  We apply the method of Lyapunov-Schmidt reduction to study large
area-constrained Willmore surfaces in Riemannian 3-manifolds asymptotic to
Schwarzschild. In particular, we prove that the end of such a manifold is
foliated by distinguished area-constrained Willmore spheres. The leaves are the
unique area-constrained Willmore spheres with large area, non-negative Hawking
mass, and distance to the center of the manifold at least a small multiple of
the area radius. Unlike previous related work, we only require that the scalar
curvature satisfies mild asymptotic conditions. We also give explicit examples
to show that these conditions on the scalar curvature are necessary.
