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Absence of Resonances near Critical Line for CC Manifolds

We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.

The method we use is more or less standard, the main point is to carry out the parametrix taking into account the parameter α(y) that appears in the Laplace-Beltrami operator and which corresponds to the sectional curvatures at the boundary. The parametrix consists of an approximation near and away from the boundary. Near the boundary we use a CC metric with polyhomogeneus metric together with the uniform estimates on the resolvent up to the critical line ℜξ = n/2 given in Proposition 2.1, and then take a suitable polyhomogeneous metric. Away from the boundary we use the hyperbolic model, for which we prove the necessary energy estimates in Proposition 1.1.
Scattering theory on CC manifolds was first studied in this generality by Borthwick [Borth]. In [Mar09] the author proves inverse theorems on CC manifolds. In this setting we have from [Mar11] combined with Theorem 1.1 of [Borth] the following Theorem 1.1. The essential spectrum of ∆ g is [α 2 0 n 2 4 , ∞) and there are no embedded eigenvalues except possibly at α 2 0 n 2 4 . We can meromorphically extend the resolvent R(ξ) := (∆ g − α 2 0 ξ(n − ξ)) −1 to the whole complex plane without some intervals (c.f. [Borth]). A vertical interval near ξ = n/2, and some horizontal intervals to the left of ℜξ = n/2. (See Figure 1) Re ξ =n/2 Figure 1. Bold lines are not covered by the meromorphic extention Let X be a compact C ∞ manifold with boundary ∂X which is equipped with a Riemannian metric g such that for any defining function "x" of ∂X x 2 g is a C ∞ non-degenerate Riemannian metric up to ∂X. It can be shown (c.f. [Grah1,Mar09]) that there exists a diffeomorphism Ψ : V −→ [0, ǫ) × R n such than g satisfies Ψ * g = dx 2 α 2 (y)x 2 + h(x, y, dy) x 2 , (1.1) for V a collar neighborhood of the boundary ∂X. Here h(x), x ∈ [0, ǫ), is a family of metrics on ∂X, and α ∈ C ∞ (∂X). To simplify notation we drop the pull-back coordinate function Ψ and call the compact manifold (X, g) with boundary ∂X and metric g a CC manifold. We let α 0 = min ∂X α and α 1 = max ∂X α. Without loss of generality we could assume that α 0 = 1, however we keep α 0 for clarity purposes. We assume that h(x), x ∈ [0, ǫ), is a family of metrics on ∂X which has a polyhomogeneous expansion near the boundary ∂X of the form where U i ∈ N 0 and h ij are symmetric 2-tensors at ∂X.
We say that the metric g is non-trapping if every geodesic approaches the boundary ∂X. Under this assumption using Propositions 1.1 and 2.1 we prove in Section 4 our main theorem Theorem 1.2. Let (X, g) be a CC manifold with metric g as in (1.1), and x a boundary defining function. If g is non-trapping, there exists C 1 , C 2 > 0 such that the weighted resolvent x 1/2 R(ξ)x 1/2 extends analytically across {ξ ∈ C : |ℑξ| > C 2 , ℜξ > n 2 } to as a bounded operator in L 2 (X). Let (M, H 0 ) be a compact Riemannian manifold, for the parametrix near ∂X we use (X 0 , g 0 ) the manifold and them take H 0 = α 2 h 0 . We are interested in the behavior of the resolvent near x = 0, thus we consider functions supported in (0, 1) x × M that could be written in local coordinates as with a i,k (x, y) polyhomogneous in x.
Throughout this note C is an arbitrary constant that can change every time it is written.
In the final section we include an application to the wave equation. To state the corollary let f 1 , f 2 ∈ Here D t = −α 0 ∂t i . Our result will only hold for high energies so we need to take v = χ(t)u and 0 < ǫ << 1 so that χ(t) is a smooth function so that We prove the following corollary Corollary 1.1. Let u be a solution to (1.6) and let v be as above. Then Acknowledgements The author acknowledges support from CNPq and thank F. Cardoso for encouragement while writing this note. The author also thanks A. Sá Barreto for suggesting the problem and P. Hislop for many helpful discussions.

Uniform estimates up to the critical line
In [CardVod] Cardoso-Vodev obtained uniform estimates for the resolvent of the Lapacian for a metric of the form where σ(r) is a one-parameter family of Riemannian metrics. The transformation x = e −r puts the AH metric (1.1) into the form of the metric studied in [CardVod] with σ(r) = e 2r h(e −r ). In that same spirit we consider the metric (2.2) g r = (α(y)) −2 dr 2 + σ(r), where the metric σ no longer has a polyhomogeneous expansion, it is actually in the class studied in [CardVod]. Combining [Mar09] and [Mar11] we can prove that the resolvent R(ξ) is analytic for ℜξ > n/2. We denote by |σ| the determinant of σ. The corresponding Laplace-Beltrami operator is We follow the notation of [CardVod] and denote |σ| 1/2 by p, we also write α instead of α(y) keeping in mind that α is a function on the boundary. If we conjugate ∆ g by p 1/2 , we obtain The method of [CardVod] under the following assumptions gives the following: Proposition 2.1. Let (X,g) be a conformally compact manifold with metric g as in (1.1), and x a boundary defining function. Then there exists c > 0 such that the weighted resolvent x 1/2 R(ξ)x 1/2 = x 1/2 (∆ g − α 2 0 ξ(n − ξ)) −1 x 1/2 extends continuously from {ℜξ > n/2, |ℑξ| ≥ 1} to {ℜξ ≥ n/2, |ℑξ| ≥ 1} on L 2 (X) and the extension satisfies for p = 0, 1, |ℑξ| ≥ 1, and 0 ≤ ℜξ − n/2 ≤ 1; where the Sobolev norm is with respect to the metric g.
The theorem follows from the previous proposition by noticing that The theorem, with r 1+δ 2 = (ln x) 1+δ 2 instead of x, follows from the last inequality by factoring out λ −2 from P. Lastly, noticing that x 1/2 < | ln x| 1+δ 2 for x > 0 sufficiently small we obtain the theorem.

Model
Writing the Laplace-Beltrami operator ∆ g in local coordinates z = (x, y), logaritmic terms appear. It is a differential operator of order two in (x∂ x , x∂ y ) with polyhomogeneous coefficients. Hence we denote by pol Diff k 0 (X) the space of polyhomogeneous differential operators of order k that could be written in local coordinates as with a i,k (x, y) polyhomogneous in x.
The variable sectional curvature gives a resolvent which lives on spaces of polyhomogeneous operators A(X), however the definition of such spaces is well known and we do not discuss these spaces in more detail here since we are just going to look at the resolvent as a linear map. We refer the interested reader to [Borth,Mar09,Mar11].
The part of the parametrix near the boundary will be given by (X 0 , g 0 ) we define next: let (M, H 0 ) be a compact Riemannian manifold then (X 0 , g 0 ) is the manifold Note that We now prove Proposition 1.1 Proof. We prove the proposition for p = 0, the other cases follow from pol Diff k 0 (X 0 ) being contained in L(H s (X 0 ), H s−k (X 0 )) ( [FrHis]). By the spectral theorem we can decompose with {µ j } j∈N0 the eigenvalues of ∆ H0 associated to an orthonormal basis {ψ j } j∈N0 of L 2 (M ) eigenvectors and counted with multiplicities.
We have the decomposition We let U J : L 2 (R, dr) −→ L 2 (R, dr) be the isometric translation for µ j = 0, we have U j P (j) 0 U j = Q, with Q = −∂ 2 r + e 2r + n 2 4 . We set k = ξ − n/2 to simplify notation.

Absence of resonances near critical line
In this section we prove Theorem 1.2.
We have I * I * ψ j = ψ j and I * I * i * ψ j = i * ψ j .
For the first step of the parametrix we note the there exist operators D R and D L in pol Diff 2 0 (X) such that ∆ g − α 2 I * ∆ g0 ψ 3 I * = xD R + (x 2 ∆ h − α 2 x 2 ∆ H0 ), Notice that (4.1) with D 1 in pol Diff 1 0 (X). We assume we can write h(x, y, dy) ∼ h 0 (y, dy) + i∈N0 where D R and D L are not necessarily the same as before.
By (4.4) we can now apply the same parametrix as the one used in [Guil1] to prove the absence of resonances exponentially close to the critical line. For ℜ(ξ) > n/2 we have We let χ 1 := 1 − i * ψ 1 , and χ 0 a smooth function with compact support on X which is equal to 1 on the support of χ 1 . We denote by D p (resp. D p 0 ) all differential operator in Diff p (X) with support in Supp(i * ψ 3 ) (resp. Diff p 0 (X 0 ) with support in Supp(ψ 3 )). We have D p x = xD p , D p 0 I * = I * D p , and D p I * = I * D p 0 .

Asymptotics of the wave equation
In this section we give a second application of the resolvent estimate given in Theorem (2.1) to the wave equation on a CC manifold. We prove Corollary 1.1.
The corollary now follows since we obtained polynomial bounds on the resolvent R(λ) and F is Schwartz.