The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

Abstract

Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.

arXiv comment"$\ge$" in the conditions $({\ast }_4)$ and $({\ast }_5)$ should be replaced by "$>$". $\gamma \ge \frac{n-1}{2}(b-a)$ in the conclusion of Theorem 1.3 should be replaced by $\gamma >\frac{n-1}{2}(b-a)$; trivial miss-calculation

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• Growth of the eigensolutions of Laplacians on Riemannian manifolds II: positivity of the initial energy
• Perturbation of a warped product metric of an end and the growth property of solutions to eigenvalue equation