symplectic manifolds. He showed, in particular, that there are symplectic mani-folds admitting compatible Riemannian metrics whose λ is arbitrarily large. Oneof the questions we want to address here is: “are there geometric bounds on λ ( M, g ) where M is a toric manifold and g is a toric K¨ahler metric on it?”. As itturns out, one can always use Bourguignon–Li–Yau’s result in the toric context,and we use it to give an explicit bound for λ in terms of moment polytope data.More precisely, we prove the following theorem. Theorem 1.1.
Let ( M n , ω ) be a toric symplectic manifold endowed with a toricK¨ahler structure whose Riemannian metric we denote by g . Let P ⊂ R n be itsmoment polytope. There is an integer, k ( P ) ≥ such that for any k ≥ k ( P ) λ ( g ) ≤ nk ( N k + 1) N k , where N k + 1 = ♯ ( P ∩ Z n /k ) . If P is integral (i.e its vertices lie in Z n ), then wehave a finer bound given by λ ( g ) ≤ n ( N + 1) N , where N + 1 = ♯ ( P ∩ Z n ) is the number of integer points in P . We will make k ( P ) explicit ahead. The Fubini-Study metric realizes the boundin the above theorem. In fact we show that this is the only toric K¨ahler metricthat does saturate this bound in the integral case. Theorem 1.2.
Let ( M n , ω ) be an integral toric symplectic manifold endowed witha toric K¨ahler structure whose Riemannian metric we denote by g . Let N + 1 bethe number of integer points in the moment polytope of M . If λ ( g ) = 2 n ( N + 1) N , then M is equivariantly symplectomorphic to CP n and this symplectomorphismtakes g into the Fubini-Study metric on CP n . It was previously known (see [7]) that the Fubini-Study metric on CP n is de-termined by the spectrum among all K¨ahler metrics on CP n compatible with thestandard complex structure. It was also proved by Tanno (see [29]) that, if aK¨ahler manifold of real dimension less than 12 has the same spectrum as CP n with the Fubini-Study metric, then it is holomorphically isometric to it. A simpleconsequence of the above theorem is that the spectrum of the Laplacian of a toricK¨ahler metric on an integral toric manifold determines if the manifold is CP n endowed with the Fubini-Study metric. Corollary 1.3.
An integral toric K¨ahler manifolds which has the same spectrumas ( CP n , ω F S , J ) is holomorphically isometric to it. ORIC ASPECTS OF THE FIRST EIGENVALUE 3
Another interesting question is that of spectrally characterizing either constantscalar-curvature, extremal or K¨ahler-Einstein toric K¨ahler metrics. In [16] theauthors prove that the equivariant spectrum determines if a toric K¨ahler metrichas constant scalar curvature. A variation of the argument there would show thatthe equivariant spectrum also determines if a metric is extremal.Going back to the first eigenvalue, there are various bounds that one can writedown for toric K¨ahler manifolds using Bourguignon–Li–Yau’s bound, see § λ –extremal , where λ –extremal meansextremal for the first eigenvalue with respect to local variations in the K¨ahlermetrics space. Hence, in general, we cannot expect a toric K¨ahler–Einstein metricto saturate fine bounds. Another natural candidate to consider is a balancedmetric when it exists, see discussion § § λ . Proposition 1.4. Let ( M, ω, T ) be a compact symplectic toric orbifold withmoment map x : M → t ∗ . Then ( M, g, J, ω, T ) is a K¨ahler–Einstein toric orbifoldwith Einstein constant λ if and only if, up to an additive constant, the momentmap satisfies (1) 2 λ h x, b i = ∆ g h x, b i ∀ b ∈ t . In this case, λ is the smallest non-vanishing eigenvalue for the K¨ahler–Einsteinorbifold toric metric. Matsushima’s theorem implies that a necessary condition for a toric K¨ahler met-ric to be K¨ahler-Einstein is that its λ be a multiple eigenvalue with multiplicityat least equal to half the dimension of the manifold. What’s more, it follows fromthe above proposition that one can see if a metric is K¨ahler- Einstein by simplychecking if its moment map coordinates are eigenfunctions for 2 λ .On a toric manifold endowed with a torus invariant metric one can considera toric version of λ namely λ T defined to be the smallest non-zero invarianteigenvalue of the Laplacian i.e. the smallest eigenvalue of the Laplacian restrictedto torus invariant functions. We clearly have λ ≤ λ T . In [4], Abreu–Freitasstudied λ T for the simplest toric manifold, namely S with the usual S action byrotations around an axis. They proved it was unbounded (both above and below)among S -invariant metrics. In this paper we generalize their results, by usingan original approach for the upper bound, on all toric manifolds. We are able toprove the following. It is possible that this result was previously known but the authors did not find a referencefor it in the literature and thus state it and prove it.
EVELINE LEGENDRE AND ROSA SENA-DIAS
Theorem 1.5.
Let ( M, ω, T ) be a compact symplectic toric orbifold, let K Tω be theset of all toric K¨ahler metrics on ( M, ω, T ) . Then, inf { λ T ( g ) | g ∈ K Tω } = 0 . and sup { λ T ( g ) | g ∈ K Tω } = + ∞ . Combining Theorem 1.1 and 1.5, we see that there are toric K¨ahler manifolds forwhich λ does not coincide with λ T . For toric K¨ahler–Einstein metrics, it followsfrom Matsushima Theorem [25] that λ = λ T as there are invariant eigenfunctionsfor λ . It would be interesting to characterize those toric K¨ahler manifolds forwhich this occurs. Given a weight vector m ∈ Z n , one could also define m -equivariant λ which we denote by λ m as the lower non-vanishing eigenvalue ofthe Laplacian restricted to the set of m -equivariant functions { f ∈ C ∞ ( M, C ) : f ( e i θ p ) = e i θ · m f ( p ) , ∀ p ∈ M, θ ∈ R n } . One could prove a similar result in this setting and again it would be interestingto understand which metrics have λ = λ m and how this depends on m . Notethat λ T = λ . Recently, in [22], Hall-Murphy proved that on any toric manifolds λ T restricted to the class of toric K¨ahler metrics whose scalar curvature is non-negative is bounded and this generalizes another result in [4].The paper is organized as follows. In section 2 we quickly review some basic factsabout toric manifolds and their toric K¨ahler metrics. The reader is encouragedto consult the references for more details and proofs. We also give a proof ofProposition 1.4. In section 3 we study λ T and generalize Abreu-Freitas’ result toprove Theorem 1.5. Section 4 deals with λ and there we prove Theorems 1.1 and1.2. Acknowledgements.
The authors would like to thank Emily Dryden and JulienKeller for interesting conversations concerning the topic of this paper and alsoStuart Hall and Tommy Murphy for sharing their preprint.2.
Background
Toric K¨ahler geometry.
