Colbois, Bruno

1994-12-21, Colbois, Bruno, Dodziuk, Jozef

We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.

1993, Buser, Peter, Colbois, Bruno, Dodziuk, Jozef

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifold M. If the manifold is compact and its sectional curvatures K satisfy 1 less-than-or-equal-to K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume of M. 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 on a and the dimension only.

1990-7-21, Buser, Peter, Colbois, Bruno, Dodziuk, Jozef