  Closed sections of totally geodesic null hypersurfaces are marginally outer
trapped surfaces (MOTS), for which a well-defined notion of stability exists.
In this paper we obtain the explicit form for the stability operator for such
MOTS and analyze in detail its properties in the particular case of
non-evolving horizons, which include both isolated and Killing horizons. We
link these stability properties with the surface gravity of the horizon and/or
to the existence of minimal sections. The results are used, in particular, to
obtain an area-angular momentum inequality for sections of axially symmetric
horizons in four spacetime dimensions, which helps clarifying the relationship
between two different approaches to this inequality existing in the literature.
