Instability of marginally outer trapped surfaces from initial data set symmetry

Let be an initial data set and let xa be a symmetry vector of . Consider a marginally outer trapped surface in and let the symmetry vector be decomposable along the unit normal to in , and along . In this note we present some basic results with regards to the stability of . The vector decomposition allows us to characterize the instability of by the nature of the zero set of the normal component to and the divergence of the component along . Further observations are made under the assumption of having a constant mean curvature, and being an Einstein manifold.