Large outlying stable constant mean curvature spheres in initial data sets

We give examples of asymptotically flat three-manifolds $(M,g)$ which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by G. Huisken and S.-T. Yau in 1996. On the other hand, we show that such surfaces cannot exist when $(M,g)$ has nonnegative scalar curvature. This result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces.