Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

Abstract

A spacelike surface S immersed in \(\mathbb {S}^4_1\) is marginally trapped if its mean curvature vector is everywhere lightlike. On any oriented spacelike surface S immersed in \(\mathbb {S}^4_1\) we show that a choice of orientation of the normal bundle \(\nu (S)\) determines a smooth map \(G: S \rightarrow \mathbb {S}^3\) which we call the null Gauss map of S. If S is marginally trapped we show that G is a conformal immersion away from the zeros of certain quadratic Hopf-differential of S and so the surface G(S) is uniquely determined up to conformal transformations of \(\mathbb {S}^3\) by two invariants: the normal Hopf differential \(\kappa \) and the schwartzian derivative s. These invariants plus an additional quadratic differential \(\delta \) are related by a differential equation and determine the geometry of S up to ambient isometries of \(\mathbb {S}^4_1\). This allows us to obtain a characterization of marginally trapped surfaces S whose null Gauss image is a constrained Willmore surface in \(\mathbb {S}^3\) in the sense of Bohle et al. (Calc Var Partial Differ Equ 32:263–277, 2008). As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.