Lens rigidity for manifolds with hyperbolic trapped set
aa r X i v : . [ m a t h . A P ] D ec LENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLICTRAPPED SET
COLIN GUILLARMOU
Abstract.
For a Riemannian manifold (
M, g ) with strictly convex boundary ∂M ,the lens data consists in the set of lengths of geodesics γ with endpoints on ∂M , to-gether with their endpoints ( x − , x + ) ∈ ∂M × ∂M and tangent exit vectors ( v − , v + ) ∈ T x − M × T x + M . We show deformation lens rigidity for manifolds with hyperbolictrapped set and no conjugate points, a class which contains all manifolds with nega-tive curvature and strictly convex boundary, including those with non-trivial topologyand trapped geodesics. For the same class of manifolds in dimension 2, we prove thatthe set of endpoints and exit vectors of geodesics (ie. the scattering data) determinesthe Riemann surface up to conformal diffeomorphism. Introduction
In this work, we study a geometric inverse problem concerning the recovery of aRiemannian manifold (
M, g ) with boundary from informations about its geodesic flowwhich can be read at the boundary. Different aspects of this problem have beenextensively studied by [Mu, Mi, Cr1, Ot, Sh, PeUh, StUh1, BuIv, CrHe], among others.It also has applications to applied inverse problems, in geophysics and tomography.Our results concern the case of negatively curved manifolds with convex boundariesand more generally manifolds with hyperbolic trapped sets and no conjugate points.In those settings we resolve the deformation lens rigidity problem in all dimensionsand in dimension 2 we show that the lens data (and actually the scattering data)determine the Riemann surface up to conformal diffeomorphism. The difference withmost previous works is allowing trapping and non-trivial topology; we obtain the firstgeneral results in that case. With this aim in view, we introduce new methods making asystematic use of recent analytic methods introduced in hyperbolic dynamical systems[Li, FaSj, FaTs, DyZw, DyGu2].1.1.
Negative curvature.
Let (
M, g ) be an n -dimensional oriented compact Rie-mannian manifolds with strictly convex boundary ∂M (ie. the second fundamentalform is positive). The incoming (-) and outgoing (+) boundaries of the unit tangentbundle of M are denoted ∂ ± SM := { ( x, v ) ∈ T M ; x ∈ ∂M, | v | g x = 1 , ∓ g x ( v, ν ) > } where ν is the inward pointing unit normal vector field to ∂M . For all ( x, v ) ∈ ∂ − SM ,the geodesic γ ( x,v ) with initial point x and tangent vector v has either infinite length orit exits M at a boundary point x ′ ∈ ∂M with tangent vector v ′ with ( x ′ , v ′ ) ∈ ∂ + SM .We call ℓ g ( x, v ) ∈ [0 , ∞ ] the length of this geodesic, and if Γ − ⊂ ∂ − SM denotes theset of ( x, v ) ∈ ∂ − SM with ℓ g ( x, v ) = ∞ , we call S g ( x, v ) := ( x ′ , v ′ ) ∈ ∂ + SM the exitpair or scattering image of ( x, v ) when ( x, v ) / ∈ Γ − . This defines the length map and scattering map ℓ g : ∂ − SM → [0 , ∞ ] , S g : ∂ − SM \ Γ − → ∂ + SM. (1.1)and the lens data is the pair ( ℓ g , S g ). The lens data do not (a priori) contain informationon closed geodesics of M , neither do they on geodesics not intersecting ∂M .If ( M, g ) and ( M ′ , g ′ ) are two Riemannian manifolds with the same boundary N and g | T N = g ′ | T N , there is a natural identification between ∂ − SM and ∂ − SM ′ since ∂ − SM can be identified with the boundary ball bundle BN := { ( x, v ) ∈ T N ; | v | g < } viathe orthogonal projection ∂SM → BN with respect to g (and similarly for ( M ′ , g ′ )).The lens rigidity problem consists in showing that, if ( M, g ) and ( M ′ , g ′ ) are twoRiemannian manifold metrics with strictly convex boundary and ∂M = ∂M ′ , then ℓ g = ℓ g ′ , S g = S g ′ = ⇒ ∃ φ ∈ Diff( M ′ ; M ) , φ ∗ g = g ′ , φ | ∂M ′ = Id . (1.2)When ( ℓ g , S g ) = ( ℓ g ′ , S g ′ ), we say that ( M, g ) and ( M ′ , g ′ ) are lens equivalent , while if S g = S g ′ we say that they are scattering equivalent .Our first result is a deformation lens rigidity statement which holds in any dimension(this follows from Theorem 4 below): Theorem 1.
For s ∈ ( − , , let ( M, g s ) be a smooth -parameter family of metricswith negative curvature on a smooth n -dimensional manifold M with strictly convexboundary, and assume that g s is lens equivalent to g for all s , then there exists afamily of diffeomorphisms φ s which are equal to Id at ∂M and with φ ∗ s g = g s . In dimension 2, we show that the scattering data determine the conformal structure(this is a corollary of Theorem 3 below):
Theorem 2.
Let ( M, g ) and ( M ′ , g ′ ) be two oriented negatively curved Riemanniansurfaces with strictly convex boundary such that ∂M = ∂M ′ and g | T ∂M = g ′ | T ∂M ′ . If ( M, g ) and ( M ′ , g ′ ) are scattering equivalent, then there is a diffeomorphism φ : M → M ′ such that φ ∗ g ′ = e ω g for some ω ∈ C ∞ ( M ) and φ | ∂M = Id , ω | ∂M = 0 . In the special case of simple manifolds, these results correspond to the much studiedboundary rigidity problem, which consists in determining a metric (up to a diffeo-morphism which is the identity on ∂M ) on an n -dimensional Riemannian manifold( M, g ) with boundary ∂M from the distance function d g : M × M → R restricted to ∂M × ∂M . A simple manifold is a manifold with strictly convex boundary such that ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 3 the exponential map exp x : exp − x ( M ) → M is a diffeomorphism at all points x ∈ M .Such manifolds have no conjugate points and no trapped geodesics (ie. geodesics en-tirely contained in M ◦ := M \ ∂M ), and between two boundary points x, x ′ ∈ ∂M there is a unique geodesic in M with endpoints x, x ′ . Boundary rigidity for simplemetrics was conjectured by Michel [Mi] and has been proved in some cases:1) If ( M, g ) and (
M, g ′ ) are conformal and lens equivalent simple manifolds, they areisometric; this is shown by Mukhometov-Romanov, Croke [Mu, MuRo, Cr2].2) If ( M, g ) and ( M ′ , g ′ ) are lens equivalent simple surfaces ( n = 2), they are isometric.This was proved by Otal [Ot] in negative curvature and by Croke [Cr1] in non-positivecurvature. For general simple metrics, Pestov-Uhlmann [PeUh] proved that the scat-tering data determine the conformal class and, combined with 1), this shows Michel’sconjecture for n = 2.3) If g and g ′ are simple metrics that are close enough to a given simple analytic metric g , and are lens equivalent, then they are isometric. This was proved by Stefanov-Uhlmann [StUh1]. All metrics C -close to a flat metric g on a smooth domain of R n is boundary rigid, this was proved by Burago-Ivanov [BuIv].4) A 1-parameter smooth family of simple non-positive curved metrics with same lensdata are all isometric, this was shown by Croke-Sharafutdinov [CrSh].Thus, Theorem 2 is similar to Pestov-Uhlmann’s result in 2) for a class of non-simplesurfaces and Theorem 1 extends 4). We emphasize that in our case, there are typi-cally infinitely many trapped geodesics (and closed geodesics) and this provides thefirst general rigidity result in presence of trapping. In fact, when there are trappedgeodesics or when the flow has conjugate points, there exist lens equivalent metricswhich are not isometric, see Croke [Cr2] and Croke-Kleiner [CrKl]. So far, only resultsof lens rigidity in very particular cases were proved in case of trapped geodesics:5) Croke-Herreros [CrHe] proved that a 2-dimensional negatively curved or flat cylinderwith convex boundary is lens rigid. Croke [Cr3] showed that the flat product metricon B n × S is scattering rigid if B n is the unit ball in R n .6) Stefanov-Uhlmann-Vasy [SUV] proved that the lens data near ∂M determine themetric near ∂M for metrics in a fixed conformal class, and more generally they recoverthe metric outside the convex core of M under convex foliations assumptions.7) For the flat metric on R n \ O where O is a union of strictly convex domains, Noakes-Stoyanov [NoSt] show that the lens data for the billiard flow on R n \ O determine O .If SM = { ( x, v ) ∈ T M ; | v | g x } is the unit tangent bundle and SM ◦ its interior, the trapped set K ⊂ SM ◦ of the geodesic flow is the set of points ( x, v ) ∈ SM ◦ suchthat the geodesic passing through x and tangent to v does not intersect the boundary ∂SM ; K is a closed flow-invariant subset of SM ◦ which includes all closed geodesics.In results 5) above, the trapped set has a simple structure, it is either two disjoint COLIN GUILLARMOU closed geodesics or an explicit smooth submanifold; in 6), it can be anything but theresult allows only to determine the metric near ∂M , which is the region of M with notrapped geodesics. In comparison, in our case (in Theorem 1 and 2), the trapped set istypically a complicated fractal set. For instance, in constant negative curvature theyhave Hausdorff dimension given in terms of the convergence exponent of the Poincar´eseries for the fundamental group (see [Su]).1.2. More general results.
As mentioned above, the results obtained in negativecurvature are particular cases of more general theorems. For t ∈ R , we denote by ϕ t the geodesic flow at time t on SM , ie. ϕ t ( x, v ) = ( x ( t ) , v ( t )) where x ( t ) is the pointat distance t on the geodesic generated by ( x, v ) and v ( t ) = ˙ x ( t ) the tangent vector.We say that the trapped set K is a hyperbolic set if there exists C > ν > y = ( x, v ) ∈ K , there is a continuous flow-invariant splitting T y ( SM ) = R X ( y ) ⊕ E u ( y ) ⊕ E s ( y ) (1.3)where E s ( y ) and E u ( y ) are vector subspaces satisfying || dϕ t ( y ) w || ≤ Ce − νt || w || , ∀ t > , ∀ w ∈ E s ( y ) , || dϕ t ( y ) w || ≤ Ce − ν | t | || w || , ∀ t < , ∀ w ∈ E u ( y ) . (1.4)Here the norm is the Sasaki norm on SM induced by g . This setting is quite naturaland ‘interpolates’ between the simple domain case (open, no trapped set) and theAnosov case (closed manifolds with hyperbolic geodesic flow). Negative curvaturenear the trapped set implies that K is a hyperbolic set, see [Kl2, § Theorem 3.
Let ( M, g ) and ( M ′ , g ′ ) be two oriented Riemannian surfaces with strictlyconvex boundary such that ∂M = ∂M ′ and g | T ∂M = g ′ | T ∂M ′ . Assume that the trappedset of g and g ′ are hyperbolic and that the metrics have no conjugate points. If ( M, g ) and ( M ′ , g ′ ) are scattering equivalent, then there is a diffeomorphism φ : M → M ′ such that φ ∗ g ′ = e ω g for some ω ∈ C ∞ ( M ) and φ | ∂M = Id , ω | ∂M = 0 . In all dimension we obtain a deformation rigidity result:
Theorem 4.
Let M be a smooth compact manifold with boundary, equipped with asmooth -parameter family of lens equivalent metrics g s for s ∈ ( − , and assumethat ∂M is strictly convex for g s for each s . Suppose that, for all s , g s have hyperbolictrapped set.1) If for all s , g s is conformal to g and has no conjugate points, then g s = g .2) If g s has non-positive curvature, then there exists a family of diffeomorphisms φ s which are equal to Id at ∂M and with φ ∗ s g = g s . ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 5
Theorem 2 and 1 follow from these results: negatively curved metrics satisfy theassumptions of both theorems since these have no conjugate points. Hyperbolicity of K is a stable condition by small perturbations of the metric, and there is structuralstability of hyperbolic sets for flows (see [HaKa, Chapter 18.2] and [Ro]), which justifiesthe study of infinitesimal rigidity in that class of metrics. Other natural examples ofsuch manifolds are strictly convex subset of closed manifold with Anosov geodesicflows.1.3. X-ray transform and Livsic type theorem.
One of the main tools for provingthe results above is a precise analysis of the X-ray transform on tensors for manifoldswith hyperbolic trapped set and no conjugate points. The X -ray transform of a func-tion f on M is defined to be the set of integrals of f along all possible geodesics withendpoints in ∂M , this is described by the operator I : C ∞ ( M ) → C ∞ ( ∂ − SM \ Γ − ) , I f ( x, v ) = Z ℓ g ( x,v )0 f ( π ( ϕ t ( x, v ))) dt where π : SM → M is the projection on the base. We prove injectivity of I : Theorem 5.
Let ( M, g ) be a Riemannian surface with strictly convex boundary, hy-perbolic trapped set and no conjugate points. Then for each p > , the operator I : L p ( M ) → L ( ∂ − SM ) is bounded and injective. We prove a similar theorem for the X-ray transform on 1-forms, and when the curva-ture is non-positive, for m -symmetric tensors (see Theorem 6 for a precise statement).We also obtain surjectivity of I ∗ and prove that I ∗ I is an elliptic pseudo-differentialoperator. An important aspect of our analysis that is somehow surprising is that, eventhough the flow has trapped trajectories, the X-ray transform still fits into a Fredholmtype problem like it does for simple domains. The main tool to show injectivity of I isa Livsic theorem of a new type. Indeed, a H¨older Livsic theorem exists on the trappedset [HaKa, Th. 19.2.4] but this is not very useful for our purpose. The result we needand prove in Proposition 5.5 is the following: if f ∈ C ∞ ( SM ) integrates to 0 alongall geodesics relating boundary points of M , then there exists u ∈ C ∞ ( SM ) satisfying Xu = f and u | ∂SM = 0. The method to prove this uses strongly the hyperbolicity of K , and a novelty here is that we make use of the theory of anisotropic Sobolev spacesadapted to the dynamic, which appeared recently in the field of hyperbolic dynamicalsystems (typically on Anosov flows [BuLi, FaSj]) and exponential decay of correlations[Li]. To perform this analysis, we use microlocal tools developed recently in jointwork with Dyatlov [DyGu2] for Axiom A type dynamical systems. Another impor-tance of this method is that it should give local uniqueness and stability estimates inany dimension for the boundary distance function in the universal cover (combiningwith methods of [StUh1, StUh3]) and allow to deal with more general questions, likeattenuated ray transform. COLIN GUILLARMOU
We also notice that a byproduct of Theorem 5 (using [DKLS, Th. 1.1]) is theexistence of many new examples with non-trivial topology and complicated trappedset where the Calder´on problem can be solved in a conformal class.1.4.
Comments.
