Large area-constrained Willmore surfaces in asymptotically Schwarzschild $3$-manifolds

Abstract

We apply the method of Lyapunov–Schmidt reduction to study 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.