Unique continuation at infinity: Carleman estimates on general warped cylinders

We obtain a vanishing result for solutions of the inequality $|\Delta u| \leq q_1 |u| + q_2 |\nabla u|$ 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.