[Paper Review] On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds

Otis Chodosh, Michael Eichmair|arXiv (Cornell University)|Mar 28, 2017

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper extends the Lyapunov-Schmidt analysis of outlying stable constant mean curvature (CMC) spheres in asymptotically flat 3-manifolds to the 'far-off-center' regime and general Schwarzschild asymptotics. It establishes sharp existence and non-existence criteria for large stable CMC spheres that depend critically on the scalar curvature's behavior at infinity, resolving a key gap in the uniqueness theory of large CMC surfaces in initial data sets with non-negative scalar curvature.

ABSTRACT

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on the behavior of scalar curvature at infinity.

Motivation & Objective

• To extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres beyond the near-center regime to the far-off-center regime.
• To generalize previous results on stable CMC spheres in asymptotically flat 3-manifolds to include general Schwarzschild asymptotics.
• To clarify the dependence of large stable CMC sphere existence on the scalar curvature's behavior at infinity.
• To resolve the uniqueness question for large stable CMC surfaces in initial data sets with non-negative scalar curvature by analyzing the far-outlying case.

Proposed method

• Adapts the Lyapunov-Schmidt reduction technique to handle CMC spheres far from the central region in asymptotically flat 3-manifolds.
• Implements a perturbative analysis around coordinate spheres in the asymptotic chart, treating the metric deviation as a small perturbation.
• Uses weighted Sobolev estimates and asymptotic expansions of the second fundamental form and mean curvature to control the nonlinear terms.
• Applies integral identities over spheres and balls in the asymptotic Euclidean chart to extract leading-order terms in the linearized problem.
• Employs a centering device based on flux integrals and symmetry reduction to handle the lack of centering information in the far-off-center regime.
• Analyzes the role of the scalar curvature's asymptotic decay rate and its interaction with the metric's higher-order terms to determine existence or non-existence of sequences of large stable CMC spheres.

Experimental results

Research questions

• RQ1 Under what conditions on the scalar curvature at infinity do large stable CMC spheres exist in asymptotically flat 3-manifolds with general Schwarzschild asymptotics?
• RQ2 Can the non-existence result for outlying stable CMC spheres in [3] be extended to the far-off-center regime under weaker assumptions on the metric expansion?
• RQ3 How does the behavior of the scalar curvature influence the existence or non-existence of divergent sequences of large stable CMC spheres with controlled mean curvature decay?
• RQ4 What is the precise role of the metric's asymptotic expansion beyond the leading-order Schwarzschild term in determining the existence of such CMC spheres?
• RQ5 Can the uniqueness of the canonical foliation as the only large stable CMC surfaces be established unconditionally by resolving the far-outlying case?

Key findings

• For a complete Riemannian 3-manifold that is $C^6$-asymptotic to Schwarzschild of mass $m>0$ with vanishing scalar curvature, every large-area stable CMC surface is a leaf of the canonical foliation.
• In the far-off-center regime, the existence of sequences of large stable CMC spheres depends critically on the scalar curvature's asymptotic behavior: non-negative scalar curvature implies non-existence of such sequences with $r_0(\Sigma_k)H(\Sigma_k)\to 0$, while negative scalar curvature may allow them.
• The paper establishes sharp non-existence results for sequences of outlying stable CMC spheres satisfying $r_0(\Sigma_k)\to\infty$ and $r_0(\Sigma_k)H(\Sigma_k)\to\eta>0$, under the assumption that the metric expansion includes a homogeneous $O(|x|^{-2})$ term.
• The analysis shows that the scalar curvature's sign and decay rate at infinity are decisive in determining the structure of large stable CMC surfaces, particularly in the absence of centering information.
• The method successfully handles the loss of centering flux integrals in the far-off-center regime by introducing a refined asymptotic analysis of the linearized problem.
• The results complete the characterization of large stable CMC surfaces in strongly asymptotically flat 3-manifolds, showing that only the canonical foliation can arise as such surfaces when scalar curvature is non-negative.