Global center stable manifold for the defocusing energy critical wave equation with potential

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension 3. We prove that the set of initial data for which solutions scatter to an unstable excited state (φ, 0) forms a finite co-dimensional path connected C¹ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with (φ, 0). It is constructed in a first step locally around any solution scattering to φ, which might be very far away from φ in the Ḣ¹ × L² (R³) norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends a previous result of ours to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from Beceanu-Goldberg to construct a local center stable manifold near any solution that scatters to (φ, 0). This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle, which is used to show that solutions that start off but near the local manifolds associated with φ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state φ.