On the wave equation on a compact Riemannian manifold without conjugate points

Bérard, Pierre

doi:10.1007/bf02028444

Cited by 183 publications

(256 citation statements)

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“…[Bér77], p.271). Now suppose that (0, y 0 ) ∈ Y α , a given boundary component, and that there exists x 0 such that…”

Section: Conic Geometrymentioning

confidence: 99%

The diffractive wave trace on manifolds with conic singularities

Ford¹,

Wunsch²

2017

Advances in Mathematics

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Abstract. Let (X, g) be a compact manifold with conic singularities. Taking ∆g to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group e −it √ ∆g arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points. IntroductionIn this paper, we consider the trace of the half-wave group U(t) def = e −it √ ∆g on a compact manifold with conic singularities (X, g). Our main result is a description of the singularities of this trace at the lengths of closed geodesics undergoing diffractive interaction with the cone points. Under the generic assumption that the cone points of X are pairwise nonconjugate along the geodesic flow, the resulting singularity at such a length t = L has the oscillatory integral representationwhere the amplitude a is to leading orderas |ξ| → ∞ and the index j is cyclic in {1, . . . , k}. Here, n is the dimension of X and k the number of diffractions along the geodesic, and χ is a smooth function supported in [1, ∞) and equal to 1 on [2, ∞). L 0 is the primitive length, in case the geodesic is an iterate of a shorter closed geodesic. The product is over the diffractions undergone by the geodesic, with D αj a quantity determined by the functional calculus of the Laplacian on the link of the j-th cone point Y αj , the factor Θ − 1 2 (Y αj → Y αj+1 ) is (at least on a formal level) the determinant of the differential of the flow between the j-th and (j + 1)-st cone points, and m γj is the Morse index of the geodesic segment γ j from the j-th to (j + 1)-st cone points. All of these factors are described in more detail below.To give this result some context, we recall the known results for the Laplace- is a well-defined distribution on R t . Moreover, it satisfies the "Poisson relation": it is singular only on the length spectrum of (X, g),L is the length of a closed geodesic in (X, g)} .(This was shown independently by Chazarain [Cha74]; see also [CdV73].) Subject to a nondegeneracy hypothesis, the singularity at the length t = ±L of a closed geodesic has a specific leading form encoding the geometry of that geodesic-the formula involves the linearized Poincaré map and the Morse index of the geodesic. The proofs of these statements center around the identificationwhere∆g is (half of) the propagator for solutions to the wave equation on X; in particular, U(t) is a Fourier integral operator.In this paper, we prove an analogue of the Duistermaat-Guillemin trace formula on compact manifolds with conic singularities (or "conic manifolds"), generalizing results of Hillairet [Hil05] from the case of flat surfaces with conic singularities. We again consider the trace Tr U(t), a spectral invariant equal to ∞ j=0 e −itλj . The Poiss...

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“…[Bér77], p.271). Now suppose that (0, y 0 ) ∈ Y α , a given boundary component, and that there exists x 0 such that…”

Section: Conic Geometrymentioning

confidence: 99%

The diffractive wave trace on manifolds with conic singularities

Ford¹,

Wunsch²

2017

Advances in Mathematics

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“…The upper bounds for R(λ) and R x (λ) are attained for round spheres and hence are sharp. Both local and integrated remainder estimates for the Weyl's law on manifolds under various geometric conditions have been actively studied in the last forty years (see [2], [8], [10], [14], [21], [24], [26], [29], [31], etc).…”

Section: Jakobson and I Polterovichmentioning

confidence: 99%

“…[10], [2], [16]) our method uses thermodynamic formalism (see, for example, [6], [20]). Let M be the universal cover of X.…”

Section: Estimates For Negatively Curved Manifoldsmentioning

confidence: 99%

“…We use the parametrix for E(t, x, y) (see [2], [32]): where r = d(x, y). The expression (3.7) is understood in the sense of generalized functions [11].…”

Section: Lower Bounds For the Spectral Function 75mentioning

confidence: 99%

“…The expression (3.7) is understood in the sense of generalized functions [11]. The coefficients u j (x, y) are the solutions of the transport equations along the geodesic joining x and y (see [2]). We recall that u j (x, x) = a j (x) as defined in (1.5).…”

Section: Lower Bounds For the Spectral Function 75mentioning

confidence: 99%

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Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Jakobson

Polterovich²

2005

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We announce asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in the local Weyl's law on Riemannian manifolds. In the negatively curved case, methods of thermodynamic formalism are applied to improve the estimates. Our results develop and extend the unpublished thesis of A. Karnaukh. We discuss some ideas of the proofs; for complete proofs see our extended paper on the subject.

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Fine Structure of Zoll Spectra

Zelditch

1997

Journal of Functional Analysis

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This paper is concerned with the wave equation on a Zoll manifold, i.e. a compact Riemannian manifold (M, g) all of whose geodesics are closed. By a result of V. Guillemin, such metrics exist in abundance on S 2 : for any odd function f on S 2 there exists a one-parameter deformation of the canonical metric g o thru Zoll metrics, g t =e f t with f t =tf+ } } } . And, as has been known for a long time, the periodicity of the geodesic flow leads to a near periodicity in the wave group U t =e it -2 . Precisely, there exists a unitary equivalence -2=R+Q &1 where [R, Q &1 ]=0, where the spectrum of R lies on an arithmetic progression and where Q &1 is a pseudodifferential operator of order &1. Hence the eigenvalues fall into cluster C k around spec(R) with widths O(k &1 ). The main result of this paper is to determine the limit distribution of the eigenvalues in the clusters. It was long ago shown by Weinsten [loc. cit] that (when properly normalized) the limit distribution has the form _ Q &1 d+ where d+ is the Liouville measure on S*M. We complete the calculation by determining _ Q &1 . It was conjectured [W.2][Ku] that up to universal constants, _ Q &1 =R({) where R is the Radon transform of (M, g) and { is the scalar curvature [W.2][Ku]; but it turns out there is a second term involving the Jacobi fields. In addition, we show that the cluster projections define almost isometric minimal immersions of (M, g) into spheres and that in the case of maximally degenerate Zoll metrics the errors are of rapid decay.

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On the wave equation on a compact Riemannian manifold without conjugate points

Cited by 183 publications

References 10 publications

The diffractive wave trace on manifolds with conic singularities

The diffractive wave trace on manifolds with conic singularities

Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Fine Structure of Zoll Spectra

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