  When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical
order of the lower bound of radial curvatures for the absence of eigenvalues of
the Laplacian is equal to $ -\frac{1}{r}$, where $r$ stands for the distance to
the center point. In this paper, we shall perturb the Riemannian metric around
a rotationally symmetric one and derive growth estimates of solutions to the
eigenvalue equation, from which the absence of eigenvalues will follows.
