Large outlying stable constant mean curvature spheres in initial data sets

TL;DR

This paper constructs examples of large, distant constant mean curvature spheres in certain asymptotically flat manifolds, answering a longstanding question, while also proving such spheres cannot exist in manifolds with nonnegative scalar curvature.

Contribution

It provides explicit examples of large stable CMC spheres in asymptotically flat manifolds and establishes their non-existence in manifolds with nonnegative scalar curvature.

Findings

Existence of arbitrarily large CMC spheres in some asymptotically flat manifolds.

Non-existence of such spheres in manifolds with nonnegative scalar curvature.

Relationship between scalar curvature and isoperimetric ratio of CMC surfaces.

Abstract

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.