The spectrum of the Laplacian of manifolds of positive curvature (Q1188200)

Let \(M\) be a complete \(n\) dimensional Riemannian manifold. Assume \(M\) has a pole at \(p_{0}\), i.e. the exponential map from \(TM_{p_{0}}\) is a diffeomorphism to \(M\). The authors show: Let \(M\) have non-negative sectional curvatures \(K_{r}\) in radial directions from the pole. Assume \(m=2\) or \(m\geq3\) and \(0\leq K_{r}\leq c_{n}(1-c_{n})r^{2}\) for all \(r>0\) where \(c_{n}=(n-2)/n\) and where \(r\) is the distance to the pole. \noindent Then there are no eigenfunctions of the Laplacian which are in \(L^{2}(M)\). Remark: This result was obtained previously by the first author under the additional assumption that outside a compact set, the metric was rotational symmetric and nonnegatively curved with no decay conditions on the curvature.