  The main aim of this thesis is to study the properties of trapped surfaces in
spacetimes with symmetries and their possible relation with the theory of black
holes. We will concetrate specially on one aspect of this possible equivalence,
namely whether the static black hole uniqueness theorems extend to static
spacetimes containing marginally outer trapped surfaces. The principal result
of this thesis states that this question has an affirmative answer, under
suitable not global-in-time conditions on the spacetime. Furthermore, in order
to solve this question, we will obtain several results which generalize known
properties of static spacetimes to the initial data setting and can be of
independent interest. Finally, we will study the Penrose inequality in static
initial data which are not time-symmetric. Our main result in this last part of
the thesis is the discovery of a counter-example of a recent version of the
Penrose inequality proposed by Bray and Khuri in 2009.
