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

TL;DR

This paper proves a radius pinching estimate for stable constant mean curvature spheres in asymptotically flat 3-manifolds with non-zero mass, aiding in establishing their uniqueness without requiring the metric to be close to Schwarzschild.

Contribution

It introduces a radius pinching estimate for stable CMC spheres in asymptotically flat manifolds with non-zero mass, removing previous metric closeness restrictions.

Findings

Radius pinching estimate holds for stable CMC spheres with non-zero mass.

This estimate enables removal of radius conditions in uniqueness proofs.

The results apply to manifolds with metrics not necessarily close to Schwarzschild.

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}=\delta_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-\tau}),\tau>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.