In this paper, we work with the existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains. These are domains whose boundary is generated by a rotation of a graph. Under some conditions on the function that generates the graph and a gap condition on the umbilicity tensor, we classify the CMC free boundary hypersurfaces as topological disks or annulus. Also, we construct some examples of free boundary minimal surfaces in the rotational ellipsoid that, in particular, satisfy our gap condition.
Comment: Some typo corrections were made throughout the text. It includes additional references and a motivation for the condition regarding the profile curve (see Remark 1) used in the results of Section 3