Formule de Poisson pour les vari�t�s riemanniennes

Chazarain, Jacques

doi:10.1007/bf01418788

Cited by 193 publications

(115 citation statements)

References 6 publications

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“…(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.…”

Section: Introductionmentioning

confidence: 93%

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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Section: Introductionmentioning

confidence: 93%

The diffractive wave trace on manifolds with conic singularities

Ford¹,

Wunsch²

2017

Advances in Mathematics

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“…Soon after my lectures in Nice, J. Chazarain [20] saw how to derive the qualitative form of the trace for the wave kernel using FIO's.…”

Section: About the Historymentioning

confidence: 99%

Spectrum of the Laplace operator and periodic geodesics: thirty years after

Verdìère¹

2007

Ann. inst. Fourier

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“…• Nous étudions ensuite la propagation des singularités pour le groupe à un paramètre engendré par ^ -, et donnons dans ce cas l'analogue de la relation de Poisson pour les variétés riemanniennes compactes avec ou sans bord [4], [11], [13].…”

Section: Pseudo-laplaciens II Par Yves Colin De Verdiereunclassified

“…Les opérateurs A,, étant à résolvante compacte, on peut étudier les singularités de la fonction spectrale comme cela a été fait pour les opérateurs elliptiques sur une variété compacte avec ou sans bord sous le nom de « relation de Poisson » ( [4], [11] et [13]); si on posê…”

Section: La Propagation Des Singularitésunclassified

Pseudo-laplaciens II

Verdìère¹

1983

Annales De L’institut Fourier

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Formule de Poisson pour les vari�t�s riemanniennes

Cited by 193 publications

References 6 publications

The diffractive wave trace on manifolds with conic singularities

The diffractive wave trace on manifolds with conic singularities

Spectrum of the Laplace operator and periodic geodesics: thirty years after

Pseudo-laplaciens II

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