On the spectral theory and dynamics of asymptotically hyperbolic manifolds
aa r X i v : . [ m a t h . SP ] D ec Ann. Inst. Fourier, Grenoble
Working version – December 14, 2020
ON THE SPECTRAL THEORY AND DYNAMICS OFASYMPTOTICALLY HYPERBOLIC MANIFOLDS by Julie ROWLETT
Sur la th´eorie spectrale et la dynamique des vari´et´esasymptotiquement hyperboliques
Abstract.
We present a brief survey of the spectral theory and dy-namics of infinite volume asymptotically hyperbolic manifolds. Be-ginning with their geometry and examples, we proceed to their spec-tral and scattering theories, dynamics, and the physical descriptionof their quantum and classical mechanics. We conclude with a dis-cussion of recent results, ideas, and conjectures.
Sur la th´eorie spectrale et la dynamique des vari´et´esasymptotiquement hyperboliques
R´esum´e.
Cet article est une pr´esentation rapide de la th´eorie spec-trale et de la dynamique des vari´et´es asymptotiquement hyperboliques`a volume infini. Nous commen¸cons par leur g´eom´etrie et quelquesexemples, on poursuit en rappelant leur th´eorie spectrale, puis con-tinuons sur des d´eveloppements r´ecents de leur dynamique. Nousconcluons par une discussion de r´esultats qui d´emontrent un rapportentre leurs m´ecaniques quantiques et classiques et enfin, nous offronsquelques id´ees et conjectures.
1. Introduction and contents
In 1984, Fefferman and Graham [31] introduced the concept of a confor-mally compact metric to generalize the Poincar´e model of hyperbolic space
Keywords: asymptotically hyperbolic, conformally compact, wave trace, negative curva-ture, resonances, length spectrum, topological entropy, dynamics, geodesic flow, primeorbit theorem, quantum and classical mechanics.2020
Mathematics Subject Classification:
JULIE ROWLETT and study conformal invariants. These metrics are also known as Poincar´emetrics. A Riemannian manifold with boundary equipped with a confor-mally compact metric is known as a conformally compact manifold. In1987, Mazzeo and Melrose [67] defined a special class of conformally com-pact manifolds, the asymptotically hyperbolic manifolds.
These are com-pact manifolds with boundary equipped with an asymptotically hyperbolicmetric: a complete metric with sectional curvatures asymptotically equalto − ANNALES DE L’INSTITUT FOURIER
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Bibliographical notes
Fefferman and Graham [31] defined conformally compact metrics to studyconformal invariants. A conformally compact manifold does not have aunique metric at the boundary; rather, the metric induces a conformal classof metrics at the boundary which is known as the “conformal infinity.” Fora real analytic conformal Riemannian manifold, they showed that one mayassociate a conformally compact manifold in one dimension higher, suchthat the conformal manifold is the conformal infinity. Based on this ambi-ent metric construction, [31] computed several scalar curvature invariantsfor conformal Riemannian manifolds.Mazzeo and Melrose studied the resolvent operator for asymptotically hy-perbolic metrics in [67]; their main results will be reviewed in § L cohomology of conformally compact manifolds andproved their Hodge theorem in [66].Selberg proved trace formulae for (weakly) symmetric spaces using abeautiful combination of techniques from harmonic analysis, group theory,and representation theory [86]. The trace formula has been generalizedto various other settings including congruence subgroups and P SL (2 , R )by Hejhal [50], [51]; and some spaces of infinite volume by Gangolli andWarner [35], [36]. Moreover, interactions between the trace formula andHecke operators in arithmetic geometry have been investigated by Arthur[4].Phillips [78] and Patterson [73] demonstrated that the existence of purepoint spectrum determines and conversely is determined by the topologicalentropy of the geodesic flow; see also joint work of Phillips and Sarnak [80]. SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX
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2. Geometry
The model geometry of an asymptotically hyperbolic manifold is an in-finite volume hyperbolic manifold which is the quotient of real hyperbolicspace by a convex cocompact group.
Definition 2.1. —
A finitely-generated discrete torsion-free group Γ of isometries of real hyperbolic space is convex cocompact if Γ does notcontain parabolic elements, contains at least one hyperbolic element, andadmits a finite sided fundamental domain with infinite volume. Examples of convex cocompact groups include Schottky groups, Fuch-sian groups, and quasifuchsian groups; the term convex cocompact firstappeared in Sullivan’s work [89]. However, in that work, convex cocompactgroups are defined to be those groups which do not contain parabolic ele-ments and may contain hyperbolic elements. In particular, he showed thatthe quotient of real hyperbolic space by a convex cocompact group is eithercompact or has infinite volume.
Definition 2.2. —
For a discrete torsion-free group Γ of isometriesof n + 1 dimensional real hyperbolic space H n +1 , the quotient H n +1 / Γ is convex cocompact if Γ is a convex cocompact group. The limit set of Γ , Λ Γ ,is the smallest closed Γ -invariant subset of ∂ H n +1 = S n ; it is equivalentlygiven by the intersection of the closure of the orbit by Γ of any point of H n +1 with ∂ H n +1 . The convex core is defined by CH (Λ Γ ) / Γ ⊂ H n +1 / Γ , where CH (Λ Γ ) is the convex hull of Λ Γ ; for a convex cocompact group Γ ,the convex core is a compact subset of H n +1 / Γ . For a discrete, finitely-generated, torsion-free group Γ of isometries of H n +1 , the quotient is conformally compact precisely when Γ is convex co-compact as defined above; see for example [11]. We recall the definition ofa conformally compact metric below. Asymptotic geometries such as asymptotically hyperbolic, asymptoti-cally conic, and asymptotically cylindrical are Riemannian manifolds with
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Definition 2.3. —
Let ( X n +1 , ∂X ) be a smooth, compact n +1 dimen-sional manifold with boundary. A boundary defining function x is a smoothfunction defined in a neighborhood N of ∂X so that x : N → [0 , ∞ ) , ∂X = x − ( { } ) , and dx = 0 on ∂X . A conformally compact manifold is a Riemannian manifold with bound-ary whose metric admits the following form.
Definition 2.4. —
Let X n +1 be a smooth compact manifold with bound-ary ∂X and boundary defining function x . If there exists a smooth metric ¯ g defined on X which is non-degenerate up to ∂X such that the metric g on the interior of X has the form g = ¯ gx , then g is a conformally compact metric. The following definition is from [11] and is related to the geometric settingof earlier work by Guillop´e and Zworski [48].