This section does not contain all the ingredientsof symplectic toric geometry needed in subsequent sections, we only lay down thenotation and refer to the classical references for this theory (in particular for proofsof what is claimed in this section) like [1, 3, 11, 14, 20, 21, 27].Let ( M n , ω, T n ) be a compact toric symplectic orbifold. It admits a momentmap x : M → P ⊂ t ∗ where t = Lie T is the Lie algebra of T and t ∗ is its dualsuch that for all a ∈ t − d h x, a i = ω ( X a , · )where X a is the vector field on M induced by the 1–parameter subgroup associatedto a . The image of x , that we denote P , is called the moment polytope. It is aconvex simple (i.e. its vertex are the intersection of exactly n –facets) polytope in t ∗ . ORIC ASPECTS OF THE FIRST EIGENVALUE 5
Definition 2.1.
Consider P ⊂ t ∗ a simple polytope, ν = { ν , . . . , ν d } a set ofvectors in t which are normal to the facets of P and inward pointing. Let Λ be thelattice in t such that T = t / Λ. If ν ⊂ Λ, the triple (
P, ν,
Λ) is called a labelledrational polytope. If each subset of vectors in ν , normals to facets meeting at avertex, forms a basis of Λ, then we say that ( P, ν,
Λ) is
Delzant .The Delzant–Lerman–Tolman correspondence states that compact toric sym-plectic orbifolds are in one to one correspondence with rational labelled polytopesand are smooth if and only if the rational labelled polytopes is Delzant.In this text, we often identify t with R n and Λ with Z n . Definition 2.2.
Let (
P, ν ) be a labelled polytope. The functions L , . . . , L d ∈ Aff( t ∗ , R ) are said to be the defining functions of ( P, ν ) if P = { x ∈ t ∗ | L k ( x ) ≥ } and dL k dx = ν k .Let ˚ P denote the interior of P . On the pre-image of the interior of the polytope˚ M = x − ( ˚ P ), the action of T is free. The action–angle coordinates ( x, θ ) =( x , . . . , x n , θ . . . , θ n ) are local coordinates on ˚ M used to (locally) identify ˚ M with ˚ P × T where the first projection coincides with the moment map and(2) ω = n X i =1 dx i ∧ dθ i . As it is shown in [2], the space of compatible T –invariant K¨ahler metrics on( M, ω, T ) is parametrized by the set of symplectic potentials which is denoted by S ( P, ν ) (up to the addition of an affine linear function). The set S ( P, ν ) is definedas the subset of functions u ∈ C ∞ ( ˚ P , R ) ∩ C ( P, R ), such that(i) u − P dk =1 L k log L k ∈ C ∞ ( P, R );(ii) the restriction of u to ˚ P is strictly convex;(iii) for each face F of P , the restriction of u to ˚ F (the relative interior of F )is strictly convex. Definition 2.3.
The Guillemin potential u o ∈ S ( P, ν ) is(3) u o = 12 d X i =1 ( L k log L k − L k ) . It corresponds to the K¨ahler toric metric on (
M, ω ) obtained via the K¨ahler re-duction of C d , see [20].Given u ∈ S ( P, ν ), the metric defined by(4) g u = n X i,j =1 u ij dx i ⊗ dx j + u ij dθ i ⊗ dθ j To recover the original convention introduced by Lerman and Tolman in the rational case,take m k ∈ Z such that m k ν k is primitive in Λ so ( P, m , . . . m d , Λ) is a rational labelled polytope.
EVELINE LEGENDRE AND ROSA SENA-DIAS where u ij = ∂ u∂x i ∂x j and ( u ij ) = ( u ij ) − , is a t –invariant K¨ahler metric on ˚ P × T ≃ ˚ M compatible with ω . Conditions ( i ) , ( ii ) , ( iii ) ensure that g u is the restrictionof a smooth metric on M . For convenience, we denote H uij = u ij , G uij = u ij , H u = ( H uij ) and G u = ( G uij ). One can prove that any toric K¨ahler structure on( M, ω ) can be written using a symplectic potential in S ( P, ν ) as above (see [2]).
Remark . Given u ∈ S ( P, ν ) the map ∂u∂x : ˚ P −→ t is a diffeomorphism (because u is strictly convex) and the coordinates z = y + i θ , where y = ∂u∂x , are local complexcoordinates on ˚ M . See [19, § A1.3]) for instance.Note that in [20], u o is defined to be u o = P di =1 ( L k log L k ). This will yield thesame metric as the metric we define via formula 4 and complex coordinates whichare related to the ones in [20] by an overall translation.Abreu [1] computed the curvature of a compatible K¨ahler toric metric, g u , interms of its symplectic potential u . The scalar curvature of g u is the pull-back by x of the function(5) scal u = − n X i,j =1 ∂ H uij ∂x i ∂x j . Moreover, the Ricci curvature is(6) ρ g u = − X i,l,k H uli,ik dx k ∧ dθ l . (See for instance [26] where the above formula is proved in the more general contextof almost K¨ahler metrics.)2.2. K¨ahler–Einstein metrics and moment map coordinates as eigen-functions of the Laplacian.
Let (
M, g u , J, ω, T ) be a compact K¨ahler toricmanifold with moment map x and denote by ∆ u , the Laplacian with respect tothe Riemannian metric g u . Recall that ( M, g u , J, ω ) is K¨ahler–Einstein if thereexists λ such that λω = ρ g u where ρ g u is the Ricci form of the Chern and Leviconnection. We say that λ is the Einstein constant. In the compact toric setting, λ > Proof.