1) First, we notice that the assumption g = g ′ on T ∂M in Theorem3 is not a serious one and could be removed by standard arguments since, by [LSU],the length function near ∂ SM := { ( x, v ) ∈ ∂SM ; h ν, v i = 0 } determines the metricon T ∂M (we would then have to change slightly the definition of S g , as in [StUh3]).2) A part of this work (in particular Section 4.3) deals with very general assumptions(no hyperbolicity assumption on K and no assumptions on conjugate point) to describesolutions of the boundary value problems for transport equations in SM .3) Contrary to the simple metric setting, the lens equivalence between two met-rics does not induce a conjugation of geodesic flows, which makes the problem moredifficult.4) As pointed out to me by M. Salo, Theorem 5 is sharp in the sense that if thereexists a flat cylinder C = (( − ǫ, ǫ ) τ × ( R /a Z ) θ , dτ + dθ ) (with a >
0) embedded ina surface with strictly convex boundary, then it is easy to check that ker I is infinitedimensional and contains all functions f compactly supported in C , depending only on τ with R ǫ − ǫ f ( τ ) dτ = 0. In this case the trapped is of course not hyperbolic.5) To prove Theorem 3, we show that the scattering map S g determines the spaceof boundary values of holomorphic functions on any surface with hyperbolic trappedset, no conjugate points. This result was first shown by Pestov-Uhlmann [PeUh] inthe case of simple domains. We use their commutator relation between flow andfiberwise Hilbert transform, but we emphasize that due to trapping, several importantaspects of their proof relating scattering map and boundary values of holomorphicfunctions are much more difficult to implement. To obtain the desired result, we needto address delicate questions which are absent in the non-trapping case: we need tosolve boundary value problems for the transport equations in low regularity spacesand understand the wavefront set of solutions, we need to describe boundary values ofinvariant distributions in SM with certain regularity only in terms of the scatteringmap S g , we also need to prove injectivity of X-ray transform on 1-forms in certainnegative Sobolev spaces. The use of the recent joint paper with Dyatlov [DyGu2] isfundamental, and hyperbolicity of the flow on K is very important to address theseproblems. The space of boundary values of holomorphic functions allows to recover( M, g ) up to a conformal diffeomorphism by the result of Belishev [Be]. We are notable to prove that the lens data determine the conformal factor. We think that it doesbut it is not an easy matter: indeed, all proofs known in the simple domain case seemto fail in our setting due to the fact that there is an infinite set of geodesics betweentwo given boundary points and the problem is that we do not know if the geodesics
ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 7 starting at ( x, v ) ∈ ∂ − SM for lens equivalent conformal metrics g ′ = e ω g and g arehomotopic. The difficulty of this question is related to the fact that small perturbationsof the metric induce large perturbations for the geodesics passing through a fixed( x, v ) ∈ ∂ − SM if ℓ g ( x, v ) is large, thus allowing for huge changes of the homotopyclass to which the geodesic belongs. Ackowledgements.
We thank particularly S. Dyatlov for the work [DyGu2], whichis fundamentally used here. Thanks also to V. Baladi, S. Gou¨ezel, M. Mazzucchelli,F. Monard, V. Millot, F. Naud, G. Paternain, S. Tapie, G. Uhlmann, M. Zworskifor useful discussions and comments. The research is partially supported by grantsANR-13-BS01-0007-01 and ANR-13-JS01-0006.2.
Geometric setting and dynamical properties
Extension of SM and the flow into a larger manifold. It is convenient toview (
M, g ) as a strictly convex region of a larger smooth manifold ( ˆ
M , ˆ g ) with strictlyconvex boundary, and to extend the geodesic vector field X on SM into a vector field X on S ˆ M which has complete flow, for instance by making X vanish at ∂S ˆ M .Let us describe this construction. Near the boundary ∂M , let ( ρ, z ) be normalcoordinates to the boundary, ie. ρ is the distance function to ∂M satisfying | dρ | g = 1near ∂M and z are coordinates on ∂M . The metric then becomes g = dρ + h ρ ina collar neighborhood [0 , δ ] ρ × ∂M of ∂M for some smooth 1-parameter family h ρ ofmetrics on ∂M and the strict convexity condition means that the second fundamentalform − ∂ ρ h ρ | ρ =0 is a positive definite symmetric cotensor. We extend smoothly h ρ from ρ ∈ [0 , δ ] to ρ ∈ [ − , δ ] as a family of metrics on ∂M satisfying − ∂ ρ h ρ > ρ ∈ [ − , M as a strictly convex region inside a larger manifold M e with strictly convex boundary as follows. First, let E = ∂M × [ − , ρ be theclosed cylindrical manifold, and consider the connected sum ˆ M := M ⊔ E where weglue the boundary { ρ = 0 } ≃ ∂M of E to the boundary ∂M of M ; then we put asmooth structure of manifold with boundary on ˆ M extending the smooth structure of M , we extend the metric g smoothly from M to ˆ M by setting ˆ g = dρ + h ρ in E . Eachhypersurface { ρ = c } with c ∈ [ − ,
0] is strictly convex. We now set the extension M e := { y ∈ ˆ M ; y ∈ M or y ∈ E and ρ ( y ) ∈ [ − ǫ, } of M for ǫ > M e , g ) is a manifold with strictly convex boundarycontaining M and contained in ˆ M . It is easily checked that the longest connectedgeodesic ray in SM e \ SM ◦ has length bounded by some L < ∞ . When ( M, g ) hasno conjugate point and hyperbolic trapped set, it is possible to choose ǫ small enoughso that ( M e , g ) has no conjugate point either (see Section 2.3), and we will do so eachtime we shall assume that ( M, g ) has no conjugate point. We denote by X the geodesicvector field on the unit tangent bundle S ˆ M of ˆ M with respect to the extended metric COLIN GUILLARMOU g . Let us define ρ ∈ C ∞ ( ˆ M ) so that near E , ρ = F ( ρ ) is a smooth nondecreasingfunction of ρ satisfying F ( ρ ) = ρ + 1 near ρ = −
1, and so that { ρ = 1 } = M e .Denote by π : S ˆ M → ˆ M the projection on the base, then the rescaled vector field X := π ∗ ( ρ ) X on S ˆ M has the same integral curves as X , it is complete and X = X in the neighborhood SM e of SM . The flow at time t of X is denoted ϕ t , and by strictconvexity of M (resp. M e ) in ˆ M , ϕ t is also the flow of X in the sense that for all y in SM (resp. in SM e ) one has ∂ t ϕ t ( y ) = X ( ϕ t ( y )) for t ∈ [0 , t ] as long as ϕ t ( y ) ∈ SM (resp. ϕ t ( y ) ∈ SM e ).We shall denote M ◦ and M ◦ e for the interior of M and M e .2.2. Incoming/outgoing tails and trapped set.
We define the incoming (-), out-going (+) and tangent (0) boundaries of SM and SM e ∂ ∓ SM := { ( x, v ) ∈ ∂SM ; ± dρ ( X ) > } , ∂ ∓ SM e := { ( x, v ) ∈ ∂SM e ; ± dρ ( X ) > } ,∂ SM = { ( x, v ) ∈ ∂SM ; dρ ( X ) = 0 } , ∂ SM e = { ( x, v ) ∈ ∂SM ; dρ ( X ) = 0 } . For each point ( x, v ) ∈ SM , define the time of escape of SM in positive (+) andnegative (-) time: ℓ + ( x, v ) = sup { t ≥ ϕ t ( x, v ) ∈ SM } ⊂ [0 , + ∞ ] ,ℓ − ( x, v ) = inf { t ≤ ϕ t ( x, v ) ∈ SM } ⊂ [ −∞ , . (2.1) Definition 2.1.
The incoming (-) and outgoing (+) tail in SM are defined by Γ ∓ = { ( x, v ) ∈ SM ; ℓ ± ( x, v ) = ±∞} = \ t ≥ ϕ ∓ t ( SM ) and the trapped set for the flow on SM is the set K := Γ + ∩ Γ − = \ t ∈ R ϕ t ( SM ) . (2.2)We note that Γ ± and K are closed set and that K is globally invariant by the flow.By the strict convexity of ∂M , the set K is a compact subset of SM ◦ since for all( x, v ) ∈ ∂SM , ϕ t ( x, v ) ∈ S ˆ M \ SM for either all t > t < ± are characterized by y ∈ Γ ± ⇐⇒ d ( ϕ t ( y ) , K ) → t → ∓∞ (2.3)where d ( · , · ) is the distance induced by the Sasaki metric. We then extend Γ ± to S ˆ M by using the characterization (2.3); the sets Γ ± are closed flow invariant subsetsof the interior S ˆ M ◦ of S ˆ M . By strict convexity of the hypersurfaces { ρ = c } with c ∈ ( − , y ∈ S ˆ M with ρ ( y ) ∈ ( − ,
0] is such that d ( ϕ t ( y ) , ∂S ˆ M ) → t → + ∞ or t → −∞ , and thus for all c ∈ (0 , K = \ t ∈ R ϕ t ( { ρ ≥ c } ) = \ t ∈ R ϕ t ( SM e ) . ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 9
Figure 1.
The manifold SM and SM e We also remark that the strict convexity of ∂M and ∂M e impliesΓ ∓ ∩ ∂SM = Γ ∓ ∩ ∂ ∓ SM, Γ ∓ ∩ ∂SM e = Γ ∓ ∩ ∂ ∓ SM e . (2.4)Using the flow invariance of Liouville measure in SM e , it is direct to check that (seethe proof of Theorem 1 in [DyGu1, Section 5.1])Vol( K ) = 0 ⇐⇒ Vol( SM e ∩ (Γ − ∪ Γ + )) = 0 . (2.5)where the volume is taken with respect to the Liouville measure.The hyperbolicity of the trapped set K is defined in the Introduction, and there isa flow-invariant continuous splitting of T ∗ K ( SM ) dual to (1.3), defined as follows: forall y ∈ K , T ∗ y ( SM ) = E ∗ ( y ) ⊕ E ∗ s ( y ) ⊕ E ∗ u ( y ) where E ∗ u ( E u ⊕ R X ) = 0 , E ∗ s ( E s ⊕ R X ) = 0 , E ∗ ( E u ⊕ E s ) = 0 . We note that E ∗ = R α where α is the Liouville 1-form.2.3. Stable and unstable manifolds.
Let us recall a few properties of flows withhyperbolic invariant sets, we refer to Hirsch-Palis-Pugh-Shub [HPPS, Sec 5 and 6],Bowen-Ruelle [BoRu] and Katok-Hasselblatt [HaKa, Chapters 17.4, 18.4] for details.For each point y ∈ K , there exist global stable and unstable manifolds W s ( y ) and W u ( y )defined by W s ( y ) := { y ′ ∈ S ˆ M ◦ ; d ( ϕ t ( y ) , ϕ t ( y ′ )) → , t → + ∞} ,W u ( y ) := { y ′ ∈ S ˆ M ◦ ; d ( ϕ t ( y ) , ϕ t ( y ′ )) → , t → −∞} which are smooth injectively immersed connected manifolds. There are local sta-ble/unstable manifolds W ǫs ( y ) ⊂ W s ( y ), W ǫu ( y ) ⊂ W u ( y ) which are properly embedded disks containing y , defined by W ǫs ( y ) := { y ′ ∈ W s ( y ); ∀ t ≥ , d ( ϕ t ( y ) , ϕ t ( y ′ )) ≤ ǫ } ,W ǫu ( y ) := { y ′ ∈ W u ( y ); ∀ t ≥ , d ( ϕ − t ( y ) , ϕ − t ( y ′ )) ≤ ǫ } for some small ǫ > ϕ t ( W ǫs ( y )) ⊂ W ǫs ( ϕ t ( y )) and ϕ − t ( W ǫu ( y )) ⊂ W ǫu ( ϕ − t ( y )) ,T y W ǫs ( y ) = E s ( y ) , and T y W ǫu ( y ) = E u ( y ) , The regularity of W u ( y ) and W s ( y ) with respect to y is H¨older. We also define W s ( K ) := ∪ y ∈ K W s ( y ) , W u ( K ) := ∪ y ∈ K W u ( y ) ,W ǫs ( K ) := ∪ y ∈ K W ǫs ( y ) , W ǫu ( K ) := ∪ y ∈ K W ǫu ( y ) . The incoming/outgoing tails are exactly the global stable/unstable manifolds of K : Lemma 2.2.
If the trapped set K is hyperbolic, then the following equalities hold Γ − = W s ( K ) , Γ + = W u ( K ) . Proof.
By (2.3), W s ( K ) ⊂ Γ − and W u ( K ) ⊂ Γ + . Then W ǫs ( K ) ∩ W ǫu ( K ) ⊂ K , andthus K has a local product structure in the sense of [HaKa, Definition p.272 ]. Nowfrom this local product structure, [HPPS, Lemma 3.2 and Theorem 5.2] show that forany ǫ > V K of K such that { y ∈ SM e ; ϕ t ( y ) ∈ V K , ∀ t ≥ } ⊂ W ǫs ( K ) (2.6)which means that any trajectory which is close enough to K is on the local stablemanifold for t large enough. The same hold for negative time and unstable manifold.A point y ∈ Γ − satisfies d ( ϕ t ( y ) , K ) → t → + ∞ , thus for t large enough the orbitreaches V K and thus ϕ t ( y ) ∈ W ǫs ( K ) for t ≫ y ∈ W s ( K ).Similarly Γ + ⊂ W u ( K ) and this achieves the proof. (cid:3) For each y ∈ K , we extend the notion of stable susbpace, resp. unstable subspace,to points on the W ǫs ( y ) submanifold, resp. W ǫu ( y ) submanifold, by E − ( y ) := T y W ǫs ( y ) if y ∈ W ǫs ( y ) , E + ( y ) := T y W ǫu ( y ) if y ∈ W ǫu ( y ) . These subbundles can be extended to subbundles E ± ⊂ T Γ ± SM e over Γ ± in a flowinvariant way (by using the flow), and we can define the subbundles E ∗± ⊂ T ∗ Γ ± SM e by E ∗± ( E ± ⊕ R X ) = 0 over Γ ± . (2.7)By [DyGu2, Lemma 2.10], these subbundles are continuous, invariant by the flow andsatisfy the following properties (we use Sasaki metric on SM ):1) there exists C > , γ > y ∈ Γ ± and ξ ∈ E ∗± ( y ), then || dϕ − t ( y ) T ξ || ≤ Ce − γ | t | || ξ || , ∓ t > ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 11
2) for ( y, ξ ) ∈ T ∗ Γ ± SM e such that ξ / ∈ E ∗± and ξ ( X ) = 0, then || dϕ − t ( y ) T ξ || → ∞ and dϕ − t ( y ) T ξ || dϕ − t ( y ) T ξ || → E ∗∓ | K as t → ∓∞ , (2.9)3) The bundles E ∗± extend E ∗ s and E ∗ u in the sense that E ∗− | K = E ∗ s and E ∗ + | K = E ∗ u .The dependance of E ∗± ( y ) with respect to y is only H¨older. The bundles E ∗± can bethought of as conormal bundles to Γ ± (this set is a union of smooth leaves parametrizedby the set K which has a fractal nature). The differential of the flow dϕ t is exponen-tially contracting on each fiber E − ( y ), the proof of Klingenberg [Kl, Proposition p.6]shows ϕ t has no conjugate points = ⇒ E − ∩ V = { } if V := ker dπ (2.10)where π : SM → M is the projection on the base. Similarly, E + ∩ V = { } in thatcase. These properties imply Lemma 2.3. If ( M, g ) has hyperbolic trapped set, strictly convex boundary, and noconjugate points, we can choose ǫ > small enough in Section 2.1 so that the extension ( M e , g ) has not conjugate points.Proof. Indeed if it were not the case, there would be (by compactness) a sequence ofpoints ( x n , v n ) ∈ SM e \ SM converging to ( x, v ) ∈ ∂ − SM ∪ ∂ SM and ( x ′ n , v ′ n ) ∈ SM e converging to ( x ′ , v ′ ) ∈ SM , and geodesics γ n passing through ( x n , v n ) and ( x ′ n , v ′ n ),with x n and x ′ n being conjugate points for the flow of the extension of g . Note that( x, v ) = ( x ′ , v ′ ) is prevented by strict convexity of ∂M . By compactness, if the lengthof γ n is bounded, we deduce that x, x ′ are conjugate points on M , which is not possibleby assumption. There remains the case where the length of γ n is not bounded, we cantake a subsequence so that the length t n → + ∞ . Then ( x, v ) ∈ Γ − , and there is w n ∈ V = ker dπ of unit norm for Sasaki metric such that dϕ t n ( x n , v n ) .w n ∈ V . Wecan argue as in the proof of [DyGu2, Lemma 2.11]: by hyperbolicity of the flow on K ,for n large enough, dϕ t n ( x n , v n ) .w n will be in an arbitrarily small conic neighborhoodof E + , thus it cannot be in the vertical bundle V . This completes the argument. (cid:3) Finally, let us denote by ι ± : ∂ ± SM → SM e , ι : ∂SM → SM e (2.11)the inclusion map, and define E ∗ ∂, ± := ( dι ± ) T E ∗± ⊂ T ∗ ( ∂ ± SM ) . (2.12) Escape rate.
