A Carleman inequality on product manifolds and applications to rigidity problems

Abstract In this article, we prove a Carleman inequality on a product manifold M × R M\times {\mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.