Definition 2.5. —
A conformally compact manifold ( X n +1 , g ) is hy-perbolic near infinity if there exists a (possibly disconnected) convex co-compact hyperbolic manifold ( X , g ) and compact sets K ⊂ X , K ⊂ X ,such that ( X − K, g ) ∼ = ( X − K , g ) . These spaces also appear in the literature as “compact perturbations ofconvex cocompact hyperbolic manifolds.” They enjoy many of the beautifulproperties of their unperturbed model spaces; [11] proved a Poisson formulaand estimates for their spectral counting function.The definition of an asymptotically hyperbolic manifold is originally dueto Mazzeo and Melrose [67].
Definition 2.6. —
A manifold with boundary ( X n +1 , ∂X ) is asymp-totically hyperbolic if there exists a boundary defining function x so that ina non-empty neighborhood of the boundary N ∼ = (0 , x ) × ∂X, the metric (2.7) g = dx + h ( x, y, dx, dy ) x where h | { x =0 } is independent of dx and is a Riemannian metric on ∂X . SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX
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Remark 2.8. —
It is straightforward to demonstrate that every asymp-totically hyperbolic manifold is conformally compact. On the other hand,the sectional curvatures of conformally compact manifolds approach −| dx | x g at the boundary: this is not necessarily a constant function. Thus, not allconformally compact manifolds are asymptotically hyperbolic; see [12].Recall that a Riemannian metric is Einstein if the Ricci and metric ten-sors satisfy the relationship Ric = − cg, for some constant c . In the case of Poincar´e Einstein metrics, those whichare both conformally compact and Einstein, c >
0, and is usually normal-ized to c = n +1 in dimension n +1. Some explicit examples of the Poincar´eEinstein metrics which arise in AdS/CFT correspondence in string theoryinclude the hyperbolic analogue of the Schwarzschild metric, and in dimen-sion n + 1 = 4, the Taub-BOLT metrics on disk bundles over S . Theseexamples may be found in Anderson [2].Mazzeo and Melrose observed that along any smooth curve in X − ∂X ap-proaching a point p ∈ ∂X, the sectional curvatures of g approach −| dx | x g . For each h in the conformal class of h | ∂X , by [67] and [68], there exists aunique (near the boundary) boundary defining function x so that(2.9) | dx | x g = 1 , and g = dx + h ( x, y, dy ) x near ∂X .With this normalization, the sectional curvatures are − O ( x ) as x →
0, soit is natural to call these spaces “asymptotically hyperbolic.” If the metricis also Einstein with Ric g = − ( n + 1) g , then the sectional curvatures are − O ( x ) in a neighborhood ∂X . When the sectional curvatures convergeto − − ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS Since conformally compact manifolds have infinite volume, it is naturalto introduce an integral renormalization. Recall that the finite part f . p . ǫ =0 f ( ǫ )is defined as f when f ( ǫ ) = f + ∞ X k =1 f k ǫ − λ k (log ǫ ) m k + o (1) , with R ( λ k ) >
0, and m k ∈ N ∪ { } . Uniqueness of f is demonstrated inH¨ormander [52]; the following definition is from Guillop´e and Zworski [46]. Definition 2.10. —
The Z f of a smooth func-tion (or density) f on X is defined, if it exists, as the finite part Z X f := f . p . ǫ =0 Z x ( p ) >ǫ f ( p ) dvol g ( p ) , where x is a boundary defining function. The 0-regularized integral also appears in the literature as the 0-renormalizedintegral. Guillop´e and Zworski also defined the 0 -trace, which they used toformulate their Selberg trace formula for Riemann surfaces [46].
Definition 2.11. —
For an operator A with smooth Schwartz kernel A ( z, y ) on X × X , the of A is0-tr ( A ) := Z X A ( z, z ) dvol g ( z ) . Both the 0-regularized integral and the 0-trace depend `a priori on thechoice of boundary defining function x in their definitions. However, in somecases, the 0-regularized integral is independent of the boundary definingfunction. The 0-volume is defined to be the 0-regularized integral of theconstant function 1, 0-vol ( X ) := Z X g . Graham [38] showed that in even dimensions (by our convention, n is odd),the 0-volume is well defined and independent of the choice of boundarydefining function used to define the 0-regularized integral. The 0-volume,also known as renormalized volume, of conformally compact manifolds is ofgeneral interest in conformal geometry; see for example Chang, Qing, andYang [20] and Chang, Gursky, and Yang [19]. SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX
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Bibliographical notes
The asymptotically cylindrical metrics or b-metrics were introduced byMelrose; an excellent reference for the geometric structure is [69]. Asymp-totically conic metrics also appear in the literature as scattering metrics,and asymptotically hyperbolic geometry has also been referred to as 0-geometry. For each of these geometries, the resolvent operator has beenconstructed as an element of a pseudodifferential operator calculus. Theseconstructions generalize H¨ormander’s parametrix construction on closedmanifolds to Riemannian manifolds with boundary whose metrics admitcertain asymptotic forms near the boundary. A novel feature of these con-structions is the introduction of an extra symbol arising from the boundary.The calculi for the asymptotically cylindrical and asymptotically hyperbolicgeometries are known, respectively, as the b-calculus and the 0-calculus.The 0-calculus was introduced by Mazzeo and Melrose in [67].Given an asymptotically hyperbolic metric, the existence of a boundarydefining function x such that the metric near the boundary has the form dx + h ( x, y, dy ) x , was first shown by Joshi and S`a Barreto while studying inverse scatteringresults for asymptotically hyperbolic manifolds [54]. However, a simplerproof of this result may be found in § conformally compact.
3. Spectral and scattering theories
In 1984, Lax and Phillips [60] published an elegant and thorough studyof the Laplace operator and its resolvent on convex cocompact hyperbolicmanifolds, and in 1987, Perry [77] generalized their results to the Lapla-cian with a short range potential. These works provided useful insights toMazzeo and Melrose [67], who studied the spectral theory of asymptoticallyhyperbolic manifolds. An important contribution to their work was madeby Guillarmou [41] nearly twenty years later. We recall their main resultsbelow.
