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

TL;DR

This paper investigates how small perturbations of a rotationally symmetric Riemannian metric affect the growth of solutions to the eigenvalue equation, establishing conditions that ensure the Laplacian has no eigenvalues.

Contribution

It introduces a method to analyze the impact of metric perturbations on eigenfunction growth, extending classical results on eigenvalue absence in symmetric manifolds.

Findings

Growth estimates for eigenfunctions under metric perturbations

Conditions guaranteeing the absence of eigenvalues

Extension of classical curvature bounds to perturbed metrics

Abstract

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.