CLASSIFICATION OF MARGINALLY TRAPPED SURFACES WITH PARALLEL MEAN CURVATURE VECTOR IN LORENTZIAN SPACE FORMS

摘要

A spatial surface in a 4D (four-dimensional) Lorentzian manifoldis called marginally trapped if its mean curvature vector is light-like at eachpoint. In this article, we completely classify marginally trapped surfaces withparallel mean curvature vector in 4D Lorentzian space forms. Our mainresults state that there are six families of such surfaces in the Minkowskispace-time, eight families in the de Sitter space-time and eight families inthe anti-de Sitter space-time. Conversely, marginally trapped surfaces withparallel mean curvature vector in 4D Lorentzian space forms are obtainedfrom these families. Explicit examples of such surfaces are presented. Inaddition we give a simple relation between marginally trapped surfaces withparallel mean curvature vector and biharmonic surfaces in 4D Minkowskispace-time E_1~4.