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N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS Mazzeo and Melrose proved that the Laplacian on an asymptoticallyhyperbolic manifold of dimension n +1 has absolutely continuous spectrum, σ ac (∆) = [ n , ∞ ), and a finite pure point spectrum which is either empty orconsists of finitely many eigenvalues, σ pp (∆) = { Λ k } mk =1 ⊂ (0 , n ). Mazzeoand Melrose constructed the resolvent kernel as an element of the so-called“0-calculus,” and showed that it is meromorphic on the half space n Re( s ) > n o , and admits a well-defined meromorphic extension to C with discrete polesof finite rank. Above, the spectral parameter s is related to the actualspectral parameter Λ by Λ = s ( n − s ) . The resonance set R consists of the poles of the meromorphically continuedresolvent R ( s ) := (∆ − s ( n − s )) − , counted according to the multiplicity, ζ ∈ R , m ( ζ ) := rank Res ζ (∆ − s ( n − s )) − . This is not quite the full story; based on Borthwick and Perry’s work con-cerning the scattering poles of asymptotically hyperbolic manifolds [14],Guillarmou [41] investigated the behavior of the resolvent at the points (cid:0) n − k (cid:1) k ∈ N . He made the important discovery that the presence of infiniterank poles at these points is determined by, and conversely determines, thetype of expansion the metric admits near the boundary; this is all beauti-fully explained in [41]. In fact, the resolvent admits a finite-meromorphicextension to C if and only if the metric g admits an expansion of the form g = x − dx + ∞ X k =0 x k h k ( y ; dy ) ! , near the boundary, where x is a boundary defining function. This is thecase for many important examples, including convex cocompact hyperbolicmanifolds and conformally compact manifolds which are hyperbolic nearinfinity. In the latter case, the result is due to Guillop´e and Zworski [48]. SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
The resolvent operator is intimately related to the scattering operator,whose definition we recall below.
Definition 3.1. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold with boundary defining function x. For Re ( s ) = n , s = n , a function f ∈ C ∞ ( ∂X ) determines a unique solution u of (∆ − s ( n − s )) u = 0 , u ∼ x n − s f + x s f , as x → , where f ∈ C ∞ ( ∂X ) . This defines the map called the scattering operator, S ( s ) : f f . Heuristically, the scattering operator which is classically a scattering ma-trix, acts as a Dirichlet to Neumann map, and physically it describes thescattering behavior of particles. Joshi and S`a Barreto [54] proved that thescattering operator extends meromorphically to s ∈ C as a family of pseu-dodifferential operators of order 2 s − n . Renormalizing the scattering op-erator as follows gives a meromorphic family of Fredholm operators withpoles of finite rank,˜ S ( s ) := Γ( s − n )Γ( n − s ) Λ n/ − s S ( s )Λ n/ − s , where Λ := 12 (∆ h + 1) / . Above, ∆ h is the Laplacian on ∂X for the metric h ( x ) (cid:12)(cid:12) x =0 . Note thatthis definition depends on the choice of boundary defining function x . Themultiplicity of a pole or zero of S ( s ) is defined to be ν ( ζ ) = − tr[Res ζ ˜ S ′ ( s ) ˜ S ( s ) − ] . The scattering multiplicities are related to the resonance multiplicities in[14] and [41] by ν ( s ) = m ( s ) − m ( n − s ) + ∞ X k =1 (cid:0) χ n/ − k ( s ) − χ n/ k ( s ) (cid:1) d k , where d k = dim ker ˜ S ( n k ) , and χ p denotes the characteristic function of the set { p } . This relationshipis originally due to Borthwick and Perry [14], but it was Guillarmou [41]who explicitly identified the numbers d k as the dimensions of the kernelsof certain natural conformal operators acting on the conformal infinity. ANNALES DE L’INSTITUT FOURIER
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On a compact Riemannian manifold, the number of eigenvalues of theLaplacian grows according to Weyl’s formula. For convex cocompact hyper-bolic manifolds of even dimension, Guillarmou [43] obtained a Weyl-typeasymptotic formula which interestingly depends on the topological entropyof the geodesic flow. If X is conformally compact and hyperbolic near in-finity, upper and lower bounds for the resonance counting function wereachieved through the work of Guillop´e and Zworski [48] and completed byBorthwick [11] who demonstrated that the number of resonances (countedaccording to the spectral parameter s ) in a ball of radius r is O ( r n +1 ).Moreover, the scattering resonance set satisfies the lower bound, N sc ( r ) > cB ( X, g ) r n +1 where c is a (universal) positive constant, and B ( X, g ) is the 0-volume of X if the dimension of X is even or the Euler characteristic of X if thedimension of X is odd. Counting estimates were obtained for surfaces byGuillop´e and Zworski [47]. Counting estimates for the scattering and resol-vent resonances of asymptotically hyperbolic manifolds in general remainsan intriguing and challenging open problem. Bibliographical notes
Guillarmou’s Weyl law is in fact not the main result of [40]. We highlyrecommend [40] to interested readers. Guillarmou develops a Birman-Kreintheory for asymptotically hyperbolic manifolds whose metrics have a cer-tain form at the boundary. These metrics include the natural examples:convex cocompact hyperbolic manifolds and conformally compact mani-folds hyperbolic near infinity. He first shows that one may define the spec-tral measure via a 0-regularized integral and demonstrates its regularityproperties on R and C . He goes on to study the determinant of the scatter-ing operator and its relationship to the dynamical zeta function. This workboth provides new tools and suggests open problems on asymptotically hy-perbolic manifolds. For example, it would be interesting to generalize theWeyl law to these asymptotically hyperbolic manifolds. SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
In [61] and [62], Lee made further contributions to the scattering the-ory and Fredholm properties of the Laplacian on asymptotically hyperbolicmanifolds; see also S`a Barreto [84]. Graham and Zworski [39] studied thescattering matrix in conformal geometry; this work is related to the num-bers d k and the conformal infinity. There is little known about the spec-tral theory of conformally compact manifolds which are not asymptoticallyhyperbolic: those which have variable curvature at infinity. The only refer-ence of which we are aware is Borthwick [12], who developed the scatteringtheory by treating the Laplacian as a degenerate elliptic operator withnon-constant indicial roots. The variability of the roots poses a significantchallenge to the analysis because it implies that the resolvent admits onlya partial meromorphic continuation.
