  We obtain a vanishing result for solutions of the inequality $|\Delta u|\le
q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped
cylindrical end of a Riemannian manifold. The appropriate decay condition at
infinity on $u$ is related to the behavior of the potential functions $q_1$ and
$q_2$ and to the asymptotic geometry of the end. The main ingredient is a new
Carleman estimate of independent interest. Geometric applications to conformal
deformations and to minimal graphs are presented.