An important quantity in the study of open dynamical systemsis the escape rate , which measures the amount of mass not escaping for long time.This quantity was studied for hyperbolic dynamical systems by Bowen-Ruelle, Young[BoRu, Yo]. First we define the non-escaping mass function V ( t ) as follows V ( t ) := Vol( T + ( t )) , with T ± ( t ) := { y ∈ SM ; ϕ ± s ( y ) ∈ SM for s ∈ [0 , t ] } . (2.13)and Vol being the volume with respect to the Liouville measure dµ . The escape rate Q ≤ V ( t ) Q := lim sup t → + ∞ t log V ( t ) . (2.14)Notice that, since ϕ t preserves the Liouville measure in SM , we haveVol( T + ( t )) = Vol( T − ( t ))since the second set is the image of the first set by ϕ t . Consequently, we also have Q = lim sup t → + ∞ t log Vol( T − ( t )) . We define J u the unstable Jacobian of the flow J u ( y ) := − ∂ t (det dϕ t ( y ) | E u ( y ) ) | t =0 where the determinant is defined using the Sasaki metric (to choose orthonormal basesin E u ). The topological pressure of a continuous function f : K → R with respectto ϕ t can be defined by the variational formula P ( f ) := sup ν ∈ Inv( K ) ( h ν ( ϕ ) + R K f dν )where Inv( K ) is the set of ϕ t -invariant Borel probability measures and h ν ( ϕ ) is themeasure theoretic entropy of the flow at time 1 with respect to ν (e.g. P (0) is just thetopological entropy of the flow).We gather two results of Young [Yo, Theorem 4] and Bowen-Ruelle [BoRu, Theo-rem 5] on the escape rate in our setting. Proposition 2.4.
If the trapped set K is hyperbolic, the escape rate Q is negative andgiven by the formula Q = P ( J u ) . (2.15) Proof.
Formula (2.15) is proved by Young [Yo, Theorem 4] and follows directly from thevolume lemma of Bowen-Ruelle [BoRu]. The pressure P ( J u ) of the unstable Jacobian J u for ϕ on K is equal to the pressure P ( J u | Ω ) of J u for ϕ on the non-wandering setΩ ⊂ K of ϕ , see [Wa, Corollary 9.10.1]. By the spectral decomposition of hyperbolicflows [HaKa, Theorem 18.3.1 and Exercise 18.3.7], the non-wandering set Ω decom-poses into finitely many disjoint invariant topologically transitive sets Ω = ∪ Ni =1 Ω i for ϕ . By [HaKa, Corollary 6.4.20], the periodic orbits of the flow are dense in Ω. By[HPPS, Proposition 7.2], each component Ω i of Ω has local product structure, andthus, according to [HaKa, Theorem 18.4.1] (see also [HPPS]), it is locally maximal;each Ω i is a basic set in the sense of Bowen-Ruelle [BoRu]. ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 13
Then we can use the result of Bowen-Ruelle [BoRu, Theorem 5] which gives thefollowing equivalence P ( J u | Ω i ) < ⇐⇒ Ω i is not an attractor for ϕ ⇐⇒ Vol( W s (Ω i )) = 0 . (2.16)where W s (Ω i ) := ∪ y ∈ Ω i W s ( y ) is the stable manifold of Ω i . Suppose that one of thesets Ω i is an attractor, then W s (Ω i ) has positive Liouville measure, implying thatVol(Γ − ) >
0, thus Vol( K ) > K , we have Vol( K ) = Vol(Ω) by [Wa, Theorem 6.15] and thus there is Ω j with positiveLiouville measure. Now we can conclude with the argument of [BoRu, Corollary 5.7]:Vol( W s (Ω j )) > W u (Ω j )) > j is an attractor for both ϕ and ϕ − by (2.16), and this implies that W u (Ω i ) = Ω i (as an attractor of ϕ ) and W u (Ω i ) isopen (as an attractor of ϕ − ), thus Ω j = K = SM since SM is connected. But underour geometric assumption ( ∂M is strictly convex) this is not possible. We concludethat Q = P ( J u | Ω ) < (cid:3) This of course implies that Vol(Γ − ∪ Γ + ) = 0. Near ∂ ± SM , we have { ϕ ∓ t ( y ) ∈ SM ; t ∈ [0 , ǫ ) , y ∈ ∂ ± SM ∩ Γ ± } ⊂ Γ ± and since for U a small open neighborhood of ∂ ± SM ∩ Γ ± the map ( t, y ) ∈ [0 , ǫ ) × U ϕ ∓ t ( y ) ∈ SM is a smooth diffeomorphism onto its image (the vector field X is transverse to ∂ ± SM near Γ ± by (2.4)), we get Vol ∂SM (Γ ± ∩ ∂ ± SM ) = 0; (2.17)where the measure on ∂SM is denoted dµ ∂SM and given by dµ ∂SM ( x, v ) = | dvol h ( x ) ∧ dS x ( v ) | with h = g | ∂M and dS x ( v ) the volume form on the sphere S x M .The flow on SM e shares the same properties as on SM and the trapped set on SM and on SM e are the same, the discussion above holds as well for SM e , and in particular Q = lim sup t → + ∞ t log Vol( { y ∈ SM e ; ϕ ± s ( y ) ∈ SM e for s ∈ [0 , t ] } ) < . (2.18)2.5. Santalo formula.
There is a measure on ∂SM which comes naturally whenconsidering geodesic flow in SM , we denote it dµ ν and it is given by dµ ν ( x, v ) := |h v, ν i| dµ ∂SM ( x, v ) = |h v, ν i| | dvol h ( x ) ∧ dS x ( v ) | . (2.19)where ν is the inward unit normal vector field to ∂M in M . This measures is also equalto | ι ∗ ( i X µ ) | . When Vol(Γ − ∪ Γ + ) = 0, then (2.17) holds and we can apply Santaloformula [Sa] to integrate functions in SM , this gives us: for all f ∈ L ( SM ) Z SM f dµ = Z ∂ − SM \ Γ − Z ℓ + ( x,v )0 f ( ϕ t ( x, v )) dt dµ ν ( x, v ) (2.20) with ℓ + defined in (2.1). Extending f to SM e by 0 in S ˆ M \ SM , (2.20) can also berewritten Z SM f dµ = Z ∂ − SM \ Γ − Z R f ( ϕ t ( x, v )) dt dµ ν ( x, v ) . (2.21)3. The scattering map and lens equivalence
In the setting of a compact Riemannian manifold (
M, g ) with strictly convex bound-ary ∂M , we define the scattering map by S g : ∂ − SM \ Γ − → ∂ + SM \ Γ + , S g ( x, v ) := ϕ ℓ + ( x,v ) ( x, v ) (3.1)where ℓ + ( x, v ) is the length of the geodesic π ( ∪ t ∈ R ϕ t ( x, v ))) ∩ M , as defined in (2.1). Definition 3.1.
Let ( M , g ) and ( M , g ) be two Riemannian manifolds with the sameboundary and such that g = g on T ∂M = T ∂M and the boundary is strictly convexfor both metrics. Let ν i be the inward pointing unit normal vector field on ∂M i and let Γ i − ⊂ SM i the incoming tail of the flow for g i . Let α : ∂SM → ∂SM be given by α ( x, v + tν ) = ( x, v + tν ) , ∀ ( v, t ) ∈ T x ∂M × R , | v | g + t = 1 . (3.2) Then ( M , g ) and ( M , g ) are said scattering equivalent if α (Γ − ) = Γ − , and α ◦ S g = S g ◦ α on ∂SM \ Γ − . Finally g and g are said lens equivalent if they are scattering equivalent and for any ( x, v ) ∈ ∂ − SM \ Γ − , the length ℓ ( x, v ) of the geodesic generated by ( x, v ) in M for g is equal to the length ℓ ( α ( x, v )) of the geodesic generated by α ( x, v ) in M for g . Let us show that for the case of surfaces, if K is hyperbolic and g has no conjugatepoints then S g determines the space E ∗ ∂, ± , this will be useful in Theorem 7 Lemma 3.2.
Let ( M, g ) be a surface with strictly convex boundary. Assume that K is hyperbolic and that the metric has no conjugate points. Then the scattering map S g determines E ∗ ∂, ± .Proof. All points in Γ + ∩ ∂SM are in some unstable leaf W u ( p ) for some p ∈ K . Theunstable leaves are one-dimensional manifolds injectively immersed in SM e and theyintersect ∂SM in a set of measure 0 in ∂SM . Above a point y ∈ W u ( p ) ∩ ∂ − SM ,the fiber E ∗ + ,∂ ( y ) is exactly one-dimensional since one has T y SM = R X ⊕ V ⊕ E + ( y )where V = ker dπ is the vertical bundle which is also tangent to ∂SM and E ∗− ( V ) = 0if there are no conjugate points (we refer the reader to the proof of Proposition 5.7below for the discussion about that fact). Take a point y ∈ W u ( p ) ∩ ∂ + SM and asequence y n → y in ∂ + SM with y n / ∈ Γ + , then by compactness (by possibly passing toa subsequence) z n := S − g ( y n ) is converging to z in Γ − ∩ ∂SM with t n := ℓ + ( z n ) → ∞ .We can write S g ( z n ) = ϕ ℓ + ( z n ) ( z n ). By Lemma 2.11 in [DyGu2] (in particular its ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 15 proof), if ξ n ∈ T ∗ z n SM satisfies ξ n ( X ) = 0 and dist( ξ n / || ξ n || , E ∗− ) > ǫ for some fixed ǫ >
0, then ( dϕ t n ( z n ) − ) T ξ n / || ( dϕ t n ( z n ) − ) T ξ n || tends to E + ( y ) ∗ ∩ S ∗ ( SM ). Then wecompute for w n ∈ T y n ( ∂SM ) dS − g ( y n ) .w n = X ( z n ) dℓ − ( y n ) .w n + dϕ − t n ( y n ) .w n and if ξ n ∈ T ∗ z n ( ∂SM ), we can define uniquely ξ ♯n ∈ T ∗ z n SM by ξ ♯n ( X ) = 0 and ξ ♯n ◦ dι = ξ n ( ι is defined in 2.11) so that ( dS g ( z n ) − ) T ξ n = ( dϕ t n ( z n ) − ) T ξ ♯n . We conclude that( dS g ( z n ) − ) T ξ n / || ( dS g ( z n ) − ) T ξ n || → E ∗ + ( y ) , n → + ∞ if ξ n is such that dist( ξ ♯n / || ξ ♯n || , E ∗− ) > ǫ . We can for instance take ξ n to be of norm 1and in the annulator of V in T ∗ ∂SM , then the desired condition is satisfied and thisshows that we can recover E ∗− ( y ) from S g . The same argument with S − g instead of S g shows that S g determines E ∗ + . This ends the proof. (cid:3) We can define the scattering operator as the pull-back by the inverse scattering map S g : C ∞ c ( ∂ − SM \ Γ − ) → C ∞ c ( ∂ + SM \ Γ + ) , S g ω − = ω − ◦ S − g . (3.3) Lemma 3.3.
For any ω ∓ ∈ C ∞ c ( ∂ ∓ SM \ Γ ∓ ) , there exists a unique function w ∈ C ∞ c ( SM \ (Γ − ∪ Γ + )) satisfying Xw = 0 , w | ∂ ∓ SM = ω ∓ (3.4) and this solution satisfies w | ∂ + SM = S g ω − (resp. w | ∂ − SM = S − g ω + ). The function w extends smoothly to SM e in a way that Xw = 0 , this defines a bounded operator E ∓ : C ∞ c ( ∂ ∓ SM \ Γ ∓ ) → C ∞ ( SM e ) , E ∓ ( ω ∓ ) := w (3.5) which satisfies the identity E + S g = E − .Proof. The function w = E ∓ ( ω ∓ ) is simply given by w ( x, v ) = E ∓ ( ω )( x, v ) = ω ∓ ( ϕ ℓ ∓ ( x,v ) ( x, v )) (3.6)in SM , and is clearly unique in SM since constant on the flow lines. It is smoothin SM since ℓ ± is smooth when restricted to ∂ ± SM \ Γ ± , by the strict convexity of ∂SM . Then E ∓ ( ω ∓ ) can be extended in SM e in a way that it is constant on the flowlines of X , satisfying X E ∓ ( ω ∓ ) = 0. The continuity and linearity of E ± is obvious, andthe identity E + S g = E − comes from uniqueness of w . Notice that supp( E ∓ ( ω ∓ )) is atpositive distance from Γ − ∪ Γ + since ω ∓ has support not intersecting Γ ∓ ∩ ∂SM . (cid:3) Denoting ω ± := ω | ∂ ± SM if ω ∈ C ∞ c ( ∂SM \ (Γ + ∪ Γ − )), we now define the space C ∞ S g ( ∂SM ) := { ω ∈ C ∞ c ( ∂SM \ (Γ + ∪ Γ − )); S g ω − = ω + } . (3.7) Using the strict convexity and fold theory, Pestov-Uhlmann [PeUh, Lemma 1.1.] prove ω ∈ C ∞ S g ( ∂SM ) ⇐⇒ ∃ w ∈ C ∞ c ( SM \ (Γ − ∪ Γ + )) , Xw = 0 , w | ∂SM = ω. (3.8)Similarly to (3.7), we define the space L S g ( ∂SM ) := { ω ∈ L ( ∂SM ; dµ ν ); S g ω − = ω + } . (3.9)We finally show Lemma 3.4.
The map S g extends as a unitary map L ( ∂ − SM, dµ ν ) → L ( ∂ + SM, dµ ν ) where dµ ν is the measure of (2.19) .Proof. Consider w − , w − ∈ C ∞ c ( ∂ − SM \ Γ − ) and w , w their invariant extension as in(3.4). Then we have0 = Z SM Xw .w + w .Xw dµ = Z SM X ( w .w ) dµ = − Z ∂ − SM ω − .ω − |h X, N i S | dµ ∂SM + Z ∂ + SM S g ω − . S g ω − |h X, N i S | dµ ∂SM where h· , ·i S is Sasaki metric and N is the unit inward pointing normal vector field to ∂SM for S . But N is the horizontal lift of ν , and so h X, N i S = h v, ν i g . This showsthat S g extends as an isometry by a density argument and reversing the role of ∂ − SM with ∂ + SM we see that S g is invertible. (cid:3) Resolvent and boundary value problem
Sobolev spaces and microlocal material.
For a closed manifold Y , the L -based Sobolev space of order s ∈ R is denoted H s ( Y ). If Z is a manifold with asmooth boundary, it can be extended smoothly across its boundary as a subset ofa closed manifold Y of the same dimension; we denote by H s ( Z ) for s ≥ L functions on Z which admit an H s extension to Y . The space H s ( Z ) is the closure of C ∞ c ( Z ◦ ) for the H s norm on Y and we denote by H − s ( Y ) the dual of H s ( Y ). We referto Taylor [Ta, Chap. 3-5] for details and precise definitions. If Z is an open manifoldor a manifold with boundary, we set C −∞ ( Z ) to be the set of distributions, definedas the dual of C ∞ c ( Z ◦ ). For α ≥
0, the Banach space C α ( Z ) is the space of α -H¨olderfunctions. We will use the notion of wavefront set of a distribution (see [H¨o, Chap.8]), the calculus of pseudo-differential operators (ΨDO in short), we refer the reader toGrigis-Sj¨ostrand [GrSj] and Zworski [Zw] for a thorough study. In particular, we shall Their result is for simple manifold, but the proof applies here without any problem since this isjust an analysis near ∂ SM where the scattering map has the same behavior as on a simple manifoldby the strict convexity of ∂M . ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 17 say that a pseudo-differential operator A on an open manifold Z with dimension n has support in U ⊂ Z if its Schwartz kernel has support in U × U . The microsupportWF( A ) (or wavefront set) of A is defined as the complement to the set of points( y , ξ ) ∈ T ∗ Z such that there is a small neighborhood U y of y and a cutoff function χ ∈ C ∞ c ( U y ) equal to 1 near y such that A χ := χAχ can be written under the form( U y is identified to an open set of R n using a chart) A χ f ( y ) = Z U y Z R n e i ( y − y ′ ) ξ σ ( y, ξ ) f ( y ′ ) dξdy ′ for some smooth symbol σ satisfying | ∂ αy ∂ βξ σ ( y, ξ ) | ≤ C α,β,N h ξ i − N for all N > α, β ∈ N n .4.2. Resolvent.
