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A relationship between volume, injectivity radius, and eigenvalues. (English) Zbl 0724.58062

Let M be a compact hyperbolic manifold of dimension n and denote by A the measure of a ball with radius equal to the injectivity radius R of M. Using the Fourier expansion of the indicator function of this ball as well as the Selberg pretrace formula the author derives the formula \[ 1=A/V+(1/AV)\sum_{k>0}| h(r_ k)|^ 2, \] where \(\delta^ 2+r^ 2_ k=\lambda_ k\) is the k-th eigenvalue of the Laplace operator and \(\delta =(n-1)/2.\) The first application of this formula is an estimation of the number N(M) of small eigenvalues for M: \[ N(M)<\alpha (n,R)(V-A)/R^ 2, \] where \(\alpha (n,R)\sim [(n- 1)\omega_{n-1}]/[2^{n+1}\omega^ 2_{n-2}]\) for large R. Moreover, the author proves a similar estimation for very small eigenvalues, i.e. eigenvalues for which \(| r_ k-\delta_ i| <\epsilon /R.\) Finally the interesting consequences in dimension two are discussed.

MSC:

58J05 Elliptic equations on manifolds, general theory
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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