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

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