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We study the dynamics of the focusing $3d$ cubic nonlinear Schr\"odingerequation in the exterior of a strictly convex obstacle at the mass-energythreshold, 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 initialdata, $Q$ 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 (followingthe work of Duyckaerts and Merle on the energy-critical problem) have provedthe existence of a specific global solution that scatters for negative timesand converges to the soliton in positive times. We prove that theseheteroclinic orbits do not exist for the problem in the exterior domain andthat all solutions at the threshold are globally defined and scatter. The maindifficulty is the control of the space translation parameter, since theGalilean transformation is not available.

Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle