A characterization of the delaunay surfaces

In this paper we use the Alexandrov Reflection Method to obtain a characterization to embedded CMC capillary annulus $\Sigma^2 \subset \mathbb{B}^3$. In especial, but using a new strategy, we present a new characterization to the critical catenoid. Precisely, we show that $\Sigma \subset \mathbb{B}^3$ being an embedded minimal free boundary annulus in $\mathbb{B}^3$ such that $\partial \Sigma$ is invariant under reflection through a coordinates planes, then $\Sigma$ is the critical catenoid. This work is part of the second author thesis which was written in 2019.