Homogeneous hypersurfaces in complex hyperbolic spaces

Abstract

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces

“…Moreover, it was proved in [2] that the tubes around W w are homogeneous precisely when w ⊥ has constant Kähler angle, that is, when ϕ ξ is independent of the vector ξ ∈ w ⊥ . Indeed, the Berndt-Brück submanifolds W 2n−k ϕ [3] are precisely those W w for which w ⊥ has constant Kähler angle ϕ and k = dim w ⊥ . We summarize all this in the following Theorem 3.1 Let g = k ⊕ p be the Cartan decomposition of the Lie algebra of the isometry group G = SU (1, n) of CH n with respect to a point o ∈ CH n .…”

“…. , n − 1} (see [3]); notice that for ϕ = 0 these are tubes around a totally geodesic CH k , k ∈ {1, . .…”

“…Homogeneous hypersurfaces in complex hyperbolic spaces were classified in [4] and their geometry was studied in [3].…”

Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes

“…However, the examples constructed in [13] and [16] are generically non-Hopf. Moreover, in CH n , n ≥ 2, there are examples of non-Hopf homogeneous hypersurfaces [5]. The observation that motivates this paper is that most of the examples in [13] and [16], and some examples in [5], share the following geometric properties: (C1) The smallest S-invariant distribution D of M that contains Jξ has rank 2.…”

Ruled hypersurfaces with constant mean curvature in complex space forms

“…A Lohnherr hypersurface (also called a fan) is the only complete ruled hypersurface of CH n with constant principal curvatures [10]. It is also the unique minimal hypersurface which is homogeneous, that is, an orbit of an isometric action on CH n [4]. Different alternative descriptions of these examples can be found in [1], [8] and [11].…”