Unique continuation at infinity: Carleman estimates on general warped cylinders

We obtain a vanishing result for solutions of the inequality $\mathrm{\mid }\mathrm{\Delta }u\mathrm{\mid }\le q_1\mathrm{\mid }u\mathrm{\mid }+q_2\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ 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.