We first define the resolvent of the flow in the physical spectral region.
Lemma 4.1.
For λ > , the resolvents R ± ( λ ) : L ( SM e ) → L ( SM e ) defined by thefollowing formula R + ( λ ) f ( y ) = Z ∞ e − λt f ( ϕ t ( y )) dt, R − ( λ ) f ( y ) = − Z −∞ e λt f ( ϕ t ( y )) dt (4.1) are bounded. They satisfy in the distribution sense in SM ◦ e ∀ f ∈ L ( SM e ) , ( − X ± λ ) R ± ( λ ) f = f, ∀ f ∈ H ( SM e ) , R ± ( λ )( − X ± λ ) f = f, (4.2) and we have the adjointness property R − ( λ ) ∗ = − R + ( λ ) on L ( SM e ) , (4.3) The expression (4.1) gives an analytic continuation of R ± ( λ ) to λ ∈ C as operator R ± ( λ ) : C ∞ c ( SM ◦ e \ Γ ∓ ) → C ∞ ( SM e ) (4.4) satisfying ( − X ± λ ) R ± ( λ ) f = f in SM e , and an analytic continuation of R ± ( λ ) χ ± and χ ± R ± ( λ ) as operators R ± ( λ ) χ ± : L ( SM e ) → L ( SM e ) , χ ± R ± ( λ ) : L ( SM e ) → L ( SM e ) (4.5) if χ ± ∈ C ∞ ( SM e ) is supported in SM e \ Γ ∓ .Proof. The proof of (4.2) is straightforward. The boundedness on L follows from theinequality (using Cauchy-Schwarz) Z SM e (cid:12)(cid:12)(cid:12) Z ±∞ e − λ | t | f ( ϕ t ( x, v )) dt (cid:12)(cid:12)(cid:12) dµ ≤ C λ Z SM e Z ±∞ e − Re( λ ) | t | | f ( ϕ t ( x, v )) | dtdµ, for some C λ > λ ), and a change of variable y = ϕ t ( x, v ) with thefact that the flow ϕ t preserves the measure dµ in SM e gives the result. The adjointproperty (4.3) is also a consequence of the invariance of dµ by the flow in SM e . The identity ( − X ± λ ) R ± ( λ ) f = f holds for any f ∈ C ∞ c ( SM ◦ e ), thus for f ∈ L ( SM e ) andany ψ ∈ C ∞ c ( SM ◦ e ) ( h· , ·i is the distribution pairing) h ( − X ± λ ) R ± ( λ ) f, ψ i = h R ± ( λ ) f, ( X ± λ ) ψ i = lim n →∞ h R ± ( λ ) f n , ( X ± λ ) ψ i = lim n →∞ h f n , ψ i if f n → f in L with f n ∈ C ∞ c ( SM e ), thus ( − X ± λ ) R ± ( λ ) f = f in C −∞ ( SM ◦ e ). Theother identity in (4.2) is proved similarly. The analytic continuation of R ± ( λ ) in (4.4)is direct to check by using that the integrals in (4.1) defining R ± ( λ ) f are integrals ona compact set t ∈ [ − T, T ] with T depending on the distance of support of f to Γ ∓ .Similarly, the extension of R ± ( λ ) χ ± f and χ ± R ± ( λ ) f for f ∈ L ( SM e ) comes from thefact that the support of t ( χ ± f )( ϕ t ( x, v )) and of t χ ± ( x, v ) f ( ϕ t ( x, v )) intersect R ± in a compact set which is uniform with respect to ( x, v ) ∈ SM e . (cid:3) We next show that the resolvent at the parameter λ = 0 can be defined if the non-escaping mass function V ( t ) in (2.13) is decaying enough as t → ∞ . Let us first definethe maximal Lyapunov exponent of the flow near Γ − ∪ Γ + : ν max = max( ν + , ν − ) , if ν ± := lim sup t → + ∞ t log sup ( x,v ) ∈T ± ( t ) k dϕ ± t ( x, v ) k . (4.6)where T ± is defined in (2.13). Proposition 4.2.
Let α ∈ (0 , , let Q be a negative real number and let ν max be themaximal Lyapunov exponent defined in (4.6) .1) The family of operators R ± ( λ ) of Lemma 4.1 extends as a continuous family in Re( λ ) ≥ of operators bounded on the spaces R ± ( λ ) : L ∞ ( SM e ) → L p ( SM e ) , if Z ∞ V ( t ) t p − dt < ∞ with p ∈ [1 , ∞ ) , (4.7) R ± ( λ ) : L p ( SM e ) → L ( SM e ) , if Z ∞ V ( t ) t p − dt < ∞ with p ∈ (1 , ∞ ) , (4.8) R ± ( λ ) : C αc ( SM ◦ e ) → H s ( SM e ) , if V ( t ) = O ( e Qt ) with s < min (cid:16) α, − Q ν max (cid:17) (4.9) where V ( t ) is the function of (2.13) . This operator satisfies ( − X ± λ ) R ± ( λ ) f = f inthe distribution sense in SM ◦ e when f is in one of the spaces where R ± ( λ ) f is well-defined.2) If ι : ∂SM → SM e is the inclusion map, then the operator ι ∗ R ± ( λ ) is a boundedoperator on the spaces L ∞ ( SM e ) → L p ( ∂SM ) , L p ( SM e ) → L ( ∂SM ) , C αc ( SM ◦ e ) → H s ( ∂SM ) (4.10) under the respective conditions (4.7) , (4.8) and (4.9) on V , p and s ; the measure usedon ∂SM is dµ ν , defined in (2.19) .3) If the condition (4.8) is satisfied and f ∈ L p ( SM e ) has supp( f ) ⊂ SM ◦ , then R ± ( λ ) f = 0 in a neighborhood of ∂ ± SM ∪ ∂ SM in SM e . ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 19
Proof.
Let us denote u + ( λ ) = R + ( λ ) f the L function given by (4.1) for Re( λ ) > f ∈ L ∞ ( SM e ). When R ∞ V ( t ) t p − dt < ∞ , the measure of Γ + ∪ Γ − is 0 and thusfor f ∈ L ∞ ( SM e ) and λ ∈ i R , the function u + ( λ ; x, v ) := R ∞ e − λ t f ( ϕ t ( x, v )) dt isfinite outside a set of measure 0 since ℓ e + ( x, v ), defined as the length of the geodesic { ϕ t ( x, v ); t ≥ } ∩ SM e , is finite on SM e \ Γ − . If λ n is any sequence with Re( λ n ) > λ , we have u + ( λ n ) → u + ( λ ) almost everywhere in SM e (using Lebesguetheorem). Moreover | u + ( λ n ) | ≤ R ∞ | f ◦ ϕ t | dt almost everywhere in SM e for all n > || R + (0)( | f | ) || L p ≤ C || f || L ∞ to get that || u + ( λ ) || L p ≤ C || f || L ∞ and u + ( λ n ) → u + ( λ ) in L p . We have for almostevery ( x, v ) | u + (0; x, v ) | = (cid:12)(cid:12)(cid:12) Z ∞ f ( ϕ t ( x, v )) dt (cid:12)(cid:12)(cid:12) ≤ || f || L ∞ ℓ e + ( x, v ) . (4.11)Notice that, in view of our assumption on the metric in SM e \ SM we have ℓ + ( x, v )+ L ≥ ℓ e + ( x, v ) ≥ ℓ + ( x, v ) for some L > x, v ) ∈ SM \ Γ − . Using the definition of V ( t ) in (2.13), the volume of the set S T of points ( x, v ) ∈ SM e such that ℓ e + ( x, v ) > T issmaller or equal to 2 V ( T − L ) with L as above (independent of T ). We apply Cavalieriprinciple for the function ℓ e + ( x, v ) in SM e \ Γ − , this gives Z SM e \ Γ − ℓ e + ( x, v ) p dµ ≤ C (cid:16) Z ∞ t p − V ( t ) dt (cid:17) (4.12)which shows (4.7) using (4.11). Notice that the same argument gives the same boundfor the L p norms of ℓ e − in SM e \ Γ + . The boundedness L p → L of (4.8) is a directconsequence of (4.12) (with ℓ − instead of ℓ + ) and the inequality Z SM e Z ∞ | f ( ϕ t ( x, v )) | dtdµ ≤ Z SM e Z ∞ SM e ( ϕ − t ( x, v )) | f ( x, v ) | dtdµ ≤ || ℓ e − || L p ′ || f || L p for all f ∈ C ∞ c ( SM ◦ e ) if 1 /p ′ + 1 /p = 1. The fact that ι ∗ R ± ( λ ) f defines a measurablefunction in L ( ∂SM, dµ ν ) when f ∈ L ( SM e ) comes directly from Santalo formula(2.21) and Fubini theorem (note that ∂ SM has zero measure in ∂SM ). This showsthe boundedness property of ι ∗ R ± ( λ ) : L p ( SM e ) → L ( ∂SM, dµ ν ). Let us now provethe boundedness of the restriction ι ∗ R ± (0) f in L p when f ∈ L ∞ . Since ℓ e + ( ϕ t ( x, v )) =( ℓ e + ( x, v ) − t ) + for t >
0, Santalo formula gives Z ∂ − SM e \ Γ − Z ℓ e + ( x,v )0 [ T, ∞ ) ( ℓ e + ( x, v ) − t ) dt |h v, ν i| dµ ∂SM e = Vol( S T ) , Z ∂ − SM \ Γ − Z ℓ + ( x,v )0 [ T, ∞ ) ( ℓ + ( x, v ) − t ) dtdµ ν ≤ Vol( S T )for T large. From this, we get for large T Z ∂ − SM \ Γ − [ T, ∞ ) ( ℓ + ( x, v )) dµ ν ≤ V ( T − L − , (4.13) and using Cavalieri principle, for any ∞ > p ≥ C p > Z ∂ − SM \ Γ − ℓ + ( x, v ) p dµ ν ≤ C (cid:16) Z ∞ t p − V ( t ) dt (cid:17) , (4.14)which shows, from (4.11) that u + | ∂SM \ Γ − ∈ L p for any 1 ≤ p < ∞ with a bound O ( || f || L ∞ ).To prove that ( − X ± λ ) R ± ( λ ) f = f in C −∞ ( SM ◦ e ) when f ∈ L p for p ∈ (1 , ∞ ) andthe condition R ∞ V ( t ) t p − dt < ∞ is satisfied, we take ψ ∈ C ∞ c ( SM ◦ e \ Γ ∓ ) and write h R ± ( λ ) f, ( X ± λ ) ψ i = lim n →∞ h R ± ( λ ) f n , ( X ± λ ) ψ i = lim n →∞ h f n , ψ i = h f, ψ i where f n ∈ C ∞ c ( SM ◦ e \ Γ ∓ ) converges in L p to f ; to obtain the second identity, we used(4.4) and the fact that ( − X ± λ ) R ± ( λ ) f n = f n in SM ◦ e \ Γ ∓ .Finally, we describe the case where the escape rate Q is negative (ie. when V ( t )decays exponentially fast). We need to prove that u + is in H s ( SM e ) for some s > f ∈ C αc ( SM ◦ e ). To prove that u + is H s ( SM e ), it suffices to prove ([H¨o, Chap. 7.9]) Z SM e Z SM e | u + ( y ) − u + ( y ′ ) | d ( y, y ′ ) n +2 s dydy ′ < ∞ if n = dim( SM ) and d ( y, y ′ ) denote the distance for the Sasaki metric on SM e . Usingthat f ∈ C α ( SM e ), we have that for all α ≥ β > C > y, y ′ ∈ SM e , ν > ν max and all t ∈ R | f ( ϕ t ( y )) − f ( ϕ t ( y ′ )) | ≤ C || f || C β e νβ | t | d ( y, y ′ ) β thus for ℓ e + ( y ) < ∞ and ℓ e + ( y ′ ) < ∞| u + ( y ) − u + ( y ′ ) | ≤ Cℓ e + ( y, y ′ ) e νβℓ e + ( y,y ′ ) d ( y, y ′ ) β . where ℓ e + ( y, y ′ ) := max( ℓ e + ( y ) , ℓ e + ( y ′ )). We then evaluate for β − s > β < α Z | u + ( y ) − u + ( y ′ ) | d ( y, y ′ ) n +2 s dydy ′ ≤ C β Z e νβℓ + ( y,y ′ ) d ( y, y ′ ) β − s ) − n dydy ′ ≤ C β Z ℓ + ( y ) >ℓ + ( y ′ ) e νβℓ + ( y ) d ( y, y ′ ) β − s ) − n dydy ′ ≤ C s,β Z SM e e νβℓ + ( y ) dy and from Cavalieri principle the last integral is finite if we choose β > < s < β < − Q/ ν . Taking ν arbitrarily close to ν max gives that u + ∈ H s ( SM e )if s < − Q/ ν max . The same argument works for u − and also for the boundary values u ± | ∂SM .To finish, the proof of part 3) in the statement of the Proposition is a direct conse-quence of the expression (4.1) for R ± ( λ ) f since the positive (reps. negative) flowoutof supp( f ) ⊂ SM ◦ intersect ∂SM in a compact region of ∂ + SM (resp. ∂ − SM ). (cid:3) ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 21
Remark.
Reasoning like in the proof Proposition 4.2, it is straightforward by usingCauchy-Schwarz to check that R ± ( λ ) extends continuously to Re( λ ) ≥ R ± ( λ ) to functions supported on SM ) R ± ( λ ) : h ℓ ± i − / − ǫ L ( SM ) → h ℓ ± i / ǫ L ( SM )for all ǫ > ℓ ± is the escape time function of (2.1). This is comparable to thelimiting absorption principle in scattering theory. The boundedness in Proposition 4.2are slightly finer and describe the L p boundedness of ℓ ± in terms of V ( t ) instead.The resolvent R ± (0) has been defined under decay property of the non-escapingmass function. In the case where K is hyperbolic, we can actually say more refinedproperties of this operator. Proposition 4.3 (Dyatlov-Guillarmou [DyGu2]) . Assume that the trapped set K ishyperbolic. There exists c > such that for all s > :1) the resolvents R ∓ ( λ ) extend meromorphically to the region Re( λ ) > − cs as abounded operator R ∓ ( λ ) : H s ( SM e ) → H − s ( SM e ) with poles of finite multiplicity.2) There is a neighborhood U ∓ of E ∗∓ such that for all pseudo-differential operator A ∓ of order with WF( A ∓ ) ⊂ U ∓ and support in SM ◦ e , A ∓ R ∓ ( λ ) maps continuously H s ( SM e ) to H s ( SM e ) , when λ is not a pole.3) Assume that λ is not a pole of R ∓ ( λ ) , then the Schwartz kernel of R ∓ ( λ ) is adistribution on SM ◦ e × SM ◦ e with wavefront set WF( R ∓ ( λ )) ⊂ N ∗ ∆( SM ◦ e × SM ◦ e ) ∪ Ω ± ∪ ( E ∗± × E ∗∓ ) . (4.15) where N ∗ ∆( SM ◦ e × SM ◦ e ) is the conormal bundle to the diagonal ∆( SM ◦ e × SM ◦ e ) of SM ◦ e × SM ◦ e and Ω ± := { ( ϕ ± t ( y ) , ( dϕ ± t ( y ) − ) T ξ, y, − ξ ) ∈ T ∗ ( SM ◦ e × SM ◦ e ); t ≥ , ξ ( X ( y )) = 0 } . Proof.
