Stability of marginally outer trapped surfaces and symmetries

We study properties of stable, strictly stable and locally outermost marginally outer trapped surfaces in spacelike hypersurfaces of spacetimes possessing certain symmetries such as isometries, homotheties and conformal Killings. We first obtain results for general diffeomorphisms in terms of the so-called metric deformation tensor and then particularize to different types of symmetries. In particular, we find restrictions at the surfaces on the vector field generating the symmetry. Some consequences are discussed. As an application we present a result on non-existence of stable marginally outer trapped surfaces in slices of FLRW.