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×RM\times {\mathbb{R}}. As applications, we prove that (1) a periodic harmonic function on R2{{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R3{{\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 R3{{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.

Keywords

• carleman inequality
• unique continuation
• minimal surface
• translator
• 35a02
• 53c42