Unique Continuation at Infinity: Carleman Estimates on General Warped Cylinders

Abstract

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.