4. Dynamics
Classical mechanics are mathematically described by the dynamics ofthe geodesic flow. We begin by recalling the definitions. Let SX denote theunit tangent bundle, and let G t be the geodesic flow on SX at time t . Definition 4.1. —
The geodesic flow on a Riemannian manifold ( X, g ) is the map G : SX × R → SX such that for v ∈ S x X , G t ( v ) = ˙ γ v ( t ) , where γ v is the geodesic from x ∈ X with initial tangent vector v , and ˙ γ v ( t ) is its tangent vector at the point γ v ( t ) . The orbit of a vector v ∈ S x X forthe geodesic flow is { G t ( v ) | t ∈ R } . The length spectrum is the set of lengths of closed geodesics. We shalluse L and L p to denote, respectively, the set of closed geodesics and the set of primitiveclosed geodesics. Letting π : SX → X be the canonical projection, forall unit vectors v ∈ SX , the restriction of π to the orbit of v under G isbijective onto the geodesic γ v ⊂ X . Thus, we see that the closed orbits ofthe flow G are in bijection with the closed geodesics of X .Under certain conditions, the geodesic flow has desirable properties. It is hyperbolic (see Anosov [3]) if for each ξ ∈ SX, T ( SX ) ξ splits into a directsum(4.2) T ( SX ) ξ = E sξ ⊕ E uξ ⊕ E ξ , ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS E sξ is exponentially contracting, E uξ is exponentially expanding, and E ξ is the one dimensional subspace tangent to the flow. This definitionwas first given for closed manifolds as in Klingenberg [58], but the samedefinition holds for open manifolds; see for example Bolton [9]. A closed G t invariant set Ω ⊂ X without fixed points is hyperbolic if the tangentbundle restricted to Ω is a Whitney sum, T Ω X = E + E s + E u of three G t invariant sub-bundles, where E is the one dimensional bundletangent to the flow, and E s , E u are, respectively, exponentially contractingand expanding. The Sinai-Ruelle-Bowen potential is a H¨older continuousfunction defined for ξ ∈ SX by,(4.3) H ( ξ ) := ddt (cid:12)(cid:12) t =0 log det dG t | E uξ . This potential gives the instantaneous rate of expansion at ξ .A vector ξ ∈ SX is non-wandering for the geodesic flow if for all neigh-borhoods U ⊂ SX with ξ ∈ U , there exists a sequence { t n } → ∞ suchthat for all n ∈ N , U ∪ G t n U = ∅ . These vectors form the non-wandering set.
The geodesic flow satisfies Smale’s
Axiom A [88] if the non-wandering set Ω is hyperbolic, and the periodicpoints of the flow are dense in Ω. The flow is topologically transitive on Ω,if for any open U and V ⊂ Ω, there exists n > G n ( U ) ∩ V = ∅ .The definition of basic set is due to Bowen [17]. A basic set Ω of an AxiomA flow G t on a Riemannian manifold ( X, g ) is a hyperbolic set in which pe-riodic orbits are dense, G t | Ω is topologically transitive, and Ω = ∩ t ∈ R G t U for some open neighborhood U ⊃ Ω. An excellent reference for the generaltheory of dynamics is the book by Katok and Hassellblatt [57].On a closed manifold with hyperbolic geodesic flow, the topological en-tropy of the geodesic flow is given by the limit(4.4) h = lim T →∞ log { γ ∈ L : l ( γ ) T } T , which was shown to exist in this setting by Bowen [16]. When X is compactand has no conjugate points, Freire and Ma˜n´e [34] showed that the topo-logical entropy h of the geodesic flow is equivalently given by the volumeentropy, the log-volume growth rate in the universal covering, h = λ ( X ) := lim sup r →∞ log Vol( B r ( x )) r . SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
Above, Vol( B r ( x )) denotes the volume of the ball of radius r and center x in the universal covering of X ; see also Manning [64].For asymptotically hyperbolic manifolds, Joshi and S`a Barreto showedthat all closed geodesics are contained in a compact subset [55]. Therefore,we may define the topological entropy of the geodesic flow by restrictingto the non-wandering set. First, we recall that for large T and small δ > , a finite set Y ⊂ SX is ( T, δ ) separated if, given ξ, ξ ′ ∈ Y, ξ = ξ ′ , there is t ∈ [0 , T ] with d ( G t ξ, G t ξ ′ ) > δ. Here the distance on SX is given by theSasaki metric. We may then define the topological entropy of the geodesicflow as follows. Definition 4.5. —
Let ( X, g ) be an asymptotically hyperbolic manifoldwith negative sectional curvatures. Let Ω ⊂ SX be the non-wandering setof the geodesic flow. We define the topological entropy of the geodesic flow to be the limit h := lim δ → lim sup T →∞ log sup { Y ⊂ Ω : Y is ( T, δ ) separated } T .
In [82], we demonstrate that this definition of topological entropy coin-cides with the definition given for the topological entropy of a hyperbolicgeodesic flow on a closed manifold (4.4).For convex cocompact hyperbolic manifolds H n +1 / Γ, Patterson [73] demon-strated that h = δ is the exponent of convergence for the Poincar´e seriesfor Γ, and Sullivan [89] showed that h is also dimension of the limit setof Γ. Yue [91] has extended their results to some complete spaces of non-constant curvature. These are known as “convex cocompact manifolds” andarise as the quotient of a complete manifold with pinched negative sectionalcurvatures by a convex cocompact group.The pressure of a function is a concept in dynamical systems arisingfrom statistical mechanics which measures the growth rate of the numberof separated orbits weighted according to the values of f ; see for exam-ple Walters [90] and Manning [64]. The rather cumbersome definition ofpressure is nonetheless useful to generalize entropy, p ( f ) = lim δ → lim sup T →∞ log sup nP ξ ∈ Y exp R T f ( G t ξ ) dt ; Y is ( T, δ ) separated o T .
In the compact setting, the topological pressure of a function f : SX → R p ( f ) = sup µ (cid:18) h µ + Z f dµ (cid:19) , ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS G t invariant measures µ, and h µ denotes the measure theoretical entropy of the geodesic flow.The dynamical theory of open manifolds was initiated in the 1970s byseveral authors. We note the work of Eberlein [25], [26], [27]; Eberlein andO’Neill [29]; and Bishop and O’Neill [8]. A further reference is Eberlein,Hamenst¨adt, and Schroeder [28]. The focus of those works is dynamics on visibility manifolds: complete, open manifolds with non-positive sectionalcurvatures. Important contributions to the dynamical theory of hyperbolicmanifolds were made by Sullivan [89], Lalley [59], Guillop´e [45], Patterson[73], and Perry [76]. The following dynamical lemma was proven in [82] using methods sug-gested by Manning [65], and we expect that it is known to experts, if notalready in the literature. Nonetheless, since it is a useful tool for dynamicson any complete manifold with pinched negative sectional curvatures, weinclude its statement here.
