Marginally trapped surfaces in S^4_1, Gauss images and conformal invariants

HULETT EDUARDO

Cordoba

Seminario; Seminario del grupo de Geometria diferencial FAMAF; 2014

Grupo de Geometria Diferencial FAMAF

In this paper we explore the geometry of marginally trapped surfaces in De Sitter 4-space S41 and its relation to Moebius surface theory in the conformal 3-sphere. Given an oriented marginally trapped surface S⊂S41 we consider its "spherical Gauss image" i.e. a surface S′ obtained varying a well defined "positive" normal null direction on S. The new surface S′ lies in the conformal 3-sphere S3 viewed as the manifold of null directions of Minkowski space R41, and so S′ has well defined local conformal or Moebius invariants which encode information of the underlying marginally trapped surface S. We derive an equation relating the conformal invariants of the spherical Gauss image S′ with an intrinsically defined geometric complex quadratic differential δ on the underlying marginally trapped surface S. This equation is used to obtain a characterization of marginally trapped surfaces whose spherical Gauss image is Willmore or constrained Willmore~cite{burstall-pedit-pinkall}. As an application we obtain integrable one-parameter deformations of two classes of marginally trapped surfaces and construct their associated families.