The area preserving Willmore flow and local maximizers of the Hawking mass in asymptotically Schwarzschild manifolds

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is $C^3-$close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type. In this paper, we prove that the leaves of this foliation are stable under small area preserving $W^{2,2}-$perturbations with respect to the area preserving Willmore flow. This implies, in particular, that the leaves are strict local area preserving maximizers of the Hawking mass with respect to the $W^{2,2}-$topology.