Lemma 4.1. —
Let ( X, g ) be a smooth, complete, n + 1 dimensionalRiemannian manifold whose sectional curvatures κ satisfy − k κ − k for some < k k . Then, the Poincar´e map about a closed orbit γ ofthe geodesic flow has n expanding eigenvalues { λ i } ni =1 which satisfy e k l ( γ ) | λ i | e k l ( γ ) for i = 1 , . . . , n, and n contracting eigenvalues { λ i } ni = n +1 which satisfy e − k l ( γ ) | λ i | e − k l ( γ ) for i = n + 1 , . . . , n, where l ( γ ) is the period (or length) of γ. The proof uses the Rauch Comparison Theorem, which can be found indo Carmo [22], and basic properties of the Lyapunov exponents found inLuis-Pesin [63].In [82], we demonstrated a “prime orbit theorem” for the geodesic flowof negatively curved asymptotically hyperbolic manifolds.
Theorem 4.2. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold with negative sectional curvatures. If the the topolog-ical entropy of the geodesic flow h > , then the length spectrum counting SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT function (4.6) N ( T ) := { γ ∈ L : l ( γ ) T } satisfies lim T →∞ T N ( T ) e hT = 1 . The proof of this result relies on the following key ingredients. First, Joshiand S`a Barreto demonstrated that the closed geodesics of any asymptot-ically hyperbolic manifold are contained in a compact subset. Next, sincethe sectional curvatures are negative, it is straightforward to generalizethe “separation lemma” of Jakobson, Polterovich, and Toth [53] to asymp-totically hyperbolic manifolds; see Lemma 3.4 of [82]. We use this lemmaand the dynamical lemma above in combination with the aforementionedreferences on the dynamics of visibility manifolds to show that, if the topo-logical entropy is positive, then the non-wandering set for the geodesic flowis a basic set. It follows from Eberlein [26] that the geodesic flow restrictedto the non-wandering set is an Axiom A flow restricted to a basic set, sowe apply the main result of Parry and Pollicott [72] to the dynamical zetafunction which completes the proof of the theorem. In fact, we are currentlyexploring further implications for the spectral theory of asymptotically hy-perbolic manifolds with negative sectional curvatures based on more refinedproperties of the dynamical zeta funtions.
There are two key objects which relate the spectral and dynamical the-ories. The first is the dynamical zeta function, which is to the geodesiclength spectrum as the Riemann zeta function is to the prime numbers.Let(4.7) Z ( s ) = exp X γ ∈L p ∞ X k =1 e − ksl p ( γ ) k , where L p consists of primitive closed orbits of the geodesic flow and l p ( γ ) isthe primitive period (or length) of γ ∈ L p . Patterson and Perry [74] definedthe following weighted dynamical zeta function,˜ Z ( s ) = exp X γ ∈L p ∞ X k =1 e − ksl p ( γ ) k q | det( I − P kγ ) | , where P kγ is the k -times Poincar´e map of the geodesic flow around the(primitive) closed orbit γ . The weighted dynamical zeta function is par-ticularly interesting for its connections to the resonances of the resolvent; ANNALES DE L’INSTITUT FOURIER
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Theorem 4.3. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold with negative sectional curvatures. Then, the dynami-cal zeta function Z ( s ) converges absolutely if and only if Re ( s ) > h , where h is the topological entropy of the geodesic flow. Moreover, if h > , then Z admits a nowhere vanishing analytic extension to an open neighborhood ofRe ( s ) > h except for a simple pole at s = h . The weighted dynamical zetafunction, ˜ Z ( s ) converges absolutely if and only if Re ( s ) > p ( − H/ , where p is the topological pressure, and H is the Sinai-Ruelle-Bowen potential. Our proof was facilitated by Joshi and S`a Barreto’s results which allowone to apply local estimates from dynamics on closed manifolds, and inparticular Chen and Manning [21], Bowen [8], and Franco [33].
Bibliographical notes
Anosov’s work was motivated by S. Smale’s lectures [88]; Anosov at-tended the lectures and subsequently answered Smale’s conjectures in animpressive doctoral thesis [3]. An Anosov flow is a special example ofSmale’s Axiom A flows; Anosov’s original definition was a flow that satis-fies “condition U” [3]. Another related flow is the Bernoulli flow which wasstudied by Lalley, who in that setting proved the prime orbit theorem [59].The prime orbit theorem for surfaces with totally geodesic boundary wasproven by Guillop´e [45]. For convex cocompact hyperbolic manifolds in alldimensions, the prime orbit theorem is due to Perry [76]. Guillarmou andNaud [42] obtained an estimate for the remainder term in Perry’s primeorbit theorem.Patterson and Perry studied the dynamical zeta function in even dimen-sions [74]. Their work includes an appendix by C. Epstein which demon-strates an interesting geometric property for conformally compact mani-folds with constant negative curvature near infinity. The metric admits a
SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT particular expansion in this neighborhood of infinity, and one may com-pute the coefficients in terms of the Riemannian curvature tensor and itsderivatives.
5. Quantum and classical mechanics
The movement of very small particles is described by quantum mechan-ics, the name of which derives from the observation that certain physicalquantities such as the energy of an electron bound into an atom or moleculecan only assume discrete values or quanta.
One of the first mathematicalformulations of quantum mechanics is matrix mechanics, due to Heisen-berg, Born, and Jordan [10] in 1925. This theory was shortly followed bywave mechanics, introduced by Schr¨odinger [85] in 1926. Two years later,Dirac [23] introduced transformation theory to unify and generalize theseformulations and to unify the particle-wave duality of energy and matterobserved in photons and electrons. In these theories, eigenvalues of certainoperators are used to describe the energy states of quantum particles. Oneof the fundamental operators considered is the Laplace operator on a Rie-mannian manifold. When the manifold has infinite volume, the resonances of the resolvent describe the quantum states.The relationship between quantum and classical mechanics remains inmany ways an open question. One theory is that all objects obey the lawsof quantum mechanics, and that classical mechanics is simply the quantummechanics of a very large collection of particles. Thus, the laws of classicalmechanics ought to follow from the laws of quantum mechanics and limitsof large quantum systems. However, this breaks down with certain chaoticsystems; see for example Gutzwiller [49]. A further crux is the Einstein-Podolsky-Rosen paradox [30]; simply put, the laws of quantum mechanicswould seem to violate the most natural and basic laws of classical mechan-ics. Mathematical physicists would like to reconcile quantum and classicalmechanics, so it is helpful to understand interactions between them. Thewave group is one bridge between the quantum and classical mechanics.
