  We study the spectrum of the Hessian of the Sherrington-Kirkpatrick model
near T=0, whose eigenvalues are the masses of the bare propagators in the
expansion around the mean-field solution. In the limit $T\ll 1$ two regions can
be identified. The first for $x$ close to 0, where $x$ is the Parisi replica
symmetry breaking scheme parameter. In this region the spectrum of the Hessian
is not trivial, and maintains the structure of the full replica symmetry
breaking state found at higher temperatures. In the second region $T\ll x \leq
1$ as $T\to 0$, the bands typical of the full replica symmetry breaking state
collapse and only two eigenvalues are found: a null one and a positive one. We
argue that this region has a droplet-like behavior. In the limit $T\to 0$ the
width of the full replica symmetry breaking region shrinks to zero and only the
droplet-like scenario survives.
