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Sur le spectre des longueurs des surfaces de Riemann. (On the length spectrum of Riemann surfaces). (French) Zbl 0585.58049
Journ. Équ. Dériv. Partielles, Saint-Jean-De-Monts 1985, No. 1, Exp. No. 9, 9 p. (1985).
Let M be a surface having infinite volume in a metric of constant curvature \(-1.\) Suppose that M has finitely generated fundamental group with no parabolic elements. The author announces an asymptotic formula for the number v(\(\ell)\) of closed geodesics with length less than \(\ell\). In fact, \(v(\ell)\sim e^{\delta \ell}/\delta \ell\), where \(\delta\) (1-\(\delta)\) is the first eigenvalue of the Laplacian.
Reviewer: H.Donnelly
MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
Full Text: Numdam EuDML