Expressing the Laplacian (i.e ∆ g = − Div g grad g ) in the action angle coor-dinates (4), we get(7) ∆ u = − n X i,j =1 (cid:20) G ij ∂ ∂θ i ∂θ j + ∂∂x i (cid:18) H ij ∂∂x j (cid:19)(cid:21) , so that(8) d ∆ u h x, b i = − n X i,j,k =1 H ij,ik b j dx k = − ρ g u ( X b , · ) ∀ b ∈ t using (6). From (8), we see that ∆ g u x is a moment map for 2 ρ g u and, in theK¨ahler–Einstein case λω = ρ g u , this implies that 2 λx − ∆ g u x = α ∈ t ∗ is constant.Thus x − α λ satisfies (1). ORIC ASPECTS OF THE FIRST EIGENVALUE 7
The converse is also a simple computation. Indeed, assuming (1), we have∆ g u x i = − n X j =1 ∂H ij ∂x j = 2 λx i for i = 1 , . . . n . Inserting this in (6), we get ρ g u ( · , · ) = − n X i,l,k =1 H li,ik dx k ∧ dθ l = 12 n X l,k =1 ∂∂x k (2 λx l ) dx k ∧ dθ l = λ n X k =1 dx k ∧ dθ k = λω, (9)as in (2). (cid:3) The first invariant eigenvalue λ T Minimizing λ T . The goal of this subsection is to show the first part of The-orem 1.5. With the notation introduced in Section 2, the first part of Theorem 1.5would follow from(10) inf u ∈S ( P,ν ) { λ ( g u ) } = 0 . An easy computation shows that for any T –invariant function Z M g u ( ∇ g u f, ∇ g u f ) dv g u = Z T n dθ ∧ · · · ∧ dθ n Z P H u ( df, df ) dx ∧ · · · ∧ dx n . Here df denotes the differential of f seen as a function on P . We fix coordinateson t ∗ and, by translating if necessary, we assume that R P x i d̟ = 0 where we haveset d̟ = dx ∧ · · · ∧ dx n . The Rayleigh characterization of the first eigenvaluetells us that for any i = 1 , . . . , n (11) λ ( g u ) ≤ R P H u ( dx i , dx i ) d̟ R P x i d̟ = R P u ii d̟ R P x i d̟ with equality if and only if x i is an eigenfunction of the Laplacian ∆ g u . Sincethe denominator does not depend on u , to show (10), it is sufficient to show thatwe can find u ∈ S ( P, ν ) with arbitrarily small u ii , as Abreu and Freitas did for S –invariant metrics on S in [4].Take any u o ∈ S ( P, ν ) and for any positive real number c > u c = u o + c x i . First, we will show that u iic decreases when c increases. We have Hess u c =Hess u o + cE i where E i = ( δ li δ ki ) ≤ l,k ≤ n and δ li being the Kronecker symbol. Inparticular,(12) det Hess u c = det Hess u o + c det M ii where M lk denotes the ( l, k )-minor matrix of Hess u o . Note that M ii is positivedefinite at each point in ˚ P since it corresponds to the restriction of the metric g u o EVELINE LEGENDRE AND ROSA SENA-DIAS (as a metric on ˚ P ) to the orthogonal space to ∂∂x i with respect to the Euclideanmetric. In particular for any c >
0, formula (12) gives det Hess u c >
0. Now, sincethe ( i, i )-minor matrices of Hess u o and Hess u c are the same we have(13) u iic = det M ii det Hess u o + c det M ii Thus, u iic → c → + ∞ .Now, we will show that u c ∈ S ( P, ν ) for all c > i ) , ( ii ) and ( iii ) of the definition, see § u c − P dk =1 L k log L k = cx i + ( u o − P dk =1 L k log L k ) is smooth since u o ∈ S ( P, ν );(ii) let x ∈ ˚ P , (Hess u c ) x is positive definite because it is the sum of a positivedefinite matrix namely Hess u o with a semi-positive definite matrix.(iii) let F be a face of P and x ∈ ˚ F . The restriction of (Hess u c ) x to the tangentspace to F is again the sum of a positive definite form namely Hess u o | F with a semi-positive definite form.Hence u c ∈ S ( P, ν ) for all c > λ ( g u c ) −→ c → + ∞ . Thisproves (10).3.2. Maximizing λ T . The goal of this subsection is to show the second partof Theorem 1.5. Let (
P, ν ) be the labelled moment polytope of a symplectictoric orbifold (
M, ω, T ). Without loss of generality, we assume, in this section,that 0 ∈ ˚ P . In particular, the defining functions L k ( x ) = h x, ν k i + c k satisfy L k (0) = c k >
0. Let u o ∈ S ( P, ν ) be the Guillemin potential, that is, u o = 12 d X i =1 L i log L i − L i and G o = Hess u o and H o = (Hess u o ) − . Choosing coordinates and an innerproduct ( · , · ), we see G o and H o as matrices. For s >
1, we denote u so , theGuillemin potential of sP which is the dilation of P by an s -factor. The definingaffine-linear functions of sP are L sk = h x, ν k i + sc k . Consider the following familyof functions on P u s = u o − u so s . We will show that for s > u s ∈ S ( P, ν ). Since u so is smooth on P when s > u s ∈ S ( P, ν ) we need to show that G s = Hess u s is positive definiteon ˚ P . This is clear since L sk ( x ) > L k ( x ) on P and G s = 12 d X k =1 (cid:18) L k − sL sk (cid:19) ν k ⊗ ν k . Given a face of the polytope F , a similar argument using only the L i ’s which donot vanish identically over F , would show that the restriction of Hess ( u s ) to F ispositive definite on F . ORIC ASPECTS OF THE FIRST EIGENVALUE 9
In [4], the authors show that for the 2–sphere, λ T ( g u s ) ր + ∞ when s goes to 1.We will use another approach to show that the same holds in higher dimension.The rough idea is that, since u s → P , the eigenvalues of the inverseof its Hessian should, in some way, tend to infinity and thus the Rayleigh quotientof any function should go to infinity. We write • G so = Hess u so , and H so = ( G so ) − • G s = Hess u s and H s = ( G s ) − .We start by proving the following simple lemma. Lemma 3.1.
For any f ∈ C ( P )(14) Z P H s ( df, df ) d̟ ≥ Z P H o ( df, df ) d̟. In particular, the variation λ T ( s ) := λ T ( g u s ) is bounded below by λ ( g u o ) Proof.
