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In this paper, we work with the existence and uniqueness of free boundaryconstant mean curvature hypersurfaces in rotational domains. These are domainswhose boundary is generated by a rotation of a graph. Under some conditions onthe function that generates the graph and a gap condition on the umbilicitytensor, we classify the CMC free boundary hypersurfaces as topological disks orannulus. Also, we construct some examples of free boundary minimal surfaces inthe rotational ellipsoid that, in particular, satisfy our gap condition.

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