  Small deformations of marginally (outer) trapped surfaces are considered by
using their stability operator. In the case of spherical symmetry, one can use
these deformations on any marginally trapped round sphere to prove several
interesting results. The concept of 'core' of a black hole is introduced: it is
a minimal region that one should remove from the spacetime in order to get rid
of all possible closed trapped surfaces. In spherical symmetry one can prove
that the spherical marginally trapped tube is the boundary of a core. By using
a novel formula for the principal eigenvalue of the stability operator, I will
argue how to pursue similar results in general black-hole spacetimes.
