On the radius pinching estimate and uniqueness of the CMC foliation in asymptotically flat 3-manifolds
arXiv:1207.0281
Abstract
In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=δ_{ij}+h_{ij} with h_{ij}=O_{4}(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 infinity satisfy the radius pinching estimate r_{1}\leq Cr_{0} , 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 [Huang-CMC] and [NERZ-CMC] in some special cases.
arXiv comment40Pages. A strengthened version of arXiv:1207.0281, "A new result on the uniqueness of the CMC foliation in asymptotically flat manifold". arXiv admin note: text overlap with arXiv:1012.3231