  We study the behavior of monogamy deficit and monogamy asymmetry for
Einstein-Podolsky-Rosen steering of Gaussian states under the influence of the
Hawking effect. We demonstrate that the monogamy of quantum steering shows an
extreme scenario in the curved spacetime: the first part of a tripartite system
cannot individually steer two other parties, but it can steer the collectivity
of the remaining two parties. We also find that the monogamy deficit of
Gaussian steering, a quantifier of genuine tripartite steering, are generated
due to the influence of the Hawking thermal bath. Our results elucidate the
structure of quantum steering in tripartite quantum systems in curved
spacetime.