First note that since H o , H so , G so , G s are symmetric matrices they have realeigenvalues. Moreover, the eigenvalues of H o G so are, strictly smaller than 1 on ˚ P .Indeed, if λ is an eigenvalue for H o G so and u is the corresponding eigenvector, then G so u = λG o u , so that d X k =1 (cid:18) L sk ( x ) − λL k ( x ) (cid:19) h ν k , u i = 0 , which is not possible if λ ≥ L sk ( x ) > L k ( x ) on P . Because H o G so issymmetric as each H o and G s are, this implies that || H o G so || <
1. Since G s = G o − s G so and s >
1, we have H s = Id n + ∞ X k =1 (cid:18) s H o G so (cid:19) k ! H o , on the interior of P . From this expression, we get that the inequality (14) holdsfor any f ∈ C ( P ) and s > (cid:3) We are now in a position to prove that for the family of metrics determined by u s , λ T is unbounded. Proposition 3.2. sup { λ T ( s ) | s > } = + ∞ .Proof. Assume that λ T ( s ) is also bounded above by a constant, say κ >
0, then,we can find a sequence s k → + such that λ T ( s k ) converges to some λ > f s k ∈ C ∞ ( M ) T = C ∞ ( P ) of eigenfunctions∆ g usk f s k = λ T ( s k ) f s k normalized such that k f s k k L = R P ( f s k ) d̟ = 1. Note that the inequality (14)implies that the Sobolev norms of { f s k } in H ( M, g u o ) are bounded above by κ +1.Indeed, combining the hypothesis and (14), we have κ > λ T ( s k ) = Z P H s ( df s k , df s k ) d̟ ≥ k∇ g uo f s k k g uo . Consequently, there exists a subsequence, that we still index by s for simplicity,of eigenfunctions f s ∈ C ∞ ( M ) T converging in the L ( M, g u o ) topology to somefunction f ∈ L ( M ). We have k f k L = 1, R P f ( x ) d̟ = 0.A straightforward calculation yields G s ( x ) = s − s d X k =1 (cid:18) L k ( x ) + sc k L k ( x ) L sk ( x ) (cid:19) ν k ⊗ ν k , and thus, for any x ∈ ˚ P , B x := lim s → + G s ( x ) s − d X k =1 (cid:18) L k ( x ) + c k L k ( x ) (cid:19) ν k ⊗ ν k is positive definite and depends smoothly on x ∈ ˚ P . For x ∈ ˚ P , let A x = lim s → + ( s − H s ( x )be the inverse of B x . Notice that ( s − H s ( x ) converges to its limit uniformly oncompact subsets in P .Let K be a compact subset of ˚ P . The integral Z K H s ( df s , df s ) d̟ can be written as(15) Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟. Now for any ǫ > (cid:12)(cid:12)(cid:12)(cid:12)Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup K || ( s − H s − A x || Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≤ ǫ Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ when s is sufficiently close to 1. On K , the symmetric bilinear form A is positivedefinite and its smallest eigenvalue is strictly positive. Hence, on K , the norm of A x is equivalent to the Euclidean norm i.e. Z K A (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≥ Γ K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ for some constant Γ K . The inequality Z K H s ( df s , df s ) d̟ ≤ κ, ORIC ASPECTS OF THE FIRST EIGENVALUE 11 implies Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≤ κ, but Z K A x (cid:18) df s √ s − , df s √ s − (cid:19) d̟ + Z K (( s − H s − A x ) (cid:18) df s √ s − , df s √ s − (cid:19) d̟ ≥ (Γ K − ǫ ) Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ and we conclude that Γ K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≤ κ. Using the Poincar´e inequality, there exists C K , a constant depending only on K , such that C K Z K (cid:12)(cid:12)(cid:12)(cid:12) df s √ s − (cid:12)(cid:12)(cid:12)(cid:12) d̟ ≥ s − Z K (cid:18) f s − R K f s R K d̟ (cid:19) d̟. However, since f s → f in the L –topology on P and on K , Z K (cid:18) f s − R K f s d̟ R K d̟ (cid:19) d̟ −→ Z K (cid:18) f − R K f d̟ R K d̟ (cid:19) d̟. But 0 ≤ Z K (cid:18) f s − R K f s d̟ R K d̟ (cid:19) d̟ ≤ s − C K κ Γ K −→ , when s → + . This implies that Z K (cid:18) f − R K f d̟ R K d̟ (cid:19) d̟ = 0 , and f is a constant on K . Since K is arbitrary, f is constant on P . But R P f = 0so that f must be identically zero which contradicts R P f = 1. (cid:3) This proposition proves the second part of Theorem 1.5. We have thus provedTheorem 1.5. 4.
Bounds on λ for toric manifolds The Bourguignon–Li–Yau bound of an integral polytope.
Considera complex projective manifold (
M, J, L ) where (
M, J ) is complex manifold and L → M is a very ample line bundle giving a fixed embedding Φ : M ֒ → CP N ≃ P ( H ( M, L )) ∗ . In [9], Bourguignon–Li–Yau gave a bound on the first eigen-value of any K¨ahler metric ω , compatible with J . The bound depends only onthe dimension of M , the K¨ahler class [ ω ] ∈ H ( M, R ) and the embedding class[Φ ∗ ω F S ] = 2 πc ( L ). The aim of this subsection is to discuss and review the result,as well as apply it to integral toric manifolds. We start by stating the main resultof [9]. Theorem 4.1 (Bourguignon–Li–Yau) . Let M n be a compact complex manifoldand let Φ : M → CP N be a holomorphic immersion such that Φ( M ) is not con-tained in any hyperplane in CP N . Then for any K¨ahler metric ω on M , compatiblewith the given complex structure (16) λ ( M, ω ) ≤ n ( N + 1) R M Φ ∗ ω F S ∧ ω n − N R M ω n , where ω F S = i ∂ ¯ ∂ log( | Z | + · · · + | Z N | ) is the Fubini-Study form on CP N . We say that an immersion Φ : M → CP N is full if its image is not contained inany hyperplane of CP N . Given a full holomorphic immersion, we set B ([Φ] , [ ω ]) = 2 n ( N + 1) R M Φ ∗ ω F S ∧ ω n − N R M ω n . It is clear that B ([Φ] , [ ω ]) only depends on the H , ( M, R )–cohomology classes [ ω ]and [Φ ∗ ω F S ].Arezzo–Ghigi–Loi generalized Theorem 4.1 to provide a bound on the first eigen-value of K¨ahler manifolds admitting a Gieseker stable bundle, see [6]. In [28],Polterovich used the Bourguignon–Li–Yau Theorem to give a bound on λ forall symplectic manifolds whose symplectic class π [ ω ] lies in H ( M, Q ). In thetoric case, this theorem can be applied directly to provide (various) bounds onthe first eigenvalue of compact toric K¨ahler manifolds. Indeed, given a toriccompact K¨ahler manifold ( M, ω, J, T ) it is known, see for e.g. [10, 20], that H dR ( M ) = H , ∂ ( M ). Hence, one can pick a symplectic form ˜ ω , compatible with J and lying in an integral and very ample class. Using Kodaira’s embeddingTheorem, we know that there exists a full embedding Φ : M → CP N such that[Φ ∗ ω F S ] = [˜ ω ]. Hence, by Theorem 4.1 λ ( ω, J ) ≤ B (Φ , [ ω ]). The input of Theo-rem 1.1 is to give a bound that depends only (and explicitly) on the polytope. Remark . Using Bourguignon–Li–Yau Theorem, we can get a finer upper boundfor λ ( M, ω ), namelyinf (cid:8) B ([Φ] , [ ω ]) | Φ : M → CP N full holomorphic immersion (cid:9) . Many natural questions arise: given Ω = [ ω ], is this infimum reached for someimmersion ? If so, is this immersion minimal or balanced ? Note that in the Rie-mannian case, there is an Embedding Theorem due to Colin-de-Verdi`ere and ElSoufi–Ilias (see [17]) concerning λ –extremal metrics. These Riemannian metricsare essentially defined as critical points of the map g λ ( g ) on the space ofRiemannian metrics with fixed total volume. In that case, the aforementioned au-thors showed that an orthonormal basis of the first eigenspace provides a minimal embedding into a sphere S N such that the standard round metric on S N pulls-backto the extremal one. Definition 4.3.
