On the radius pinching estimate and uniqueness of the CMC foliation in asymptotically flat 3-manifolds

- In this paper, we do not require the metric to be close to Schwarzschild manifold in any sense or to satisfy RT conditions.
- We find a natural way to calculate the flux integral∫Σ(H−He)(v⋅b)dμe in the last section and manage to relate it to the mass of the asymptotically flat end.
- We find a more subtle estimate on the second fundamental form, Lemma 4.11, which plays an important role in proving the uniqueness.

In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric to have the form gij=δij+hij with hij=O4(r−1) and R=O(r−3−τ), τ>0. We do not require the metric to be close to Schwarzschild metric in any sense or to satisfy RT conditions. We prove that, when the mass is not 0, stable CMC spheres that separate a certain compact part from the infinity satisfy the radius pinching estimate r1≤Cr0, which in many cases is critical to prove the uniqueness of the CMC spheres. As applications of this estimate, we remove the radius conditions of the uniqueness result in [8] and [15] in some special cases.