Part 1) and 2) are stated in Proposition 6.1 of [DyGu2], (they actually followfrom Lemma 4.3 and 4.4 of that paper), while part 3) is proved in Lemma 4.5 of[DyGu2]. (cid:3)
We can now combine Propositions 4.2 and 4.3 and obtain
Proposition 4.4.
Assume that the trapped set K is hyperbolic. Then we get for all p < ∞ :1) The resolvent R ± ( λ ) has no pole at λ = 0 , and it defines for all s ∈ (0 , / abounded operator R ± (0) on the following spaces R ± (0) : H s ( SM e ) → H − s ( SM e ) , R ± (0) : L ∞ ( SM e ) → L p ( SM e ) that satisfies − XR ± (0) f = f in the distribution sense, and for f ∈ C ( SM e ) one has ∀ y ∈ SM \ Γ ∓ , ( R ± (0) f )( y ) = Z ±∞ f ( ϕ t ( y )) dt. (4.16) which is continuous in SM \ Γ ∓ and satisfies R ± (0) f | ∂ ± SM = 0 if supp( f ) ⊂ SM .2) As a map H s ( SM e ) → H − s ( SM e ) for s ∈ (0 , / , we have R + (0) = − R − (0) ∗ . (4.17)
3) If f ∈ C ∞ c ( SM ◦ e ) , the function u ± := R ± (0) f has wavefront set WF( u ± ) ⊂ E ∗∓ , (4.18) the restriction u ± | ∂SM := ι ∗ u ± makes sense as a distribution satisfying u ± | ∂SM ∈ L p ( ∂SM ) , WF( u ± | ∂SM ) ⊂ E ∗∓ ,∂ . (4.19)
4) Let α > , then for f ∈ C α ( SM ) extended by on SM e \ SM as an element in H s ( SM e ) for s < min( α, / , we have R ± (0) f ∈ H s ( SM e ) for s < min( α, − Q/ ν max ) ,where ν max is the maximal Lyapunov exponent (4.6) . Moreover u ± | ∂SM ∈ H s ( ∂ ± SM ) for such s .Proof. Recall that for Re( λ ) > f ∈ C ∞ c ( SM ◦ e ) and ψ ∈ C ∞ c ( SM ◦ e ) , h R + ( λ ) f, ψ i = Z ∞ e − λt h f ◦ ϕ t , ψ i dt. By Proposition 4.2, then as λ → λ ) ≥ R ± ( λ ) f → R ± (0) f in L p (thus in the distribution sense). This implies that theextended resolvent R ± ( λ ) of Proposition 4.3 can not have poles at λ = 0 by density of C ∞ c ( SM ◦ e ) in any H s ( SM e ). The same argument shows that R ± ( λ ) is holomorphic in { Re( λ ) > Q } . The expression (4.16) comes from Proposition 4.2, which also impliesthe continuity of R ± (0) f outside Γ ∓ and its vanishing at ∂ ± SM when supp( f ) ⊂ SM .Part 2) and (4.17) follows by continuity by taking λ → H s ( SM e ) functions instead of L ( SM e )).For part 3), the wavefront set property of u ± := R ± (0) f if f ∈ C ∞ c ( SM ◦ e ) followsfrom the wavefront set description (4.15) of the Schwartz kernel of R ± (0) and thecomposition rule of [H¨o, Theorem 8.2.13]. The fact that u ± restricts to ∂SM as adistribution which satisfies (4.19) comes from [H¨o, Theorem 8.2.4] and the fact that N ∗ ( ∂SM ) ∩ E ∗± = 0, if N ∗ ( ∂SM ) ⊂ T ∗ ( SM e ) is the conormal bundle to ∂SM . The L ( ∂SM ) boundedness of the restriction follows from (4.10).For part 4), the fact that the extension of f by 0 is in H s ( SM e ) for s ∈ (0 , /
2) isproved in [Ta, Proposition 5.3], and the rest is proved in Proposition 4.2. (cid:3)
ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 23
In fact, if f ∈ C ∞ c ( SM ◦ e ) has support in SM , the expression (4.16) vanishes in aneighborhood of ∂ + SM (resp. ∂ − SM ) in SM e \ SM ◦ , and thus R ± (0) f vanishes to all order at ∂ ± SM. (4.20)We also want to make the following observation: the involution A : ( x, v ) ( x, − v )on SM e and SM is a diffeomorphism and thus acts by pullback on distributions, itallows to decompose distributions u on SM ◦ e into even and odd parts u = u ev + u od where u od := (Id − A ∗ ) u . If f ∈ C ∞ c ( SM ◦ e ) is even, it is direct from the expression(4.16) that( R ± (0) f ) ev = ± ( R + (0) − R − (0)) f, ( R ± (0) f ) od = ( R + (0) + R − (0)) f (4.21)and this extends by continuity to distributions. Similarly if f is odd, ( R ± (0) f ) ev = ( R + (0) + R − (0)) f and ( R ± (0) f ) od = ± ( R + (0) − R − (0)) f .4.3. Boundary value problem.
First, we extend the boundary value problem ofLemma 3.3 to the case of L ( ∂ ∓ SM ) boundary data. Lemma 4.5.
Assume that R ∞ tV ( t ) dt < ∞ if V is the function (2.13) . The map E ∓ of (3.5) can be extended as a bounded operator L ( ∂ ∓ SM, dµ ν ) → L ( SM e ) , satisfying X E ∓ ( ω ∓ ) = 0 in the distribution sense for ω ∓ ∈ L ( ∂ ∓ SM, dµ ν ) .Proof. Using the expression (3.6), Santalo formula and Cauchy-Schwarz inequality, wesee that there is
C > ω ∓ ∈ C ∞ c ( ∂ ∓ SM ) ||E ∓ ( ω ∓ ) || L ( SM e ) ≤ C ( || ω ∓ || L ( ∂ − SM,dµ ν ) + || ℓ e ± || L ( ∂ − SM,dµ ν ) || ω ∓ || L ( ∂ − SM,dµ ν ) )where we used that there is C ′ > | ℓ e ∓ ( x, v ) | ≤ C ′ on ∂ ∓ SM . Using (4.14),we deduce the announced boundedness. The fact that X E ∓ = 0 on L follows fromthe same identity on C ∞ c ( ∂ ∓ SM ). (cid:3) In the case of a hyperbolic trapped set, using the resolvents R ± (0), we are able toconstruct invariant distributions in SM with prescribed value on ∂ − SM and we candescribe (partly) its singularities. Proposition 4.6.
Assume that K is hyperbolic, then:1) For ω − ∈ L ( ∂ − SM, dµ ν ) satisfying supp( ω − ) ⊂ ∂ − SM and WF( ω − ) ⊂ E ∗ ∂, − , WF( S g ω − ) ⊂ E ∗ ∂, + . (4.22) the function E − ( ω − ) ∈ L ( SM e ) has wave-front set which satisfies WF( E − ( ω − )) ∩ T ∗ ( SM e \ K ) ⊂ E ∗− ∪ E ∗ + , (4.23) the restriction E − ( ω − ) | ∂ − SM makes sense as a distribution in L ( ∂ − SM ) and is equalto E − ( ω − ) | ∂ − SM = ω − .2) If ω − ∈ H s ( ∂ − SM ) for some s > with supp( ω − ) ⊂ ∂ − SM , if (4.22) holds and S g ω − ∈ H s ( ∂ + SM ) , then E − ( ω − ) ∈ H s ( SM e ) . If π : SM e → M e is the projection onthe base and π ∗ the pushforward defined in (5.9) then π ∗ ( E − ( ω − )) ∈ H s + 12loc ( M e ) . (4.24) Proof.
Let U ′ ⊂ ∂ − SM be an open neighborhood of supp( ω − ) whose closure does notintersect ∂ SM and let U be the open neighborhood of supp( ω − ) in SM e defined by U = ∪ −∞
Putting u = R − (0)( Xψ − ), we have u = 0 near ∂ − SM and thus all point ( y, ξ ) / ∈ E ∗− with y / ∈ Γ + is not in WF( u ) by (4.25). This implies thatWF( w ) ∩ T ∗ ( SM e \ Γ + ) ⊂ E ∗− (4.26)and in particular w is smooth in SM e \ (Γ − ∪ Γ + ), which implies that E − ( ω − ) = w , asmentioned above. By ellipticity and the equation Xw = 0, we haveWF( w ) ⊂ { ξ ∈ T ∗ ( SM e ); ξ ( X ) = 0 } (4.27)and w smooth near ∂ + SM \ Γ + , then as above we can use [H¨o, Theorem 8.2.4] to deducethat the restriction ω + := w | ∂ + SM makes sense as a distribution. Moreover it can beobtained as limits of restrictions E − ( ω ( n ) − ) | ∂ + SM where ω ( n ) − ∈ C ∞ c ( ∂ − SM ) is a sequenceconverging in L to ω − (since also E − ( ω ( n ) − ) has wave-front set contained in a uniformregion not intersecting the conormal to ∂ + SM ). Then, as E − ( ω ( n ) − ) | ∂ + SM = S g ω ( n ) − ,we deduce from Lemma 3.4 that S g ω − = ω + . By our assumptions on ω − , we thushave ω + ∈ H s ( ∂ + SM ) and WF( ω + ) ⊂ E ∗ ∂, + . Notice also that supp( ω + ) ⊂ ∂ + SM .Then proceeding as above, but using the flow in backward direction, we can write E − ( ω − ) = E + ( ω + ) = ψ + − R + (0)( Xψ + ) where ψ + ∈ H s ( SM e ) is defined similarly to ψ − but has support near supp( ω + ) and WF( ψ + ) ⊂ E ∗ + . Then using similar argumentsas above , WF( E − ( ω − )) ∩ T ∗ ( SM e \ Γ − ) ⊂ E ∗ + and combining with (4.26) this givesWF( E − ( ω − )) ∩ T ∗ ( SM e \ K ) ⊂ E ∗− ∪ E ∗ + . Let us now prove that w ∈ H s ( SM e ) if s >
0. By point 2) in Proposition 4.3 appliedto R ± (0)( Xψ ± ), we obtain that A ± w ∈ H s ( SM e ) for some s > A ± is any 0-thorder ΨDO with WF( A ± ) contained in a small enough neighborhood V ± of E ∗± . Thenif B is any 0-th order ΨDO with WF( B ) contained outside an open neighborhood V of T ∗ K ( SM e ), B w ∈ H s ( SM e ). By (4.27), we have B w ∈ C ∞ ( SM e ) if B is any0-th order ΨDO with WF( B ) contained outside a small conic neighborhood V ofthe characteristic set { ξ ∈ T ∗ ( SM e ); ξ ( X ) = 0 } . Therefore, it remains to prove that B w ∈ H s ( SM e ) if B is any 0-th order ΨDO with wave-front set contained in theregion V := ( V ∩ V ) \ ( V − ∪ V + ). But this property will follow from propagation ofsingularities. Indeed, let ( y, ξ ) ∈ V , then the following alternative holds:1) if y / ∈ K , there is T > T ( y, ξ ) / ∈ V or Φ − T ( y, ξ ) / ∈ V
2) if y ∈ K , by (2.9) there is T > − T ( y, ξ ) ∈ V − or Φ T ( y, ξ ) ∈ V + .We can apply [DyZw, Proposition 2.5] (recall that Xw = 0), we obtain B w ∈ H s ( SM e ) and this concludes the proof of w ∈ H s ( SM e ).To conclude, the 1 / w in an open neighborhood W of SM e so that Xw = 0 in W and w ∈ H s ( W ), the averaging lemma implies thatits average in the fibers π ∗ w restricts to M e as an H s +1 / function. (cid:3) Combining Proposition 4.6 with (3.8), we obtain (using notation (3.9)) the followingexistence result for invariant distributions on SM with prescribed boundary values.This will be fundamental for the resolution of the lens rigidity for surfaces. Corollary 4.7.
Assume that the trapped set K is hyperbolic. There exists an openneighborhood U of ∂SM ∩ (Γ − ∪ Γ + ) in SM ◦ e such that for any ω ∈ L S g ( ∂SM ) ,satisfying WF( ω ) ⊂ E ∗ ∂, − ∪ E ∗ ∂, + , there exists w ∈ L ( SM e ) such that the restriction w | ∂SM makes sense as a distribution and Xw = 0 in SM ∪ U, w | ∂SM = ω, WF( w ) ∩ T ∗ ( SM e \ K ) ⊂ E ∗− ∪ E ∗ + . If ω ∈ H s ( ∂SM ) for s > , then w ∈ H s ( SM e ) and π ∗ w ∈ H s +1 / ( M e ) .Proof. We decompose ω = ω + ω where ω ∈ C ∞ S g ( ∂SM ) with supp( ω ) ⊂ ∂SM \ (Γ − ∪ Γ + ) and ω supported near ∂SM ∩ (Γ − ∪ Γ + ). We apply (3.8) to ω , this produces w ∈ C ∞ ( SM ) which is flow invariant and with boundary value ω . Then, we applyProposition 4.6 to ω | ∂ − SM , this produces w = E − ( ω | ∂ − SM ) satisfying Xw = 0 in SM e and w | ∂ − SM = ω | ∂ − SM . Then set w = w + w . The wavefront set property of w and the regularity of π ∗ w follows from Proposition 4.6. (cid:3) X -ray transform and the operator ΠWe start by defining the X -ray transform as the map I : C ∞ c ( SM \ Γ − ) → C ∞ c ( ∂ − SM \ Γ − ) , If ( x, v ) := Z ∞ f ( ϕ t ( x, v )) dt. From the expression (4.16), we observe that If = ( R + (0) f ) | ∂ − SM \ Γ − . (5.1)Then I can be extended to more general space. For instance, Santalo formula impliesdirectly that as long as Vol( K ) = 0 (and no other assumption on K ), I : L ( SM ) → L ( ∂ − SM ; dµ ν ) . For our purposes, as we shall see later, there is an important condition on the non-escaping mass function which allows to use
T T ∗ type arguments and relate I ∗ I to thespectral measure at 0 of the flow. This condition is ∃ p ∈ (2 , ∞ ] , Z ∞ t pp − V ( t ) dt < ∞ , (5.2)if V is the function defined in (2.13). It is always satisfied if K is hyperbolic. We have ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 27
Lemma 5.1.
Assume that (5.2) holds for some p > , then the X-ray transform I extends boundedly as an operator I : L p ( SM ) → L ( ∂ − SM, dµ ν ) . Proof.