The (even) wave kernel is the Schwartz kernel of the fundamental solutionto (cid:18) ∂ t + ∆ − n (cid:19) U ( t, w, w ′ ) = 0 , ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS U (0 , w, w ′ ) = δ ( w − w ′ ) , ∂∂t U (0 , w, w ′ ) = 0 . Due to the semi-group property with respect to time, the wave kernel isalso referred to as the wave group with the notationcos (cid:16) t p ∆ − n / (cid:17) . The wave group on asymptotically hyperbolic manifolds was constructed byJoshi and S`a Barreto in [55] as an element of an operator calculus definedon a certain manifold with corners obtained by blowing up R + × X × X along two submanifolds (the diagonal is blown up first, followed by thesubmanifold where the diagonal meets the corner).They demonstrated that the wave group for an asymptotically hyper-bolic manifold has a well-defined 0-trace, and that the singular support of0-tr cos( t p ∆ − n /
4) is contained in the set of lengths of closed geodesics.In effect, they generalized Duistermaat and Guillemin [24] to the asymp-totically hyperbolic setting.
The leading term in the following trace formula follows immediately fromJoshi and S`a Barreto [55]. The main point of the result is the long timeremainder estimate which allows one to use the trace formula to uncover arapport between the size of the topological entropy and the existence of purepoint spectrum thereby establishing a quantitative relationship between theclassical and quantum mechanics. This long-time estimate is based on thetechniques of Jakobson, Polterovich, and Toth [53] using the stationaryphase method augmented by estimates using the iterative construction ofthe wave group in [55] and properties of the wave group demonstrated byB´erard [7]. The estimate is in
Ehrenfest time, in which the oscillation ofthe test function and the size of its support satisfy a certain relationship;see for example Zelditch [92].
Theorem 5.1. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold with negative sectional curvatures. As a distributionalequality in D ′ ((0 , ∞ )) , (5.1) cos( t p ∆ − n /
4) = X γ ∈L p ∞ X k =1 l ( γ ) δ ( t − kl ( γ )) q | det( I − P kγ ) | + A ( t ) . SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
The remainder A is smooth and satisfies the following estimate: there exists ǫ > such that for any t > , there exists a C > such that (5.2) (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ A ( t ) cos( λt ) ρ ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) C for all λ > and ρ ∈ C ∞ ([ t , ǫ ln λ ]) . The constant C depends only on t and || ρ || ∞ ; C is independent of λ . On a convex cocompact hyperbolic manifold X n +1 = H n +1 / Γ, Perry [75]and Guillarmou Naud [42] proved the Selberg trace formula. The dynam-ical side of their trace formula is the following distributional equality in D ′ ((0 , ∞ )),(5.3)0-tr cos( t p ∆ − n /
4) = X γ ∈L p ∞ X k =1 l ( γ ) δ ( | t | − kl ( γ )) q | det( I − P kγ ) | + A ( X ) cosh t (sinh | t | ) n +1 , where A ( X ) = ( n !!2 − n +1)2 ( − π ) − n +12 X ) if n + 1 is even0 if n + 1 is odd.In the case of surfaces, the trace formula was originally demonstrated byGuillop´e and Zworski [46]; that result is somewhat more general, since theyallow the surface to have cusps.It is interesting to note that the remainder in this Selberg trace formulahas markedly different properties from the remainder in the DuistermaatGuillemin trace formula for closed manifolds [24]. Specifically, there areexamples of convex co-compact hyperbolic manifolds whose geodesic flowhas topological entropy h > n ; see for example Canary, Minsky, and Tay-lor [18] who construct examples of hyperbolic 3-manifolds. By the primeorbit theorem for these manifolds [76], Lemma 4.1, and a straightforwardestimate, the renormalized wave trace lies only in D ′ ((0 , ∞ )) rather than S ′ ((0 , ∞ )). On the other hand, the wave trace on a compact manifold isalways a Schwartz distribution.Borthwick’s Poisson formula [12] was the key to our resonance wave traceformula [82]. Corollary 5.4. —
Let ( X, g ) be an n + 1 dimension conformally com-pact manifold hyperbolic near infinity with negative sectional curvatures. ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS Then, we have the distributional equality in D ′ ((0 , ∞ )) , (5.5)12 X s ∈R m ( s ) e ( s − n/ t + 12 ∞ X k =1 d k e − kt = X γ ∈L p ∞ X k =1 l ( γ ) δ ( t − kl ( γ )) q | det( I − P kγ ) | + C ( t ) . Above, the numbers d k are determined by natural conformal operatorsacting on the conformal infinity; see [42] . The long time asymptotics of theremainder are given by (5.2). Our final and most interesting result in [82] is a quantitative relationshipbetween the topological entropy of the geodesic flow and the existence ofpure point spectrum, in the spirit of Patterson [73] and Phillips [78]. Thekey ingredients in the proof are the long-time remainder estimate in thetrace formula and a Littlewood counting estimate used in analytic numbertheory; see for example Rubinstein and Sarnak [83], Phillips and Rudnik[79], and Karnauk [56]. The idea was inspired by Jakobson, Polterovich,and Toth [53] who used this counting technique (which they called the
Dirichlet box principle ) to estimate the remainder in Weyl’s law on surfaceswith pinched negative curvature.
Corollary 5.6. —
Let ( X, g ) be an n + 1 dimension conformally com-pact manifold hyperbolic near infinity with negative sectional curvaturesand topological entropy h for the geodesic flow. Let < k k besuch that the sectional curvatures κ satisfy − k κ − k . If h > nk , then σ pp (∆) = ∅ , and moreover, there is Λ = s ( n − s ) ∈ σ pp (∆) with s > h + n (1 − k ) / . Conversely, if h nk , then σ pp (∆) = ∅ . The contrapositive of the corollary implies σ pp (∆) = ∅ ⇒ h > nk , σ pp (∆) = ∅ ⇒ h nk . Bibliographical notes
The relationship between the pure point spectrum and the topologicalentropy of the geodesic flow in [82] is not sharp like the constant curvaturecase. It would be interesting to construct examples and study the effectsof varying the metric and the curvature. In the case of conformally com-pact metrics hyperbolic near infinity, our result seems to imply that if the
SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT topological entropy is large, one cannot destroy pure point spectrum byperturbing the metric on a compact set, or that such a perturbation wouldnecessarily change the curvature pinching constants. What are the physicalimplications? For readers interested in the physical aspects, we recommendNaud’s work concerning Riemann surfaces [71].
