Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.