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 $Tll 1$ 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 $Tll x leq 1$ as $Tto 0$, 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 $Tto 0$ the width of the full replica symmetry breaking region shrinks to zero and only the droplet-like scenario survives.
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