6. The horizon
During the preparation of this survey, Borthwick and Perry applied ourtrace formulae [82] to obtain inverse scattering results for conformally com-pact manifolds hyperbolic near infinity [15]. Their strongest result is in thecase of surfaces which have a common resonance set and the same modelat infinity; the set of all such surfaces is compact in the C ∞ topology. Inhigher dimensions, they assume the sectional curvatures are negative anddepending on whether the dimension is even or odd, they assume eithera common resonance set or a common scattering phase and obtain com-pactness results. It may be possible to obtain further results via our traceformulae and Corollary 5.6; in any case, there are many open problems forthe spectral theory and dynamics of asymptotically hyperbolic manifolds.We conclude this survey with the following conjectures and ideas. On a compact manifold X with Laplace spectrum { λ k } ∞ k =1 , the wavetrace is formally, (6.1) ∞ X k =1 e i √ λ k t = X γ ∈L p ∞ X k =1 l ( γ ) δ ( | t | − kl ( γ )) q | det( I − P kγ ) | + A ( t ) ∈ S ′ ((0 , ∞ )) . The principle term in the dynamical side,(6.2) X γ ∈L p ∞ X k =1 l ( γ ) δ ( | t | − kl ( γ )) q | det( I − P kγ ) | may have exponential growth depending on the curvature bounds and thetopological entropy of the geodesic flow. Since its sum with the remainderterm A ( t ) lies in S ′ ((0 , ∞ )), the remainder term cannot lie in S ′ ((0 , ∞ )) un-less the principle term (6.2) is also in S ′ ((0 , ∞ )). In contrast, the remainderin the renormalized wave trace for convex co-compact hyperbolic manifolds(5.3) has exponential decay for large t . It is therefore not unreasonable toexpect the following. ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS Conjecture 6.1. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 dimensional manifold with negative sectional curvatures. As a distribu-tional equality in D ′ ((0 , ∞ )) , (6.3) cos( t p ∆ − n /
4) = X γ ∈L p ∞ X k =1 l ( γ ) δ ( t − kl ( γ )) q | det( I − P kγ ) | + A ( t ) . The remainder A is smooth on (0 , ∞ ) , and there exist C , k > whichdepend only on n , k , and k such that | A ( t ) | Ce kt ∀ t > , where the sectional curvatures κ satisfy − k κ − k . We propose a new approach to wave trace remainder estimates using thehigher wave invariants. It is unclear how feasible this approach is, however,it is of independent interest that Zelditch’s results [94] extend to asymp-totically hyperbolic manifolds.On a compact manifold, the singular support of the wave trace for t > t √ ∆) = e ( t ) + X γ e γ ( t ) , where e γ ( t ) ∼ a − ( γ )( t − l ( γ ) + i − + ∞ X k =0 a k ( γ )( t − l ( γ ) + i k log( t − l ( γ ) + i . The wave invariants are the coefficients a k ( γ ) in the singularity expansionof the wave trace. By [55] and [24], the renormalized wave trace for anasymptotically hyperbolic manifold has an analogous singularity expansion0-tr cos( t p ∆ − n / ∼ e ( t ) + X γ e γ ( t ) . The principle wave invariant, a − ( γ ) = l p ( γ ) p | det( I − P γ ) | , where l p ( γ ) is the length of the primitive period of γ . The higher waveinvariants a k for k > non-degenerate if the Poincar´emap P γ at γ is any symplectic sum of non-degenerate elliptic, hyperbolic,or loxodromic parts. The elliptic eigenvalues come in complex conjugatepairs of modulus one, e ± iα j ; the hyperbolic eigenvalues come in inverse SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT pairs of real eigenvalues, e ± λ j (these could also be negative); and the lox-odromic eigenvalues come in 4-tuplets, e ± µ j ± iν j ; where the Floquet expo-nents, α j , λ j , µ j , ν j ∈ R . The non-degeneracy condition is equivalent to theindependence over Q of the Floquet exponents together with π . To studythe wave invariants, it is useful to introduce Fermi normal coordinates ( s, y )along the geodesic γ . Invariant contractions against ∂∂s and against the Ja-cobi eigenfields Y j , Y j , with coefficients given by invariant polynomials inthe components y jk are Fermi-Jacobi polynomials.
The
Floquet invariants, β i = (1 − ρ i ) − , where { ρ i } n +2 i =1 (in dimension n + 1) are the eigenvaluesof P γ .The first main result of [94] is that the wave invariants may be writtenin the form a γk = F k, − ( D ) · Ch ( x ) | x = P γ , where Ch ( x ) = i σ p | det( I − x ) | is a character of the metaplectic representation (with σ a certain Maslovindex), and where F k, − ( D ) is an invariant partial differential operator onthe metaplectic group M p (2( n + 1) , R ) which is canonically fashioned fromthe germ of the metric g at γ . In particular, upon close inspection of [94],we have the following. Note that the statement of the result Theorem 6.2. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold, and let γ be a non-degenerate closed geodesic. Thewave invariants, a k ( γ ) = Z γ I γ ; k ( s ; g ) ds, where I γ ; k satisfies i. I γ ; k ( s, g ) is a homogeneous Fermi-Jacobi-Floquet polynomial of weight − k − in the data { y ij , ˙ y ij , D βs,y g } with | β | k + 4 ; ii. The degree of I γ ; k in the Jacobi field components is at most k + 6 ; iii. At most k + 1 indefinite integrations over γ occur in I γ ; k ; iv. The degree of I γ ; k in the Floquet invariants β j is at most k + 2 .Remark 6.4. — The exact expression for the wave invariants is givenin § k and applyingLemma 4.1 is one approach to Conjecture 6.1.The wave invariants at a closed geodesic γ are expressed by Guillemin[43], [44] and Zelditch [93], [94] as non-commutative residues of the wave ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS t = l ( γ ), a γk = res D kt e it √ ∆ | t = l ( γ ) := Res s =0 Tr D kt e it √ ∆ ∆ − s | t = l ( γ ) . Since this residue is invariant under conjugation by microlocal unitaryFourier integral operators, the wave invariants may be calculated by puttingthe wave group into a microlocal quantum Birkhoff normal form around γ and by determining the residues of the resulting wave group of the nor-mal form. Both Guillemin [44] and Zelditch [93] computed the form forelliptic closed geodesics; subsequently, Zelditch [94] computed the form fornon-degenerate closed geodesics. In this case, the form includes transverseelliptic harmonic oscillators ˆ I ej = ( D y j + y j ), real hyperbolic action oper-ators ˆ I hj , and loxodromic action operators ˆ I ch,Rej , ˆ I ch,Imj . Theorem 6.3. —
