Gap results and existence of free boundary CMC surfaces in rotational domains

In this paper, we work with the existence and uniqueness of free boundary constant mean curvature surfaces in rotational domains. These are domains whose boundary is generated by a rotation of a graph. We classify the free boundary CMC surfaces as topological disks or annulus under some conditions on the function that generates the graph and a gap condition on the umbilicity tensor. Also, we construct some examples of free boundary CMC surfaces in the rotational ellipsoid that, in particular, satisfy our gap condition.