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

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.