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Let M be an n-dimensional Riemannian manifold which possesses a pole o, namely, the exponential map expo : ToM → M at o is a diffeomorphism. In the case that the metric of M is rotationally symmetric, Donnelly’s theorem [1] implies that the Laplacian of M has no eigenvalue greater than (n − 1)2/4 if the curvature K of M satisfies K ≤ 0 and the decay conditions ∫∞ 0 r|K + 1| dr < +∞ and limr→∞ r|K + 1| = 0, where r denotes the distance from o. Also, in the rotationally symmetric case, Donnelly and Garofalo [3] proved that if the radial curvature K satisfies 0 ≥ K(r) and K(r) increases, then the Laplacian has no eigenvalue. As for a manifold whose metric is not necessarily rotationally symmetric, Donnelly also proved the following.

A NOTE ON THE ABSENCE OF EIGENVALUES ON NEGATIVELY CURVED MANIFOLDS