Given a compact symplectic toric manifold (
M, ω, T ) with mo-ment polytope P we say that P ⊂ t ∗ is integral if its vertices lie in the dual of thelattice Λ ⊂ t of circle subgroups of the torus T . ORIC ASPECTS OF THE FIRST EIGENVALUE 13
It is well known that integral polytopes correspond to symplectic toric manifoldswhose cohomology class is integral. Moreover, any such toric manifold admits acompatible toric K¨ahler structure [11, 20] and an explicit equivariant holomorphicfull embedding into some projective space, see for e.g. [10, Theorem 6.1.5]. Wewill apply Bourguignon–Li–Yau Theorem using this embedding. For the sake ofcompleteness we recall some facts about this embedding.Any symplectic toric manifold admits compatible complex structures (see [11,20]). An example of such a complex structure is given by the Guillemin metric g = g u o corresponding to the Guillemin potential u o (see Definition 2.3). Moreover,any two such compatible complex structures are biholomorphic (see Remark 4.4below).We start with a smooth K¨ahler toric manifold ( M, g, ω, J ) whose cohomologyclass [ ω ] is integral and such that g = g u for some symplectic potential u ∈ S ( P, ν ).On the underlying toric variety (
M, J ) the class [ ω ] corresponds to an ample divisorwhich is then very ample [10, Theorem 6.1.5]. More precisely, ( M, J ) carriesa holomorphic line bundle L whose first Chern class is [ ω ] and which defines afull holomorphic embedding Φ u : M ֒ → CP N where N + 1 is the number oflattice points in P . The embedding is associated to a basis of H ( L ) namely { e mz , m ∈ Λ ∗ ∩ P } where is a reference holomorphic section of L and may bedefined by(17) Φ u ( z ) = [ e m · z : · · · : e m N · z ] , where m , · · · , m N are the lattice points in P and z = y + i θ are local holomorphiccoordinates on ˚ M , see Remark 2.4.One can express such an embedding in action-angle coordinates via z = y + i θ = ∂u∂x + i θ and for convenience we denote each coordinate(18) Φ u ( x, θ ) = [ e m · ∂u∂x e m · i θ : · · · : e m N · ∂u∂x e m N · i θ ] , because e m · z = e m · ∂u∂x e m · i θ . Remark . It is known, see for example [12], that for two distinct symplecticpotentials u, u o ∈ S ( P, ν ) the map γ u,u o : P × T −→ P × T defined by γ u,u o ( x, e i θ ) = ( (cid:0) ∂u o ∂x (cid:1) − ∂u∂x , e i θ ) extends as an equivariant diffeomor-phism on M sending J u to J u o and γ ∗ u,u o ω = ω + dd c h where h ∈ C ∞ ( M ) T . Remark . It has been proved by Guillemin in [20] that [Φ ∗ u o ω F S ] = [ ω ]. More-over, considering the diffeomorphism γ u,u o ∈ Diffeo( M ) of Remark 4.4, we have γ ∗ u,u o Φ u o = Φ u Since the space of symplectic potentials is convex (i.e u t = tu + (1 − t ) u o ∈ S ( P, ν )for t ∈ [0 , γ u,u o lies in the connected component of the identity inDiffeo( M ). In particular, it preserves cohomology classes and [Φ ∗ u ω F S ] = [ ω ]. Remark . Together with Φ u comes an embedding φ : T ֒ → T N +1 induced bythe linear map φ ∗ : t → R N +1 , taking θ ∈ t to( θ · m , . . . , θ · m N ) ∈ R N +1 which is clearly injective. The maps Φ u are φ –equivariant embeddings.It follows from Remark 4.5, that the class of [Φ ∗ u ω F S ] does not depend on thechosen symplectic potential u ∈ S ( P, ν ). It also follows that for any u ∈ S ( P, ν ),Φ u is a full holomorphic embedding iff Φ u o is.Let us consider the Guillemin potential u o . Proposition 4.7.
Given a symplectic toric manifold, the map Φ u above is a welldefined full holomorphic embedding. This is a well known fact. We write the details down here for the reader’sconvenience. See [20] and [10] for more on this.
Proof.
Since u o = P dk =1 ( L k log L k − L k ), then substituting y = ∂u o ∂x = 12 d X k =1 (log L k ( x )) ν k , we get thatΦ u ( x, θ ) = [Π di =1 L k ( x ) m · νk e i m · θ : · · · : Π di =1 L k ( x ) mN · νk e i m N · θ ] . The homogeneous coordinates in this last expression are not smooth on M because y = ∂u∂x blows up when x approaches the boundary and θ is only well-definedmodulo a lattice. To see it is globally defined on M , we multiply each homogeneouscoordinate of Φ u o by the function Π di =1 L i ( x ) c i / where L i ( x ) = x · ν i + c i and weget(19) Φ u o ( x, θ ) = h(cid:16) Π di =1 ( L i ( x )) Li ( m )2 (cid:17) e i m · θ i m ∈ P ∩ Z n , where we identify t ≃ R n and Λ ∗ ≃ Z n . • The homogeneous coordinates of Φ u o do not vanish simultaneously. It is clearthat the homogeneous coordinates of Φ u o do not vanish over the interior of P . Let x ∈ ∂P . Without loss of generality assume that L ( x ) = · · · = L r ( x ) = 0 andsuppose that x is in the interior of the face F defined by L ( x ) = · · · = L r ( x ) = 0.Because P is integral there is also a point m ∈ F ∩ Z n so that L ( m ) = · · · = L r ( m ) = 0 and the m -th coordinate does not vanish. • Φ u o is globally defined on M . The application θ : ˚ M → R n / Z n is well definedon the interior of P but not on the boundary. Let x ∈ ∂P say L ( x ) = · · · = L r ( x ) = 0. Then, we should check thatΦ u o ( x, θ ) = Φ u o x, θ + r X l =1 α l ν l ! , because the Lie algebra of the subgroup of T n that fixes the points in x − ( F ) isspanned by ν , · · · ν r . The only homogeneous coordinates that do not vanish at x are those corresponding to m ’s such that L i ( m ) = 0 for i = 1 , · · · , r . Let m a besuch a point. Then P rl =1 α l ν l = P rl =1 α l ( − c l ) because m a · ν l = − c l is independent ORIC ASPECTS OF THE FIRST EIGENVALUE 15 of m a , that is each component of Φ u o ( x, θ + P rl =1 α l ν l ) is (cid:16) e i P rl =1 α l ( − c l ) (cid:17) times thecorresponding component of Φ u o ( x, θ ) and thus Φ u o ( x, θ + P rl =1 α l ν l ) = Φ u o ( x, θ ). • Φ u o is holomorphic because on the interior of P it coincides with[ e m · z ] m ∈ P ∩ Z n , which is expressed in terms of complex coordinates as a holomorphic function. • Φ u o is an injective immersion. It is clear over the interior of P thanks toRemark 4.6. Over the boundary, we may assume that P is standard aroundone of its vertices so that L = x − a , · · · , L n = x n − a n . In this case byreordering if necessary we may assume that m − m = (1 , , · · · , , · · · , m n − m =(0 , · · · , , u o and its derivative it is enough to proveinjectivity of z → ( e ( m − m ) z , · · · , e ( m n − m ) z ) . But the above is simply z → ( e z , · · · , e z n )which is injective modulo 2 π i Z n as expected and has injective derivative. (cid:3) Applying the Bourguignon–Li–Yau theorem to ( M n , ω, g u , J, T ), we get that(20) λ ( ω, g u ) ≤ n ( N + 1) N Z M Φ ∗ u ω F S ∧ ω n − ω n = 2 n ( N + 1) N where N + 1 is the number of lattice points in P . Remark . Observe that taking kP for k ∈ N ∗ and k ≥ n . However, the left hand side decreases quicklyas well since λ ( kω, kg u ) = k λ ( ω, g u ). Hence, in each rays of K¨ahler cone in H , ( M, Z ) there is an optimal class, the primitive class, on which we may applythe bound B (Φ u , [ ω P ]). Remark . The Bourguignon–Li–Yau bound is an integer if and only if N = 2, N = n or N = 2 n . The two first cases imply M is a projective space and the lastone gives λ ( ω, g u ) ≤ n + 1. Note that, in this last case, the first eigenvalue of aK¨ahler–Einstein metric, which is 2 λ by Proposition 1.4 where λ = 2 πc ( M ) / [ ω ],cannot reach this bound whenever [ ω ] is integral.4.2. A bound on λ for toric manifolds. Let ( M n , ω, g, J ) be a compact toricK¨ahler manifold. The cohomology class of ω π is integral if and only if P is integral.In this subsection we will not assume that P is integral. We start by defining aninteger k ( P ) associated to P . Let k be a fixed integer. Consider the lattice Z n /k ∩ P . Definition 4.10.