Let f ∈ L p ( SM ), then using H¨older with p ′ + p = 1 and rp ′ = p − p − > Z ∂ − SM (cid:12)(cid:12)(cid:12) Z ℓ + ( y )0 f ( ϕ t ( y )) dt (cid:12)(cid:12)(cid:12) dµ ν ( y ) ≤ Z ∂ − SM (cid:16) Z ℓ + ( y )0 | f ( ϕ t ( y )) | p dt (cid:17) /p ℓ + ( y ) /p ′ dµ ν ( y ) ≤ (cid:16) Z ∂ − SM Z ℓ + ( y )0 | f ( ϕ t ( y )) | p dtdµ ν ( y ) (cid:17) /p || ℓ + || /p ′ L r/p ′ ( ∂ − SM,dµ ν ) ≤ || f || L p ( SM ) || ℓ + || /p ′ L r/p ′ ( ∂ − SM,dµ ν ) where we have used Santalo formula to obtain the last line. Since ℓ + ∈ L q ( ∂ − SM, dµ ν )when R ∞ t q − V ( t ) dt by (4.14), we deduce the result. (cid:3) Assume that R ∞ t p/ ( p − V ( t ) dt < ∞ for some p ∈ (2 , ∞ ). Note that by Sobolevembedding I : H s ( SM ) → L ( ∂ − SM, dµ ν ) is bounded if s = n − np for the p ∈ (2 , ∞ )of Lemma 5.1. Since H − s ( SM ) is defined as the dual of H s ( SM ) and L p ′ is dual to L p for p ∈ (2 , ∞ ) if 1 /p + 1 /p ′ = 1, the adjoint of I , denoted I ∗ , is bounded as operators(for s as above) I ∗ : L ( ∂ − SM, dµ ν ) → L p ′ ( SM ) , I ∗ : L ( ∂ − SM, dµ ν ) → H − s ( SM ) . (5.3)In fact, a short computation gives Lemma 5.2. If (5.2) holds true, then I ∗ = E − .Proof. Let ω − ∈ C ∞ c ( ∂ − SM \ Γ − ), then E − ( ω − ) ∈ C ∞ ( SM ) and its support does notintersect Γ − ∪ Γ + . By Green’s formula, we have for f ∈ C ∞ c ( SM ◦ ) Z SM f E − ( ω − ) dµ = Z SM − X ( R + (0) f ) . E − ( ω − ) dµ = Z ∂ − SM If.ω − |h X, N i| dµ ∂SM where S is Sasaki metric and N the inward pointing unit normal to ∂SM in SM . Likein the proof of Lemma 3.4, |h X, N i S | = |h v, ν i| . Using density of C ∞ c ( SM ◦ ) in L p ( SM )and of C ∞ c ( ∂ − SM \ Γ − ) in L ( ∂ − SM, dµ ν ), we get the desired result. (cid:3) To describe the properties of I and I ∗ , it is convenient to define the operatorΠ := I ∗ I : L p ( SM ) → L p ′ ( SM ) , when Z ∞ t pp − V ( t ) dt < ∞ . (5.4)for p ∈ (2 , ∞ ). We prove the following relation between Π and the resolvents: Lemma 5.3.
Assuming (5.2) , the operator
Π = I ∗ I of (5.4) is equal on L p ( SM ) to Π = R + (0) − R − (0) Proof.
Since h R + (0) f, f i = −h f, R − (0) f i by (4.17), it suffices to prove the identity h I ∗ If, f i L ( ∂ − SM,dµ ν ) = 2 h R + (0) f, f i for all f ∈ C ∞ c ( SM \ (Γ − ∪ Γ + )) real valued. We write u = R + (0) f and compute, usingGreen’s formula, Z SM u.f dµ = − Z SM u.Xudµ = − Z SM X ( u ) dµ = Z ∂ − SM u |h v, ν i| dµ ∂SM and this achieves the proof. (cid:3) With the assumption of Lemma 5.3, the operator Π can also be extended as abounded operator Π e on SM e Π e := R + (0) − R − (0) : L p ( SM e ) → L ( SM e ) , (5.5)satisfying Π e f | SM = Π f for all f ∈ L p ( SM ) extended by 0 on SM e \ SM . As above,one directly sees that Π e = I e ∗ I e if we call I e : L p ( SM e ) → L ( ∂ − SM e ; |h v, ν i| dµ ∂SM e )the X-ray transform on SM e , defined just as on SM and satisfying the same properties.In particular this shows that Π e : L p ( SM e ) → L p ′ ( SM e ) is bounded. We summarizethe discussion by the following: Proposition 5.4.
Assume that (5.2) holds for p ∈ (2 , ∞ ) . Then we obtain1) the operator Π e is bounded and self-adjoint as a map Π e : L p ( SM e ) → L p ′ ( SM e ) , /p + 1 /p ′ = 1 , it satisfies for each f ∈ L p ( SM e ) X Π e f = 0 (5.6) in the distribution sense and Π e f is given, outside a set of measure , by the formula Π e f ( x, v ) = Z ∞−∞ f ( ϕ t ( x, v )) dt. (5.7)
2) If the trapped set K is hyperbolic, the operator Π e : H s ( SM e ) → H − s ( SM e ) isbounded for all s ∈ (0 , / . For each f ∈ C ∞ c ( SM ◦ e ) , the expression (5.7) holds in SM e \ (Γ + ∪ Γ − ) , we have WF(Π e f ) ∈ E ∗− ∪ E ∗ + , the restriction ω ± := (Π e f ) | ∂ ± SM makes sense as a distribution, is in L ( ∂ ± SM, dµ ν ) with wave-front set WF( ω ± ) ⊂ E ∗ ∂, ± , (5.8) and S g ω − = ω + where S g is the scattering map (3.3) . Finally ω ± ∈ H s ( ∂ ± SM ) for all s < − Q/ ν max with ν max defined in (4.6) .Proof. The boundedness and the self-adjoint property have already been proved. Theproperty (5.6) is clear from the properties of R ± (0) given in 1) of Proposition 4.4.The expression of Π e f follows from (4.16) (and the proof of Proposition 4.2 for theextension to L p functions). The wavefront set property of w follows from (4.18), and the ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 29 wavefront set and regularity properties (5.8) of the restrictions ω ± are consequences ofProposition 4.4. The fact that ω − ∈ L ( ∂ − SM, dµ ν ) comes from Lemma 5.1. Finally, S g ω − = ω + since, by (5.7), ω + = ω − ◦ S − g on ∂ + SM \ Γ + , this also implies that ω + ∈ L ( ∂SM, dµ ν ) by Lemma 3.4. The last statement in the Proposition is a consequenceof 3) in Proposition 4.2. (cid:3) Next, we describe the kernel of Π e restricted to smooth functions supported in SM . Proposition 5.5.
Assume that K is hyperbolic. Let f ∈ C ∞ ( SM ) extended by in SM e \ SM , if Π e f = 0 in SM , there exists u ∈ C ∞ ( SM ) vanishing at ∂SM such that Xu = f . If f vanishes to infinite order at ∂M , then u also does so.Proof. First, the extension of f by 0 can be viewed as an element in H s ( SM e ) for s < / f ) ⊂ N ∗ ( ∂SM ) where N ∗ ( ∂SM ) is the conormal bundle of ∂SM in SM e . By the composition law of wave-front set in [H¨o, Theorem 8.2.13] and (4.15),we deduce that WF( R ∓ (0) f ) ⊂ N ∗ ∂SM ∪ E ∗± ∪ B ∓ B ± := ∪ t ≥ { ( ϕ ± t ( y ) , ( dϕ ± t ( y ) − ) T ξ ) ∈ T ∗ SM ◦ e ; y ∈ ∂ SM, ξ ∈ N ∗ ( ∂SM ) } Clearly, by strict convexity, B ± projects down to M e \ M ◦ . Now, the function ℓ ± issmooth in SM \ ( ∂ SM ∪ Γ − ∪ Γ + ) and from the expression (4.16) and the smoothnessof f , we then get that R ∓ (0) f is smooth in SM \ ( ∂ SM ∪ Γ ± ) and ( R ± (0) f ) | ∂ ± SM = 0.To analyze the regularity at ∂ SM , we decompose f = f ev + f od , we get by (4.21) that( R ± (0) f ev ) ev = ± Π e f = 0 and similarly ( R ± (0) f od ) od = 0. Now the argument of[SaUh, Lemma 2.3] shows that ( R ± (0) f ev ) od | SM and ( R ± (0) f od ) ev | SM are both smoothnear ∂ SM , which implies that R ± (0) f is smooth near ∂ SM in SM . Since R + (0) f = R − (0) f if Π e f = 0, we deduce that ( R ± (0) f ) | SM ∈ C ∞ ( SM \ K ) and ( R ± (0) f ) | ∂SM =0. From the wavefront set description above and the fact that E ∗ + ∩ E ∗− = { } over K , we conclude that ( R ∓ (0) f ) | SM ∈ C ∞ ( SM ). It just suffices to set u = R + (0) f toconclude the proof. The fact that f vanishes to all order at ∂SM implies that R ± (0) f vanishes to all order at ∂ ± SM by (4.20), and thus u vanishes to all order at ∂SM . (cid:3) The operators I and Π . Here we deal with the analysis of X-ray transformacting on functions on M . The projection π : SM e → M e on the base induces apull-back map π ∗ : C ∞ c ( M ◦ e ) → C ∞ c ( SM ◦ e ) , π ∗ f := f ◦ π and a push-forward map π ∗ defined by duality π ∗ : C −∞ ( SM ◦ e ) → C −∞ ( M ◦ e ) , h π ∗ u, f i := h u, π ∗ f i . (5.9)Push-forward corresponds to integration in the fibers of SM e when acting on smoothfunctions. The pull-back by π also makes sense on M and gives a bounded operator π ∗ : L p ( M ) → L p ( SM ) for all p ∈ (1 , ∞ ). When (5.2) holds for some p ∈ (2 , ∞ ), wedefine the X-ray transform on functions as the bounded operator (see Lemma (5.1)) I := I π ∗ : L p ( M ) → L ( ∂ − SM, dµ ν ) . (5.10)The adjoint I ∗ : L ( ∂ − SM, dµ ν ) → L p ′ ( M ) is bounded if 1 /p ′ + 1 /p = 1 and it is givenby I ∗ = π ∗ I ∗ . The operator Π is simply defined as the bounded self-adjoint operatorfor p ∈ (2 , ∞ ) and 1 /p ′ + 1 /p = 1Π := I ∗ I = π ∗ Π π ∗ : L p ( M ) → L p ′ ( M ) . (5.11)Similarly, we define the self-adjoint bounded operatorΠ e := π ∗ Π e π ∗ = ( I e π ∗ ) ∗ I e π ∗ : L p ( M e ) → L p ′ ( M e ) . (5.12)We first want to mention some boundedness result which holds in a general setting (nocondition on conjugate points are required) and says that Π is always regularizing if V ( t ) decays sufficiently. Lemma 5.6.
Assume that (5.2) holds for p > , then I ∗ and I are bounded as maps I ∗ : L ( ∂ − SM, dµ ν ) → H − n − + np loc ( M ◦ ) , I : H n − − np comp ( M ◦ ) → L ( ∂ − SM, dµ ν ) . and the same property holds for I e with M e replacing M .Proof. It suffices to prove the boundedness for I ∗ . By Sobolev embedding, I ∗ : L ( ∂ − SM, dµ ν ) → H − n + np loc ( M ◦ ) is bounded, and using Lemma 5.2, we have XI ∗ = 0as operators. Then applying [GeGo, Theorem 2.1] as in the proof of Proposition 4.4,we gain 1 / π ∗ , this ends the proof. (cid:3) If V ( t ) = O ( t −∞ ), the Sobolev exponents are H − / − ǫ comp ( M ◦ ) and H / ǫ loc ( M ◦ ) for all ǫ >
0, and if K = ∅ we get I ∗ I : H − / ( M ◦ ) → H / ( M ◦ ). Following the method of[Gu], we prove Proposition 5.7.
Assume that the geodesic flow on SM has no conjugate points andthat the trapped set K is hyperbolic. The operator Π e = π ∗ Π e π ∗ is an elliptic pseudo-differential operator of order − in M ◦ e , with principal symbol σ (Π e )( x, ξ ) = C n | ξ | − g for some constant C n = 0 depending only on n .Proof. First we choose the extension ( M e , g ) so that the geodesic flow on M e hasnon-conjugate points. Once we know the wavefront set of the Schwartz kernels ofthe resolvent R ± (0), the proof is very similar to Theorem 3.1 and Theorem 3.4 in[Gu], therefore we do not write all details but refer to that paper where this is donecarefully for Anosov flows. It suffices to analyze χ Π e χ ′ where χ, χ ′ ∈ C ∞ c ( M ◦ e ) arearbitrary functions. Its Schwartz kernel is given by χ ( x ) χ ′ ( x ′ )(( π ⊗ π ) ∗ Π e )( x, x ′ ) ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 31 where Π e = R + (0) − R − (0) is identified with its Schwartz kernel. We write for ǫ ≥ R + (0) = Z ǫ e tX dt + e ǫX R + (0)where e tX is the pull-back by the flow at time t . Using (4.15) and the computationof WF( e ǫX ) which follows from [H¨o, Theorem 8.2.4], the composition law of wavefrontset [H¨o, Theorem 8.2.14] can be used like in the proof of [Gu, Theorem 3.1]: we obtainWF( π ∗ ( χ ) e ǫX R + (0) π ∗ ( χ ′ )) ⊂ (cid:16) { ( ϕ t ( y ) , ( dϕ t ( y ) − ) T η, y, − η ); t ≤ − ǫ, η ( X ( y )) = 0 }∪ { ( ϕ − ǫ ( y ) , η, y, − dϕ − ǫ ( y ) T η ); ( y, η ) ∈ T ∗ ( SM ) \ { }}∪ ( E ∗− × E ∗ + ) (cid:17) ∩ { ( y, η, y ′ , η ′ ); ( π ( y ) , π ( y ′ )) ∈ U × U ′ } . where U := supp( χ ) and U ′ = supp( χ ′ ); here the wave-front set of an operator meansthe wave-front set of the Schwartz kernel of the operator. By applying the rule ofpushforward of wave-front sets (given for example in [FrJo, Proposition 11.3.3.]), weget WF( π ∗ e ǫX R π ∗ ) ⊂ S ∪ S ∪ S where S := { ( π ( y ) , ξ, π ( y ′ ) , ξ ′ ) ∈ T ∗ ( U × U ); ( y, dπ ( y ) T ξ, y ′ , dπ ( y ′ ) T ξ ′ ) ∈ E ∗− × E ∗ + } S := { ( π ( ϕ t ( y )) , ξ, π ( y ) , ξ ′ ) ∈ T ∗ ( U × U ); ∃ t ≤ − ǫ, ∃ η, η ( X ( y )) = 0 ,dπ ( y ) T ξ ′ = − η, dπ ( ϕ t ( y )) T ξ = ( dϕ t ( y ) − ) T η } S := { ( π ( ϕ − ǫ ( y )) , ξ, π ( y ) , ξ ′ ) ∈ T ∗ ( U × U ); ( d ( π ◦ ϕ − ǫ )( y )) T ξ = − dπ ( y ) T ξ ′ } if we set T ∗ ( U × U ) := T ∗ ( U × U ) \ { } . We let V = ker dπ ⊂ T ( SM e ) be the verticalbundle, and H be the horizontal bundle (cf. [Pa, Chapter 1.3]), and V ∗ , H ∗ ⊂ T ∗ ( SM e )their dual defined by H ∗ ( V ) = 0 and V ∗ ( H ) = 0 ( V ∗ is dual to V and H ∗ is dual to H for the Sasaki metric). By (2.10), the absence of conjugate points for the flow in M e implies that T ( SM e ) = R X ⊕ V ⊕ E ± at Γ ± and thus E ∗± ∩ H ∗ = { } . This impliesthat S = ∅ . Similarly, it is direct to see that S = ∅ is equivalent to the absenceof conjugate points for the flow (see the proof of [Gu, Theorem 3.1] for details). Thelast part is S . The proof is exactly the same as in [Gu, Theorem 3.1] thus we do notrepeat it but simply summarize the argument: the projection of S on M ◦ e is containedin ∆ ǫ ( M ◦ e × M ◦ e ) := { ( x, x ′ ) ∈ M ◦ e × M ◦ e ; d g ( x, x ′ ) = ǫ } where d g is the Riemanniandistance. The operator L ǫ = R ǫ π ∗ e tX π ∗ dt is explicit for small ǫ > L ǫ f ( x ) := Z ǫ Z S x M e f ( ϕ t ( x, v )) dvdt, This operator has singular support ∆ ǫ ( M ◦ e × M ◦ e ) ∪ ∆ ( M ◦ e × M ◦ e ) and thus, ǫ > has singular support on the diagonal∆ ( M ◦ e × M ◦ e ). Now the kernel ψ ( x, x ′ ) L ǫ ( x, x ′ ) is that of an elliptic pseudo-differentialoperator of order − ψ ∈ C ∞ c ( M ◦ e × M ◦ e ) is supported close enough to the diagonal { x = x ′ } and equal to 1 in a neighborhood of the diagonal: the analysis is purely local and exactly the same as in [PeUh, Lemma 3.1], which also shows that the symbol ofthis ΨDO is C n | ξ | − g for some C n >
0. It is direct to see (from R + (0) ∗ = − R − (0)) thatΠ e = 2 π ∗ R + (0) π ∗ , and we have then proved the claim. (cid:3) Since the Schwartz kernel of Π e on M ◦ is the restriction of the kernel of Π e to M ◦ × M ◦ , we deduce that in the case of hyperbolic trapped set and no conjugatepoints, Lemma 5.6 gives that Π e : H − / ( M ◦ ) → H / ( M ◦ ) and the T T ∗ argumentshows that for any compact domain O ⊂ M ◦ with non-empty interior and smoothboundary, we have I : H − / ( O ) → L ( ∂ − SM ; dµ ν ) , I ∗ : L ( ∂ − SM ; dµ ν ) → H / ( O ) . (5.13)We can use Proposition 5.7 to prove the regularity property on elements in ker I . Corollary 5.8.
