Perturbation of a warped product metric of an end and the growth property of solutions to eigenvalue equations (Q428128)

Let \((M,g)\) be a non-compact complete Riemannian manifold and let \(\Delta\) be the Laplace-Beltrami operator; this operator is essentially self-adjoint. Assume that \(E\) is an end and that \(g|_E=dr^2+f(r)^2g_{\partial E}\) where \(r\) is the radial distance and \(g_{\partial E}\) is the induced metric on \(\partial E\). Under suitable conditions on the warping function, the author shows that \(\Delta\) has no eigenvalues and that the spectrum of \(\Delta\) is \([0,\infty)\). The conditions imposed imply that \(E\) is an expanding end. The author notes that if the manifold is rotationally symmetric and if the end shrinks at infinity, then infinitely many eigenvalues appear; similarly, if the end is asymptotically cylinder and \(f\) is non constant near infinity, then again infinitely many eigenvalues appear.