Let P be a Delzant polytope. Set L min i,k = min { L i ( m ) , m ∈ P ∩ Z n /k } . Let P k be the polytope defined by the inequalities L i ( x ) ≥ L min i,k i.e. P k = { x ∈ P : L i ≥ L min i,k , i = 1 , · · · d } . Note that if P k is a non-empty polytope with d facets then these facets are parallelto those of P .We want to show that as k tends to infinity the lattice Z n /k ∩ P becomes finerand eventually P k will look combinatorially like P . Lemma 4.11.
Let P be a simple compact Delzant polytope P = { x ∈ R n : L i ≥ , i = 1 , · · · d } . Let L min i,k and P k be defined as above then L min i,k → as k tends to infinity and P k has the same combinatorial type as P for k large enough.Proof. Assume without loss of generality that i = 1 i.e. we want to prove that L min1 ,k → k tends to infinity. Choose a vertex of P say a = ( a , · · · , a n ) on thefirst facet of P that is L ( a ) = 0. Assume furthermore that the first n facets of P meet at a . This is possible as we can simply relabel the facets.. Since P is Delzantthen there is A ∈ SL ( n, Z ) such that AP is standard at a . That is AP = { x ∈ R n : x − ˜ a ≥ , · · · , x n − ˜ a n ≥ , ˜ L n +1 ( x ) ≥ , · · · ˜ L d ( x ) ≥ } , where ˜ L n +1 , · · · , ˜ L d are affine functions and ˜ a = (˜ a , · · · , ˜ a n ) = Aa . Note that˜ L i (˜ a ) > i = n + 1 , · · · , d because { ˜ a } is the intersection of the first n facets of AP . Pick a k = ( a ,k , · · · , a n,k ) ∈ Z n in the following way a i,k − k ≤ ˜ a i ≤ a i,k k , ∀ i = 1 , · · · , n. Now a k /k satisfies the first n inequalities in the definition of AP by construction.On the other hand | a k /k − ˜ a | ≤ n/k so that when k is sufficiently large and i > n ,˜ L i ( a k ) is close to ˜ L i (˜ a ) > L i ( a k ) > k . This implies a k /k ∈ AP therefore a k /k ∈ Z n /k ∩ AP . But A − a k /k ∈ Z n /k ∩ P where we haveused the fact that A ∈ SL ( n, Z ). Also | A − a k /k − a | ≤ || A − || n/k and thereforetends to zero as k tends to infinity. It follows that L ( A − a k k ) → L ( a ) = 0 because L is a continuous function. Since L min1 ,k ≤ L ( A − a k k ) and is positive, we prove ourclaim. (cid:3) Definition 4.12.
We define k ( P ) to be smallest integer k ≥ P k hasthe same combinatorial type as P .For any integer k , we set N k = ♯ ( Z n /k ∩ P ) −
1. Note that if P is integral, k ( P ) = 1. Recall that we write L i ( x ) = x · ν i + c i . Lemma 4.13.
Let P be a Delzant polytope and k ≥ k ( P ) then kP k is an integralDelzant polytope such that N k + 1 = ♯ (( kP k ) ∩ Z n ) .Proof. For i ∈ { , . . . , d } , we denote ˜ c i,k = c i − L min i,k so that P k = { x ∈ R n | x · ν i +˜ c i,k ≥ , for i = 1 , . . . , d } . Then kP k = { x ∈ R n | ( x/k ) · ν i + ˜ c i,k ≥ , for i = 1 , . . . , d } = { x ∈ R n | x · ν i + k ˜ c i,k ≥ , for i = 1 , . . . , d } . (21) ORIC ASPECTS OF THE FIRST EIGENVALUE 17
Therefore, kP k has the same normal inward vectors as P and is Delzant iff P is.We are using the fact that that k ≥ k ( P ) so that kP k , P k and P have the samecombinatorial type. Moreover k ˜ c i,k ∈ Z . Indeed, the compacity of P implies thatthere exists m/k ∈ P such that m ∈ Z n and L min i,k = L i ( m/k ) and thus k ˜ c i,k = k ( c i − L min i,k ) = k ( c i − L i ( m/k )) = k ( c i − (( m/k ) · ν i + c i )) = − m · ν i ∈ Z . Hence, kP k is an integral Delzant polytope. Finally, m ∈ (( kP k ) ∩ Z n ) if and only if mk ∈ (( P k ) ∩ ( Z n /k )) ⊂ ( P ∩ ( Z n /k )), thus ♯ (( kP k ) ∩ Z n ) ≤ ♯ ( P ∩ ( Z n /k )). Now anypoint in ( P ∩ ( Z n /k )) \ ( P k ∩ ( Z n /k )) would contradict the minimality of L min i,k forsome i ∈ { , . . . , d } . Hence ♯ (( kP k ) ∩ Z n ) = ♯ (( P k ) ∩ ( Z n /k )) = ♯ ( P ∩ ( Z n /k )). (cid:3) We are now in a position to prove Theorem 1.1
Proof. (of Theorem 1.1) At this point we can apply Bourguignon–Li–Yau’s theo-rem 4.1 to conclude that(22) λ ( M, ω ) ≤ n ( N k + 1) R P Φ ∗ u,k ω F S ∧ ω n − N k R P ω n , where Φ u,k is the embedding (18) associated to kP k . Now it is well known thatthe symplectic class associated to kP k is k times the one associated to P k . Then,see Remark 4.5, we have [Φ ∗ u,k ω F S ] = k [ ω k ] where ω k is the symplectic form de-termined by P k . Recall that P k = { x ∈ P : L i ≥ L min i,k , i = 1 , · · · d } has the samecombinatorial type as P , by assumption on k . In particular, P and P k share thesame normal inward vectors (hence the same fan) but the constant parts of theaffine-linear functions defining P k are c i − L min i,k for i = 1 , ..., d .Let E i be the cohomology class that is Poincar´e dual to the divisor in M whoseimage under the moment map is the facet F i . Then12 π i [ ∂ ¯ ∂ log L i ] = E i (see [20], Theorem 6.2 for a statement of this elementary fact).It is well know (see [20]) that [ ω ]2 π = P di =1 c i E i . Applying this to P k we get(23) [Φ ∗ u,k ω F S ]2 π = k ( c i − L min i,k ) E i = k [ ω ] − d X i =1 L min i,k E i ! . Replacing in equation (22) we see that λ ( M, ω ) ≤ n ( N k + 1) N k k − k R P P di =1 L min i,k E i ∧ ω n − R P ω n ! . Because E i is Poincar´e dual to the pre-image of the i -th facet, which we denoteby D i , we have that Z P E i ∧ ω n − = Z D i ω n − = vol n − ( D i ) > . It follows that λ ( M, ω ) ≤ nk ( N k + 1) N k . It is clear from the above argument that if P is integral one can take k = 1 andthe bound becomes λ ( M, ω ) ≤ n ( N + 1) N .