There exists a microlocally elliptic Fourier integraloperator W from the conic neighborhood of R + γ in T ∗ ( N γ ) to the corre-sponding cone in T ∗ + S in T ∗ ( S × R n +1 ) such that W √ ∆ W − ≡ D s + 1 l ( γ ) p X j =1 α j ˆ I ej + q X j =1 λ j ˆ I hj + c X j =1 µ j ˆ I ch,Rej + ν j ˆ I ch,Imj + p ( ˆ I e , . . . , ˆ I ep , ˆ I h , . . . , ˆ I hq , ˆ I ch,Re , ˆ I ch,Im , . . . , ˆ I ch,Rec , ˆ I ch,Imc ) D s + . . . + p k +1 ( ˆ I e , . . . , ˆ I ch,Imc ) D ks + . . . where the numerators p l ( ˆ I e , . . . , ˆ I ep , ˆ I h , . . . , ˆ I ch,Imc ) are polynomials of de-gree l + 1 in the variables ( ˆ I e , . . . , ˆ I ch,Imc ) , and the k th remainder term liesin the space ⊕ k +2 j =0 O k +2 − j ) Ψ − j . Here, O m Ψ r is the space of pseudodiffer-ential operators of order r whose complete symbols vanish to order m at ( y, η ) = (0 , . Since the preceding results imply that the wave invariants at a non-degenerate closed geodesic determine, and conversely are determined bythe Birkhoff normal form of the Laplacian, we have the following.
Theorem 6.4. —
Let ( X, g ) be an asymptotically hyperbolic n + 1 di-mensional manifold, and let γ be a non-degenerate closed geodesic so thatthe only closed geodesics of the same length are γ, γ − . Then the entirequantum Birkhoff normal form around γ (and hence the classical Birkhoffnormal form) is determined by the wave invariants of γ and its iterates,and conversely, the wave invariants of γ and its iterates are determined bythe quantum normal form coefficients. SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
Proofs:
To prove Theorems 6.2–6.4, it suffices to recall that by [55], theclosed geodesics of an asymptotically hyperbolic manifold lie in a compactsubset. The proofs then follow immediately from the proofs of TheoremI, Theorem B, and Theorem II of [94]. Rather than reproducing these ar-guments, we refer the reader to [94]. In fact, if (
X, g ) is any Riemannianmanifold of dimension n + 1, and γ is a non-degenerate closed geodesiccontained entirely in the interior of X , these results also hold. The inverse spectral results in Guillemin [43], [44]; and Zelditch [93],and [94] use a characterization of the wave invariants via the spectral zetafunction. Recall that on a compact Riemannian manifold (
X, g ) with non-zero Laplace spectrum { λ k } ∞ k =1 , ζ ( s ) := ∞ X k =1 λ − sk . By the Weyl asymptotic formula, the zeta function is well defined andholomorphic when Re( s ) ≫
0. Based on the relationship between ζ andthe heat kernel via the Mellin transform (see, for example [81]), it followsthat ζ extends to a meromorphic function on C . Moreover, s = 0 is aregular value which is used to define the zeta regularized determinant ofthe Laplacian, det ∆ := e − ζ ′ (0) . Guillemin [44] introduced the the zeta function ζ l ( s ) := tr( e il √ ∆ )∆ s , which he used to characterize the wave invariants. Guillemin’s zeta func-tion is well defined and holomorphic when Re( − s ) ≫ C with poles at {− , } ∪ N . Zelditch observedthat the wave trace invariants are the residues of this zeta function at itspoles; this observation played a key role in the proofs of both Guillemin’s[44] and Zelditch’s [93], [94] inverse spectral results.On an asymptotically hyperbolic manifold, resonances play the role of theeigenvalues { λ k } . The “scattering resonance set” defined in § R sc := { s ∈ R} ∪ ∞ [ k =1 n n − k with multiplicity d k o . ANNALES DE L’INSTITUT FOURIER
N THE SPECTRAL THEORY AND DYNAMICS OF A.H. MANIFOLDS ζ sc ( s ) := X λ ∈R sc , λ = n n − λ ) s . A preliminary step in the study of the spectral zeta function and its relationto wave invariants for asymptotically hyperbolic manifolds is the following.
Lemma 6.5. —
Let ( X, g ) be a conformally compact manifold hyper-bolic near infinity of dimension n + 1 . Then, the scattering zeta functionconverges absolutely for Re ( s ) > n + 1 . Proof:
The proof follows immediately from Borthwick’s counting esti-mate [12], { λ ∈ R sc : | λ | r } = O ( r n +1 ) . One may also use the 0-regularized integral to define a “wave zeta func-tion” in the spirit of Guillemin [44], ζ w ( l, s ) := 0-tr( e il √ ∆ )∆ s . In light of Guillarmou’s Weyl law [40], Borthwick’s counting estimates andpreliminary results for the “relative zeta function” [12] which is related toM¨uller’s spectral zeta function [70], it is not unreasonable to expect resultsin the spirit of M¨uller generalize to conformally compact manifolds hyper-bolic near infinity. It may also be possible to extend the inverse spectralresults for the wave invariants [44] to these spaces via the wave zeta func-tion. These spectral zeta functions, the dynamical zeta functions, countingestimates for asymptotically hyperbolic manifolds, the Poisson formula forasymptotically hyperbolic manifolds, and the dynamical theory of confor-mally compact manifolds are some of many open problems to be explored.
Acknowledgments
I would like to thank G´erard Besson and the organizers of the Interna-tional Conference on Spectral Theory and Geometry, June 1–5, 2009 atthe Institut Fourier. I am also grateful to Yves Colin de Verdi`ere, Vin-cent Grandjean, Werner M¨uller, Peter Sarnak, Samuel Tapie, and SteveZelditch for discussions and correspondence, and to the anonymous refereefor constructive comments.
SUBMITTED ARTICLE : ARXIV˙VERSION˙GRENOBLE4.TEX JULIE ROWLETT
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