Foliations of asymptotically flat manifolds by stable constant mean curvature spheres (Q6633916)

The paper investigates the geometry of asymptotically flat Riemannian manifolds of dimension \(n \geq 3\), with a focus on foliations by stable constant mean curvature (CMC) spheres. Specifically, the paper aims at the following four issues:\N\N-- First of all, the paper provides a streamlined, conceptually simpler proof for the existence of asymptotic foliations using Lyapunov-Schmidt reduction.\N\N-- Next, the paper explores the relationship between the geometric and Hamiltonian centers of mass for such manifolds.\N\N-- Next, the paper establishes uniqueness results for large stable CMC spheres under various decay assumptions on the manifold's scalar curvature.\N\N-- Finally, the paper extends results to dimensions \(n \geq 3\) and generalizes prior findings in dimension \(n = 3\).\N\NThe appendices provide technical definitions, detailed proofs, and a survey of related literature. The main results of the paper include Theorems 9, 10, 12 and 14. Theorem 9 establishes the existence of foliations by stable CMC spheres for all asymptotically flat manifolds and connects the foliation to a well-defined geometric center of mass and it is proved in Section 2. Theorem 10 proves the equivalence between geometric and Hamiltonian centers of mass under specified conditions and its proof is in Section 3. Theorem 12 demonstrates the global uniqueness of large stable CMC spheres in three-dimensional manifolds with non-negative scalar curvature and it is proved in Section 4. The last result is Theorem 14 which extends Theorem 12 to cases where scalar curvature changes sign, under weaker centering assumptions and this result is also proved in Section 4.\N\NIn summary, the paper provides valuable new results that will enhance the reader's understanding of asymptotically flat Riemannian manifolds and their geometry.