This proves Theorem 1.1. (cid:3)
The equality case in Bourguignon–Li–Yau’s bound.
The goal of thissubsection is to study K¨ahler toric metrics which saturate the bound in Theorem1.1. We give a quick overview of Bourguignon, Li and Yau’s proof in [9] as thisproof will be important to us.
Sketch of proof of Theorem 4.1, see [9] . Recall that the first eigenspace of ( CP N , ω F S )has a basis given by the functions [ Z ] Ψ ij ( Z ) − δ ij N +1 where for i, j ∈ { , , . . . , N } ,Ψ ij ( Z ) = Z i Z j P Nk =0 | Z k | is one component of the SU ( N + 1) moment map Ψ : CP N ֒ → su ∗ N +1 . The mainstep of the proof in [9], is to show that, given a full embedding Φ : M ֒ → CP N ,there exists a unique B ∈ SL ( N + 1 , C ) such that B ∗ = B > R M ω n Z M (Ψ ij ◦ B ◦ Φ)( p ) ω n = δ ij N + 1 . Said differently, ( B ◦ Φ) ∗ ω F S is ( ω n /n !)–balanced, (see [15]). To simplify thenotation we write ω –balanced instead of ( ω n /n !)–balanced.Denote f Bij = Ψ ij ◦ B ◦ Φ − δ ij N +1 ∈ C ∞ ( M ). The Rayleigh principle implies that(25) λ ( M, ω ) Z M | f Bij | ω n n ! ≤ Z M |∇ ω f Bij | ω n n !with equality if and only f Bij is an eigenfunction of ∆ for the eigenvalue λ ( M, ω ).Taking the sum over i, j = 0 , . . . , N the left hand side of (25) gives λ ( M, ω ) NN + 1 Z M ω n n !thanks to (24). Noticing that g ( ∇ f, ∇ f ) ω n = n Re( df ∧ d c f ∧ ω n − ), that the form P i,j df Bij ∧ d c f Bij is real and that, on CP N ,(26) X i,j d Ψ ij ∧ d c Ψ ij = 2 ω F S , the right hand side of (25) gives X i,j Z M |∇ ω f Bij | ω n n ! = X i,j Z M df Bij ∧ d c f Bij ∧ ω n − ( n − n − Z M ( B ◦ Φ) ∗ ω F S ∧ ω n − . Finally, B ∗ ω F S and ω F S are in the same cohomology class on CP N , this concludesthe proof. (cid:3) Remark . The equality case in (16) implies the equality case in each inequality(25) and then that each function f Bij is an eigenfunction of ∆ for the first eigenvalue.
ORIC ASPECTS OF THE FIRST EIGENVALUE 19
In the toric context and for the embedding Φ u we get the following refinementof Bourguignon–Li–Yau’s result on the existence of balanced metrics. Lemma 4.15.
Let ( M, ω, g u , J u , T ) be a toric K¨ahler manifold with integral poly-tope P ⊂ t ∗ and corresponding embedding Φ u : M ֒ → CP N . There exists a diagonalmatrix B = diag( α , . . . , α N ) ∈ GL ( N + 1 , R ) with α i > and tr B = 1 satisfyingthe condition (24) .Proof. First, observe that when i = j , the function Ψ ij ◦ Φ u integrates to 0 on M since it does on each orbit of T . Hence to prove the lemma, we only have to provethat there exists α = ( α , . . . , α N ) ∈ R N +1 > such that, for i = 0 , . . . , N , ψ u,i ( α ) := α i Z M | Z i | P Nj =0 α j | Z j | ω n n ! = 1 N + 1where Z i = Φ u,m i ( x, θ ) see (18). Let Σ be the simplex defined byΣ := ( X ∈ R N +1 | N X i =0 X i = 1 , X i > ) . Because M is full in CP N , P Nj =0 α j | Z j | does not vanish identically on M (oth-erwise the M would be contained in a positive codimension subvariety). Since P Ni =1 ψ u,i ( α ) = vol( M ) < + ∞ and each component ψ u,i ( α ) ≥ α ∈ R N +1 ,by the dominated convergence lemma, the maps ψ u,i can be extended continuouslyto Σ. Hence, we see ψ u = ( ψ u, , . . . , ψ u,N ) as a continuous map ψ u : Σ −→ Σfrom the closed simplex Σ to itself. It is obvious that ψ u maps ∂ Σ to ∂ Σ.To prove the lemma we need to prove that ψ u is surjective which will follow ifwe prove that the restriction ψ u : ∂ Σ → ∂ Σ has non-trivial degree.Like in [9], instead of integrating on M , we integrate on CP N with the measure dµ u defined to be the push forward of the measure on M defined by the metric.This is possible because Φ u is a full embedding. Hence ψ u,i ( α ) = α i Z CP N | Z i | P Nj =0 α j | Z j | dµ u . Now if we consider the volume form dµ o induced by the Fubini-Study metric on CP N , the corresponding map ψ o : ∂ Σ → ∂ Σ with components ψ o,i ( α ) = α i Z CP N | Z i | P Nj =0 α j | Z j | dµ o and its restriction ψ o : ∂ Σ → ∂ Σ are bijections. Now ψ t = tψ u + (1 − t ) ψ o is afamily of continuous maps from Σ to itself preserving the boundary. The degreeof the ψ t does not depend on t and is non trivial for t = 0. (cid:3) Remark . A straightforward corollary of this lemma is that the ω –balancedmetric is toric as soon as ω is toric. Theorem 1.2 is a consequence of two propositions we state below and whichwe prove using the following observation. Assume that the first eigenvalue of(
