Extrinsically homogeneous real hypersurfaces with three distinct principal curvatures in \(H_n(\mathbb{C})\) (Q1769145)

Let \(H_n(\mathbb{C})\) be a complex hyperbolic space of complex dimension \(n\) \((n\geq 2)\) endowed with the metric of constant holomorphic sectional curvature \(4c\), and \(G\) be the identity component of the group of all isometries of \(H_n(\mathbb{C})\). A submanifold \(M\) in \(H_n(\mathbb{C})\) is said to be extrinsically homogeneous if \(M\) is an orbit under a closed subgroup of \(G\). In this paper, the authors prove: Let \(L\) be a connected closed subgroup of \(G\). Assume that every real hypersurface given as an orbit under \(L\) has three distinct principal curvatures and the structure vector field is not principal. Then any of such orbits is isometrically congruent to an orbit under the Berndt subgroup.