Assume that the trapped set K is hyperbolic, the metric has no con-jugate points. Let f ∈ L p ( M ) + H − / ( M ◦ ) for some p > satisfying I f = 0 . Then f ∈ C ∞ ( M ) and f vanishes to all order at ∂M .Proof. First, I f = 0 in L ( ∂ − SM ; dµ ν ) implies that I e f = 0 if I e = I e π ∗ is the X-raytransform on functions on M e and f is extended by 0 in M e \ M . Thus Π e f = 0 in M ◦ e . This implies, by ellipticity of Π e in M ◦ e that f is smooth, and since it is equal to0 in M ◦ e \ M , we deduce that f vanishes to all order at ∂M . (cid:3) X-ray on symmetric tensors.
For any m ∈ N , symmetric cotensors of order m on M ◦ e can be viewed as functions on SM ◦ e via the map π ∗ m : C ∞ c ( M ◦ e , ⊗ mS T ∗ M ◦ e ) → C ∞ c ( SM ◦ e ) , ( π ∗ m f )( x, v ) := f ( x )( ⊗ m v ) . The dual operator is defined by π m ∗ : C −∞ ( SM ◦ e ) → C −∞ ( M ◦ e , ⊗ mS T ∗ M ◦ e ) , h π m ∗ u, f i := h u, π ∗ m f i Next, we define the operator D := S ◦ ∇ : C ∞ c ( M ◦ e , ⊗ mS T ∗ M e ) → C ∞ c ( M ◦ e , ⊗ m +1 S T ∗ M ◦ e )by composing the Levi-Civita connection ∇ with the symmetrization of tensors S : ⊗ m +1 T ∗ M ◦ e → ⊗ m +1 S T ∗ M ◦ e . The divergence of m -cotensors is the adjoint differentialoperator, which is given by D ∗ f := −T ( ∇ f ) where T : ⊗ mS T ∗ M → ⊗ m − S T ∗ M denotesthe trace map defined by contracting with the Riemannian metric: T ( q )( v , . . . , v m − ) := n X i =1 q ( e i , e i , v , . . . , v m − ) (5.14)if ( e , . . . , e n ) is a local orthonormal basis of T M e . Each u ∈ L ( SM e ) function canbe decomposed using the spectral decomposition of the vertical Laplacian ∆ v in thefibers of SM e (which are spheres) u = ∞ X k =0 u k , ∆ v u k = k ( k + n − . (5.15) ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 33 where u k are L sections of a vector bundle over M e ; see [GuKa2, PSU].When (5.2) holds for some p ∈ (2 , ∞ ), we define just as for m = 0 the X-raytransform on ⊗ mS T ∗ M as the bounded operator for all p ∈ (2 , ∞ ) I m := I π ∗ m : L p ( M ; ⊗ mS T ∗ M ) → L ( ∂ − SM, dµ ν ) . (5.16)The adjoint I ∗ m : L ( ∂ − SM, dµ ν ) → L p ′ ( M ; ⊗ mS T ∗ M ) is bounded if 1 /p ′ + 1 /p = 1and it is given by I ∗ m = π m ∗ I ∗ . The operator Π m is simply defined as the boundedself-adjoint operator for p ∈ (2 , ∞ ) and 1 /p ′ + 1 /p = 1Π m := I ∗ m I m = π ∗ Π π ∗ m : L p ( M ; ⊗ mS T ∗ M ) → L p ′ ( M ; ⊗ mS T ∗ M ) . (5.17)As for m = 0, we set Π em := π ∗ Π e π ∗ m , which can also be seen as ( I em ) ∗ I em on if I em = I e π ∗ m is the X-ray transform on m cotensors on M e . Repeating the arguments of[Gu, Theorem 3.5] but adapted to our case we get directly Proposition 5.9.
Assume that the geodesic flow on M has no conjugate points andthat the trapped K is hyperbolic. For m ≥ , the operator Π em is a pseudo-differentialoperator of order − on the bundle ⊗ mS T ∗ M ◦ e , which is elliptic on ker D ∗ in the sensethat for all ψ ∈ C ∞ c ( SM ◦ e ) there exist pseudo-differential operators Q, S, R on M ◦ e with respective order , − , −∞ so that Qψ Π em ψ = ψ + Dψ Sψ D ∗ + R (5.18)The only difference with [Gu, Theorem 3.5] is that the flow is not hyperbolic every-where anymore, but using that the bundle E ∗± are transverse to the annihilator H ∗ ofthe vertical bundle V = ker dπ , the proof reduces to be the same, just as we explainedin the proof of Proposition 5.7 for m = 0. We do not repeat the arguments, as it doesnot bring anything new. The same result as (5.13) also holds for I m and I ∗ m since Π m is a ΨDO of order −
1: if
O ⊂ M ◦ is any compact domain (with non-empty interior)with smooth boundary, I m : H − / ( O , ⊗ mS T ∗ M ) → L ( ∂ − SM ; dµ ν ) . (5.19)5.3. Injectivity of X-ray transform on symmetric tensors.
In this section, weuse the Pestov identity and the smoothness property in Corollary 5.8 to prove injec-tivity of X-ray transform on functions and 1-forms in case of hyperbolic trapping. Theproof is basically the same as in the simple domain setting, once we have proved thesmoothness of elements in ker I m ∩ ker D ∗ . Theorem 6.
Let ( M, g ) be a compact Riemannian manifold with strictly convex bound-ary. Assume that the geodesic flow has no conjugate points, that the trapped set K ishyperbolic.1) Let f ∈ L p ( M ) + H − / ( M ◦ ) with p > such that I f = 0 , then f = 0 .2) Let f ∈ C ∞ ( M ; T ∗ M ) + H − / ( M ◦ ; T ∗ M ) such that I f = 0 , then there exists ψ ∈ C ∞ ( M ) + H / ( M ◦ ) vanishing at ∂M such that f = dψ .3) Assume that the sectional curvatures of g are non-positive, then if for m > , f m ∈ C ∞ ( M ; ⊗ mS T ∗ M ) satisfies I m f m = 0 , then f m = Dp m − for some p m − ∈ C ∞ ( M ; ⊗ m − S T ∗ M ) which vanishes at ∂M .Proof. Let us first show 1) and 2). Using Hodge decomposition we write f = dψ + f ′ with f ′ ∈ C ∞ ( M, T ∗ M ) + H − / ( M ◦ , T ∗ M ) satisfying D ∗ f ′ = 0 and ψ ∈ C ∞ ( M ) + H / ( M ◦ ) satisfying ψ | ∂M = 0. This can be done by taking ψ := ∆ − D δf where ∆ − D is the inverse of the Dirichlet Laplacian on ( M, g ) and δ := d ∗ = D ∗ on 1-forms. Noticethat f ′ is smooth near ∂M since f is (using ellipticity of ∆ D ). Since I dψ = 0 weget Π f ′ = 0 and Π e f ′ = 0. By applying (5.18) to f ′ with ψ = 1 on M , we get that f ′ ∈ C ∞ ( M ◦ ) thus f ′ ∈ C ∞ ( M ). Since also Π f = 0, Corollary 5.8 then implies that f and f ′ are smooth. By Proposition 5.5, we see that there exists u j ∈ C ∞ ( SM )for j = 0 , Xu = π ∗ f and Xu = π ∗ f ′ , with u j vanishing to all orderon ∂SM . Now since the functions u j are smooth and vanish at the boundary ∂SM ,Pestov’s identity [PSU, Proposition 2.2. and Remark 2.3] holds here in the same wayas it does for simple manifolds with boundary or for closed manifolds: ||∇ v Xu j || L = || X ∇ v u j || L − h R ∇ v u j , ∇ v u j i + ( n − || Xu j || L (5.20)where ∇ v is the covariant derivative in the vertical direction of SM , mapping functionson SM to sections of the bundle E → SM with fibers E ( x,v ) := { w ∈ T x M ; g x ( w, v ) = 0 } ,R is the curvature tensor acting on E by R ( x,v ) w := R ( w, v ) v ∈ E ( x,v ) , and X actson sections of E by differentiating parallel transport along the geodesic (see Section2 of [PSU]). Then the proof of Lemma 11.2 of [PSU] and Proposition 7.2 of [DKSU]is based on Santalo’s formula (2.20) and thus applies as well in our setting (ie. theboundary is strictly convex, there is no conjugate points and Γ + ∪ Γ − has Liouvillemeasure 0), then for all Z ∈ C ∞ ( SM, E ) || XZ || L − h RZ, Z i ≥ Z = 0. In particular, since ∇ v Xu = ∇ v f = 0, we deducefrom (5.20) that f = 0, and since ||∇ v Xu || L = ( n − || f || L , we deduce from (5.20)that ∇ v u = 0 and thus u = π ∗ ψ ′ for some smooth function ψ ′ on M which vanishesto all order at ∂M ; this implies that Xu = π ∗ dψ ′ . Notice that if D ∗ f ′ = 0, then D ∗ f ′ = ∆ g ψ ′ = 0 and therefore ψ ′ = 0 since ψ ′ vanishes at ∂M . Thus f ′ = 0.Finally, the case with m > g is non-positive uses the proofof [CrSh] (in the closed case) and [PSU, Section 11] (in the case of simple domains).If I m f = 0, we also have I em f m = 0 and thus Π e π ∗ m f m = 0. By Proposition 5.5, thereexists u = − R + (0) π ∗ m f m = − R − (0) π ∗ m f m smooth in SM such that Xu = π ∗ m f m and u | ∂SM = 0. Non-positive curvature implies that the flow is 1-controlled in the sense of ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 35 [PSU] and once we know that Xu = π ∗ m f m with u smooth and vanishing at ∂M , theproof of Theorem 11.8 in [PSU] (that proof is detailed in Section 9 and 11) based onPestov identity applies verbatim in our case . We do not repeat it here as it does notbring anything new. (cid:3) We get Theorem 4 and Theorem 1 as a direct corollary:
Proof of Theorem 4 . We only prove 2) since the conformal case 1) is easier anda direct consequence of point 1) in Theorem 6. If the metrics are lens equivalent,Γ ± ∩ ∂ ± SM are the same for all metrics, and for a fixed y := ( x, v ) ∈ ∂ − SM \ Γ − , thegeodesic γ s ( y ; t ) with t ∈ [0 , ℓ + ( y )] depends smoothly on s (by general ODE arguments)and by differentiating ∂ s ℓ + ( y ) = 0, we obtain that q s := ∂ s g s is a smooth symmetric2-tensors satisfying I s q s = 0 if I s is the X-ray for g s on symmetric 2 cotensors. Theargument is standard and detailed in [Sh, Section 1.1]. Applying Theorem 6 with m = 2 in non-positive curvature shows that q s = D s p s for some smooth 1-form p s vanishing at ∂M . The tensor p s can be written as p s = (∆ D s ) − D ∗ s q s if ∆ D s := D ∗ s D s with Dirichlet condition at ∂M (this is invertible, see [Sh]). Then we argue like in theproof of [GuKa1, Theorem 1]: by ellipticity of ∆ D s and smootness in s , p s is smoothin s . Then one can construct a smooth family of diffeomorphisms φ s which are theidentity on ∂M so that φ − s ∂ s φ s = p s and φ = Id (here we view p s as a vector field).This concludes the proof. (cid:3) Proof of Theorem 1 . A negatively curved manifold with strictly convex boundaryhas hyperbolic trapped set K (see [Kl2, § (cid:3) Invariant distributions with prescribed push-forward.
We will show theexistence of invariant distributions on SM with prescribed push-forward. This corre-sponds essentially to surjectivity of I ∗ and of I ∗ on ker D ∗ . Proposition 5.10.
We make the same assumptions as in Theorem 6.1) For any f ∈ H s ( M ) for s > , there exists w ∈ ( ∩ u< H u ( SM e )) ∩ L ( SM e ) suchthat Xw = 0 in SM ◦ e and π ∗ w = f in M . Moreover, if f ∈ C ∞ ( M ) , w has wavefrontset satisfying WF( w ) ⊂ E ∗ + ∪ E ∗− and its boundary value ω = w | ∂SM satisfies (5.8) and ω ∈ L S g ( ∂SM ) , and w ∈ H s ( SM e ) for some s > .2) Let f ∈ C ∞ ( M ; T ∗ M ) satisfying D ∗ f = 0 , then there exists w ∈ L p ′ ( SM e ) suchthat Xw = 0 in SM ◦ e and π ∗ w = f in M , with WF( w ) ⊂ E ∗ + ∪ E ∗− and ω := w | ∂SM satisfies (5.8) and is in L S g ( ∂SM ) .Proof. Let Y be a closed manifold extending smoothly M e across its boundary, extendthe metric smoothly to Y (and still call the extension g ). Let ψ ∈ C ∞ c ( Y ) withsupport in M e which is equal to 1 on a neighborhood of M and write ψ := π ∗ ( ψ ) its lift to SY . Using Proposition 5.7, define the elliptic ΨDO of order − YP = ψ Π e ψ + (1 − ψ )(1 + ∆ g ) − / (1 − ψ ) : H − s ( Y ) → H − s +1 ( Y )bounded for all s ≥
0; here ∆ g is the Laplacian on ( Y, g ). Thus there exists
C > K : H − s ( Y ) → H − s +1 ( Y ) a bounded ΨDO (of order −
1) such that for all f ∈ H − s ( Y ) || P f || H − s ( Y ) ≥ C || f || H − s ( Y ) − || Kf || H − s +1 ( Y ) and thus the range of P is closed. Consequently, by Banach closed range theorem, P ∗ : H s − ( Y ) → H s ( Y ) has closed range. Note that P ∗ has the same form as P ,and to prove its surjectivity, it suffices to prove injectivity of P . If P f = 0, then f ∈ C ∞ ( Y ) by ellipticity of P , and (1 − ψ ) f = 0 since (1 + ∆ g ) − / is injective, and h Π e ( ψ f ) , ψ f i L = 0. This implies that I e ( ψ f ) = 0 and by Theorem 6 applied with M e instead of M , we get ψ f = 0, thus f = 0. We deduce that if f ∈ H s ( M ), takingan extension ˜ f ∈ H s ( Y ) supported in the region where ψ = 1, there exists a unique u ∈ H s − ( Y ) such that P ∗ u = ˜ f . Note that if f is smooth, u is smooth by ellipticityof P ∗ . In particular, we get ψ Π e ( ψ u ) = ˜ f and taking w := Π e ( ψ u ), we get Xw = 0in SM e , π ∗ w = f in M , and by Proposition 5.4, we obtain the desired regularity for w and the properties of its restriction w | ∂SM and (5.8). This proves 1).The proof of 2) is essentially the same as in [DaUh, Lemma 2.2] once we knowProposition 5.9 and the kernel of I . We just recall very briefly the argument andrefer to [DaUh, Lemma 2.2] for details. First, by [KMPT, Corollary 3.3] (see alsothe last remark of that paper for the manifold case) there is a bounded extension op-erator E : ker D ∗ | L ( M,T ∗ M ) → ker D ∗ | L ( M ◦ e ,T ∗ M e ) which restricts continuously to E :ker D ∗ | C ∞ ( M,T ∗ M ) → ker D ∗ | C ∞ c ( M ◦ e ,T ∗ M e ) then if r M : L ( M e , T ∗ M e ) → L ( M, T ∗ M ) isthe restriction to M , we get from Proposition 5.9 that r M Π e ψ Q ∗ E = Id + r M R ∗ E asa map on ker D ∗ | L ( M,T ∗ M ) with R smoothing on M ◦ e . This implies that the range ofId + r M R ∗ E is closed with finite codimension, and the same holds on ker D ∗ | C ∞ ( M,T ∗ M ) .Then r M Π e ψ Q ∗ E (ker D ∗ | C ∞ ( M,T ∗ M ) ) has closed range in ker D ∗ | C ∞ ( M,T ∗ M ) with finitecodimension and thus r M Π e ψ Q ∗ ( C ∞ ( M ◦ e , T ∗ M e )) has closed range with finite codi-mension in ker D ∗ | C ∞ ( M,T ∗ M ) . The kernel of the adjoint is trivial by using Theorem 6just as in [DaUh, Lemma 2.2.]. This shows that there is u ∈ C ∞ ( M e , T ∗ M e ) such that r M Π e u = f , and thus setting w := Π e π ∗ u we get the result. (cid:3) Determination of the conformal structure for surfaces
In this Section, we will study the lens rigidity for surfaces with strictly convexboundary, no conjugate points and hyperbolic trapped set. To recover the conformalstructure from the scattering map, we shall use most of the results proved abovetogether with the approach of Pestov-Uhlmann [PeUh] which reduces the scatteringrigidity to the Calder´on problem on surfaces.
ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 37
For the oriented Riemannian surface M e with boundary, the unit tangent bundle SM e is a principal circle bundle, with an action S × SM e → SM e , e iθ . ( x, v ) = ( x, R θ v )where R θ is the rotation of angle + θ . This induces a vector field V generating thisaction, defined by V f ( x, v ) = ∂ θ ( f ( e iθ . ( x, v )) | θ =0 . We then define the vector field X ⊥ := [ X, V ] and the basis (
X, X ⊥ , V ) is an orthonormal basis of SM e for the Sasakimetric. The space SM e splits into SM e = V ⊕ H where V = R V = ker dπ is thevertical space, and H = span( X, X ⊥ ) the horizontal space which can also be definedusing the Levi-Civita connection (see for example [Pa]). Following Guillemin-Kazhdan[GuKa1], there is an orthogonal decomposition (Fourier series in the fibers) L ( SM ◦ e ) = M k ∈ Z Ω k , with V w k = ikw k if w k ∈ Ω k (6.1)where Ω k is the space of L sections of a complex line bundle over M ◦ e . Similarly, onehas a decomposition on ∂SML ( ∂SM ) = M k ∈ Z Ω ′ k , with V ω k = ikω k if ω k ∈ Ω ′ k (6.2)using Fourier analysis in the fibers of the circle bundle.6.1. Hilbert transform and Pestov-Uhlmann commutator relation.
The Hilberttransform in the fibers is defined by using the decomposition (6.1): H : L ( SM ◦ e ) → L ( SM ◦ e ) , H ( X k ∈ Z w k ) = − i X k ∈ Z sign( k ) w k . with sign(0) := 0 by convention. It is skew-adjoint and Hu = Hu , thus we can extendcontinuously H to C −∞ ( SM ◦ e ) → C −∞ ( SM ◦ e ) by the expression h Hu, ψ i := −h u, Hψ i , ψ ∈ C ∞ c ( SM ◦ e )where the distribution pairing is h u, ψ i = R SM e uψdµ when u ∈ L ( SM ◦ e ). Similarly,we define the Hilbert transform in the fibers on ∂SMH ∂ : C ∞ ( ∂SM ) → C ∞ ( ∂SM ) , H ∂ ( X k ∈ Z ω k ) = − i X k ∈ Z sign( k ) ω k and its extension to distributions as for SM e . For smooth w ∈ C ∞ c ( SM ◦ e ) we have that( Hw ) | ∂SM = H ∂ ω, with ω := w | ∂SM (6.3)thus the identity extends by continuity to the space of distributions in SM ◦ e withwave-front set disjoint from N ∗ ( ∂SM ) since, by [H¨o, Theorem 8.2.4], the restrictionmap C ∞ ( SM ◦ e ) → C ∞ ( ∂SM ) obtained by pull-back through the inclusion map ι of(2.11) extends continuously to the space of distributions on SM ◦ e with wavefront setnot intersecting N ∗ ( ∂SM ). By [Gu, Lemma 3.5], we see that WF( Hu ) ⊂ WF( u ) for all u ∈ C −∞ ( SM e ) and the same holds for H ∂ and u ∈ C −∞ ( ∂SM ). The followingcommutator relation between Hilbert transform and flow follows easily from the Fourierdecomposition and was proved by Pestov-Uhlmann [PeUh, Theorem 1.5]:if w ∈ C ∞ ( SM ◦ e ) , [ H, X ] w = X ⊥ w + ( X ⊥ w ) (6.4)where w = π π ∗ ( π ∗ w ) and π ∗ w ( x ) = R S x M e w ( x, v ) dS x ( v ) for smooth w . Notice that w ∈ C ∞ ( SM ◦ e ) w ∈ C ∞ ( SM ◦ e ) extends continuously to C −∞ ( SM ◦ e ) since π is asubmersion (the pullback π ∗ extends to distributions), then the relation (6.4) extendscontinuously to C −∞ ( SM ◦ e ). We also have, for any w ∈ C −∞ ( SM ◦ e ) X ⊥ w = 12 π π ∗ ( ∗ d ( π ∗ w )) . (6.5)where ∗ : T ∗ M e → T ∗ M e is the Hodge-star operator on 1-forms. We use the odd/evendecomposition of distributions with respect to the involution A ( x, v ) = ( x, − v ) on SM e , SM and ∂SM , as explained in the end of Section 4.2. The operator X mapsodd distributions to even distributions and conversely. The operator H maps odd(resp. even) distributions to odd (resp. even) distributions, we set H ev w := H ( w ev )and H od w := H ( w od ). We write similarly H ∂, ev and H ∂, od for the Hilbert transform on(open sets of) ∂SM and the relation (6.3) also holds with H ∂, ev replacing H ∂ if w iseven. Taking the odd part of (6.4), we have for any w ∈ C −∞ ( SM ◦ e ) H od Xw − XH ev w = 12 π π ∗ ( ∗ d ( π ∗ w )) = X ⊥ w . (6.6)6.2. Determination of the conformal structure from scattering map.
For func-tions ω ∈ C ∞ ( ∂SM ), the function π ∗ ω is smooth on ∂M , given by the expression π ∗ ω ( x, v ) = π R S x M e ω ( x, v ) dS x ( v ) and thus if w ∈ C ∞ ( SM ◦ e ) and ω = w | ∂SM , onehas π ∗ ω = ( π ∗ w ) | ∂M . As above, the restriction map C ∞ ( SM ◦ e ) → C ∞ ( ∂SM ), ex-tends continuously to the space of distributions on SM ◦ e with wavefront set includedin E ∗ + ∪ E ∗− (since this does not intersect N ∗ ( ∂SM )). Therefore, for w ∈ C −∞ ( SM ◦ e )with WF( w ) ⊂ E ∗ + ∪ E ∗− , we have π ∗ ω = ( π ∗ w ) | ∂M , with ω := w | ∂SM (6.7)in the distribution sense (in fact, as in the proof of Proposition 5.10, it is easily checkedthat π ∗ w ∈ C ∞ ( M ◦ e )).For an oriented Riemannian surface ( M, g ) with boundary, the space of holomorphicfunctions can be described as follows: f = f + if is holomorphic if ∗ df = df where ∗ is the Hodge star operator. We use the notation P ( f ) ∈ C ∞ ( M ) for the uniquesolution of ∆ g P ( f ) = 0 with P ( f ) = f on ∂M . Theorem 7.
Let ( M, g ) and ( M ′ , g ′ ) be two oriented Riemannian surfaces with thesame boundary N , and g | T N = g ′ | T N . For both surfaces, assume that the boundary isstrictly convex, the trapped set are hyperbolic, that (5.2) holds, and the metrics have
ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 39 no conjugate points. If ( M, g ) and ( M ′ , g ′ ) are scattering equivalent, then there existsa diffeomorphism φ : M → M ′ with φ | ∂M = Id and such that φ ∗ g ′ = e η g for some η ∈ C ∞ ( M ) satisfying η | ∂M = 0 .Proof. We shall follow the method of Pestov-Uhlmann [PeUh] and we will need touse most of the results from the previous sections. We work on (
M, g ) but all theresults below apply as well on ( M ′ , g ′ ). For f ∈ C ∞ ( N ), the harmonic extension P ( f )admits a harmonic conjugate P ( f ∗ ) if ∗ d P ( f ) = d P ( f ∗ ) or equivalently P ( f + if ∗ ) isholomorphic. We are going to prove the following statement: let f ∗ ∈ C ∞ ( N ), then2 π ( S ∗ g − Id)( H ∂, ev ω ) = ( S ∗ g − Id) π ∗ f ∗ (6.8)holds for some ω ∈ L S g ( ∂SM ) satisfying WF( ω − ) ⊂ E ∗ ∂, − and WF( ω + ) ⊂ E ∗ ∂, + , if andonly if I ∗ ω − = P ( f ) with P ( f − if ∗ ) holomorphic (6.9)where π ∗ E − = I ∗ (see Lemma 5.2) and ω ± := ω | ∂ ± SM .Let us prove the first sense. Let f ∈ C ∞ ( N ) so that P ( f ) admits a harmonicconjugate. Using Proposition 5.10, there exists w ∈ L ( SM e ) ∩ C ∞ ( SM e \ (Γ + ∪ Γ − ))satisfying Xw = 0 in M ◦ e in the distribution sense with π ∗ w = P ( f ) in M and ω := w | ∂SM ∈ L S g ( ∂SM ) , WF( ω ) ⊂ E ∗ ∂, + ∪ E ∗ ∂, − (6.10) ω − := ω | ∂ − SM , where E ∗± ,∂ ⊂ T ∗ Γ ± ( ∂SM ) are the bundles defined by (2.12) for themanifold M and π ∗ is the pushforward defined by (5.9) on SM . From (6.6) and usingthat H ev w is smooth in SM \ (Γ − ∪ Γ + ), we get XH ev w = − π π ∗ ( ∗ d P ( f )) (6.11)as smooth functions on SM \ (Γ − ∪ Γ + ). Now, for any ψ ∈ C ∞ ( SM \ (Γ + ∪ Γ − )), IXψ = ( S ∗ g − Id)( ψ | ∂SM \ (Γ − ∪ Γ + ) )as a function on ∂ − SM \ Γ − . Applying I to (6.11) and using that P ( f − if ∗ ) isholomorphic then gives ( I is the X-ray transform on 1-forms)2 π ( S ∗ g − Id)(( H ev w ) | ∂SM ) = − I ( ∗ d P ( f )) = I ( d P ( f ∗ )) = IXπ ∗ ( P ( f ∗ )) = ( S ∗ g − Id) π ∗ f ∗ as smooth functions on ∂ − SM \ Γ − which are globally in L ( ∂ − SM, dµ ν ). Using (6.3)we thus obtain the identity (6.8).Next, we prove the converse. Conversely, let f ∗ ∈ C ∞ ( N ), let q ∈ C ∞ ( M ) with q | ∂M = f ∗ and let χ ∈ C ∞ c ( SM ◦ ) which is equal to 1 in { ρ > ǫ } with ǫ > ρ as in Section 2.1), thus on K . We write w := χ E − ω − and w := (1 − χ ) E − ω − and by (6.6), we get for j = 1 , HXw j − XHw j = π ∗ ( ∗ dπ ∗ w j ) . (6.12) By Proposition 4.6, WF( w ) ⊂ E ∗ + ∪ E ∗− thus π ∗ w ∈ C ∞ ( M ) (using ( E ∗− ∪ E ∗ + ) ∩ H ∗ = { } if H ∗ ⊂ T ∗ ( SM ◦ e ) is the annulator of the vertical bundle V = ker dπ ), and π ∗ w ∈ H / ( SM ◦ e ) with support containing K . We claim that we can apply I to(6.12) and view the result as a measurable function in ∂ − SM \ Γ − : for j = 2 we canapply I since all terms are smooth in SM \ (Γ − ∪ Γ + ) and we get a smooth function on ∂ − SM \ Γ − that is in L ( ∂ − SM ) and for j = 1 the only possible trouble is I ( ∗ dπ ∗ w )but this makes sense since I : H − / ( M ◦ , T ∗ M ) → L ( ∂ − SM, dµ ν ) is bounded just as I in (5.13) (see the remark after Proposition 5.9). Therefore, applying I to (6.12) andsumming for j = 1 ,
2, we obtain almost everywhere on ∂ − SM ( S ∗ g − Id)( H ∂, ev ω ) = IXH E − ( ω − ) = − π I ( ∗ dπ ∗ w + ∗ dπ ∗ w ) , this term is in L ( ∂ − SM, dµ ν ) and equal to π ( S ∗ g − Id) π ∗ f ∗ = π I ( dq ) by our assump-tion. Since we know that this term is smooth on ∂ − SM we obtain in L ( ∂ − SM, dµ ν ) I ( ∗ dI ∗ ω − + dq ) = 0 . By Theorem 6 one has ∗ dI ∗ ω − + dq = dψ for some ψ ∈ C ∞ ( M ) + H / ( M ◦ ) satisfying ψ | ∂M = 0. Applying first d and then d ∗ to that equation and using ellipticity, we get ψ − q ∈ C ∞ ( M ) and I ∗ ω − ∈ C ∞ ( M ) and both functions are harmonic conjugate,which means that (6.9) holds with f := ( I ∗ ω − ) | ∂M .We can finally finish the proof. All that we said above applies also on ( M ′ , g ′ ) and weshall put prime for objects related to g ′ . Let α : SM ′ → SM be the map (3.2), so that α ◦ S g ′ = S g ◦ α by assumption. Remark that for each ω ∈ C ∞ ( ∂SM ), ( ω ◦ α ) k = ω k ◦ α in the Fourier decomposition (6.2), and thus α ∗ ( H ∂, ev ω ) = H ′ ∂, ev ( α ∗ ω ) . (6.13)This identity extends to ω ∈ L ( ∂SM ) by continuity. Let f ∗ ∈ C ∞ ( N ) and assumethat there exists f ∈ C ∞ ( N ) so that P ( f + if ∗ ) is holomorphic in ( M, g ), then wehave proved that there is ω ∈ L S g ( ∂SM ) satisfying (6.8), π ∗ ω = f and (6.10). Using α ◦ S g ′ = S g ◦ α and π ◦ α = π , together with (6.13), we get( S ∗ g ′ − Id)( H ′ ∂, ev ω ′ ) = ( S ∗ g ′ − Id) π ∗ f ∗ . (6.14)with ω ′ := α ∗ ω . We can use Lemma 3.2 which implies that WF( ω ′ ) ⊂ E ′ ∗ ∂, + ∪ E ′ ∗ ∂, − ,and since ω ′ ∈ L S g ( SM ′ ), we get by (6.9) applied with ( M ′ , g ′ ) that I ′ ∗ ( ω ′ ) − i P ′ ( f ∗ )is holomorphic in ( M ′ , g ′ ). Since I ′ ∗ ( ω ′ ) | ∂M = π ∗ ω = f , we have shown that allboundary value of a holomorphic function on ( M, g ) is also the boundary value of oneon ( M ′ , g ′ ). Exchanging the role of ( M, g ) and ( M ′ , g ′ ), we show that the space ofboundary values of holomorphic functions on ( M, g ) and (
M, g ′ ) are the same. Theexistence of the conformal diffeomorphism φ : M → M ′ then follows from the work ofBelishev [Be]. (cid:3) ENS RIGIDITY FOR MANIFOLDS WITH HYPERBOLIC TRAPPED SET 41
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