# zbMATH — the first resource for mathematics

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: