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

Abstract

A spacelike surface $S\subset \mathbb{S}^4_1$ is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface $S \subset \mathbb{S}^4_1$ we show that a choice of orientation of the normal bundle $\nu(S)$ determines a smooth map $G: S \to \mathbb{S}^3$ which we call the null Gauss map of $S$. We show that if $S$ is marginally trapped then $G$ is a conformal immersion away 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 $s$. We show that 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 C.Bohle, G. Peters and U.Pinkall [arXiv:math/0411479]. 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.