M, ω, g u , J u , T ) reaches the Bourguignon, Li and Yau’s bound, i.e λ ( g u ) = n ( N +1) N .By Remark 4.14 and Lemma 4.15, there exists a set of N + 1 real positive numbers { α k } k ∈ P ∩ Z n such that for each m, k ∈ P ∩ Z n , the function Ψ mk − δ mk / ( N + 1) isan eigenfunction of eigenvalue n ( N +1) N whereΨ mk = α k α m Z m Z k P j | α j Z j | is seen as a function of ( x, θ ). Here we write Z m = e ( u x + i θ ) · m , where u x = ∂u∂x . We assume, without loss of generality, that 0 ∈ P ∩ Z n . Hencethe image of Φ u , see (18), lies in the set where Z = 0 and we work on this set.We normalize the α ’s so as to have α = 1 instead of P k ∈ P ∩ Z n α k = 1 as in theprevious lemma. We recall that(27) Z M Ψ mk ω n n ! = δ mk Z M ω n n ! . Observe that for each pair m, k ∈ P ∩ Z n (28) ∆Ψ mk = ∆( α m Z m Ψ k ) = α m Z m ∆Ψ k − α m h dZ m , d Ψ k i since ∆( Z m ) = 0 where h· , ·i denotes the inner product induced by g u on thecotangent bundle of M .When k = 0 = m , identity (28) becomes2 n ( N + 1) N Ψ m = 2 n ( N + 1) N α m Z m (Ψ − N + 1 ) − α m h dZ m , d Ψ i = 2 n ( N + 1) N Ψ m − nα m Z m N − α m h dZ m , d Ψ i , and we get 2 nα m Z m N = − α m h dZ m , d Ψ i . Developing the right hand side in action angle coordinates, using (4), we have2 nα m Z m N = − α m h dZ m , d Ψ i = − α m n X i,j =1 H ij ∂ x i Z m ∂ x j Ψ = 4 α m Z m P k ∈ W P ns,t =1 u ts m s k t | α k Z k | (cid:0)P k ∈ W | α k Z k | (cid:1) where W = P ∩ Z n . Dividing both sides by 2 α m Z m , we end up with(29) nN = 2 P k ∈ W P ns,t =1 u ts m s k t | α k Z k | (cid:0)P k ∈ W | α k Z k | (cid:1) = − ( d Ψ )( m ) ORIC ASPECTS OF THE FIRST EIGENVALUE 21
Proposition 4.17.
Let ( ω, g u , J u , T ) be a toric K¨ahler structure on CP n . Assumethat λ = 2( n + 1) i.e. assume that the first eigenvalue of the Laplacian reaches theBourguignon–Li–Yau bound. Then, the toric K¨ahler metric g u is the Fubini-Studymetric on CP n .Proof. The moment polytope P of CP n is a simplex and if ( ω, g u , J u , T ) saturatesthe bound then P is primitive as explained in Remark 4.8. So one can suitablynormalize it so that it has integer vertices 0 , e , · · · e n where e i is the vector in R n whose i -th component is 1 and all others are zero. In the notation above, N = n and we identify W = P ∩ Z n with { , , . . . , n } . Equation (29) implies that, forall m = 1 , . . . , n ,(30) 1 = 2 P nm,k =1 u mk | α k Z k | (1 + P nk =1 | α k Z k | ) = − ∂ Ψ ∂x m where Z k = e ( u k + i θ k ) for k = 1 , . . . , n . HenceΨ = K − n X k =1 x k for some constant K . The additive constant is fixed to be K = 1 by the integrationconstraint (27).In the case m = k = 0, identity (28) gives2( n + 1)(Ψ mm − /n + 1) = α m Z m n + 1)Ψ m − α m h dZ m , d Ψ m i that is − − α m h dZ m , d Ψ m i . Developing the right hand side using (30), we get − − α m h dZ m , d Ψ m i = 2 | α m Z m | (cid:18) u mm P i | α i Z i | − (cid:19) = − ∂ Ψ mm ∂x m . Therefore, for each m > ∂ Ψ mm ∂x m = 1In the case 0 = m = k = 0, identity (28) gives 2( n + 1)Ψ mk = α m Z m n +1)Ψ k − α m h dZ m , d Ψ k i , that is0 = − α m h dZ m , d Ψ k i . Developing the right hand side using (30), we get0 = − α m h dZ m , d Ψ k i = 2 α m α k Z m Z k (cid:18) u mk P i | α i Z i | − (cid:19) = − α m α k Z m Z k | α m Z m | (cid:18) ∂ Ψ mm ∂x k (cid:19) . Therefore, for each m, k > k = m , we have ∂ Ψ mm ∂x k = 0 . Together with (31) it gives(32) Ψ mm = x m where again the additive constant is fixed by the integration constraint (27) onΨ mm . We conclude from Remark (4.14) that the components of the moment mapare eigenfunction for the same eigenvalue and thus, by Proposition 1.4 ( g u , J u )is K¨ahler-Einstein. By uniqueness of extremal toric metric [18], ( g u , J u ) is theFubini-Study metric. (cid:3) We can also prove that if a toric manifold admits a toric K¨ahler metric for whichthe embedding given by the integral points of the moment polytope saturates theBourguignon–Li–Yau bound, then that manifold must be CP n and it follows fromthe above proposition that the metric must be the Fubini-Study metric. Proposition 4.18.
Let ( M, ω, g u , J u ) be a toric K¨ahler manifold with integralpolytope P . Assume that λ ( g u ) = n ( N +1) N with N = ♯ ( P ∩ Z n ) − . Then P is thestandard simplex and M is (equivariantly symplectomorphic to) CP n .Proof. Again we assume that the origin lies in P ∩ Z n , more precisely, up to anintegral invertible affine transformation, we may assume that P is standard at theorigin i.e. the facets that meet at 0 have normals e , · · · , e n . In particular, thevertices of P adjacent to the origin, say m , · · · m n , are each an integral multipleof an element of a dual basis of e , · · · , e n respectively.Under the hypothesis of the proposition and with respect to the notation above,Equations (28) and (29) hold for points in P ∩ Z n . Suppose there is m ∈ P ∩ Z n a vertex distinct from the origin and from m , · · · m n . Then, there exist a , · · · a n such that m = P nl =1 a l m l .Equation (29) holds for m as well as for m , · · · m n . So we must have nN = − ( d Ψ )( m ) = − n X l =1 a l ( d Ψ )( m l ) = ( n X l =1 a l ) nN which implies P nl =1 a l = 1. So m lies on a facet of the simplex of vertices0 , m , · · · m n which contradicts convexity unless P is that simplex. (cid:3) To sum up we proved Theorem 1.2.
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E-mail address : [email protected] Rosa Sena-Dias, Centro de An´alise Matem´atica, Geometria e Sistemas Dinˆamicos,Departamento de Matem´atica, Instituto Superior T´ecnico, Av. Rovisco Pais,1049-001 Lisboa, Portugal
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