Ruled hypersurfaces with constant mean curvature in complex space forms

Abstract

We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their classification, by virtue of a result of Lohnherr and Reckziegel.2010 Mathematics Subject Classification. 53B25, 53C42, 53C55. Key words and phrases. Complex space form, complex projective space, complex hyperbolic space, ruled hypersurface, constant mean curvature, minimal hypersurface.

Some problems on ruled hypersurfaces in nonflat complex space forms

“…Moreover, A(H) = 0 on U, from where ab(3H 2 + c) = 0. Since a and b are not zero in U, M has constant mean curvature on U and, as M is a ruled, it must be minimal (see [5]), which concludes the proof.…”

“…Motivated by some recent results concerning the classification of ruled hypersurfaces in nonflat complex space forms having constant mean curvature [5] or having constant scalar curvature [10], we firstly focus on studying ruled hypersurfaces in CP n or CH n whose shape operators have constant squared norm, proving that there are no such hypersurfaces in CP n , whereas any possible example in CH n must be strongly 2-Hopf.…”