Perturbation of a warped product metric of an end and the growth property of solutions to eigenvalue equation

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.