  We study the dynamics of the focusing $3d$ cubic nonlinear Schr\"odinger
equation in the exterior of a strictly convex obstacle at the mass-energy
threshold, namely, when $ E_{\Omega}[u_0] M_{\Omega}[u_0] = E_{\R^3}[Q]
M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(\Omega)}
\left\|u_0\right\|_{L^{2}(\Omega)}< \left\| \nabla Q \right\|_{L^2(\R^3)}
\left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(\Omega)$ is the initial
data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$
is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following
the work of Duyckaerts and Merle on the energy-critical problem) have proved
the existence of a specific global solution that scatters for negative times
and converges to the soliton in positive times. We prove that these
heteroclinic orbits do not exist for the problem in the exterior domain and
that all solutions at the threshold are globally defined and scatter. The main
difficulty is the control of the space translation parameter, since the
Galilean transformation is not available.
