Constant Harmonic Mean Curvature Foliation in Asymptotic Schwarzschild Spaces-II

TL;DR

The paper advances the theory of foliations by hypersurfaces of constant harmonic mean curvature in asymptotically Schwarzschild spaces to arbitrary dimensions $n\ge3$, establishing existence of a CHMC foliation and identifying its center of mass with the ADM center of mass. It develops a parabolic approach via the volume-preserving harmonic mean curvature flow starting from coordinate spheres, and proves long-time existence and exponential convergence to CHMC surfaces, with detailed curvature control under a decay regime. A key innovation is the treatment of the non-self-adjoint linearization L by introducing a self-adjoint symmetrization S, proving invertibility and quantitative bounds that enable the foliation construction and center-of-mass equivalence. In 3D, with stronger decay $\delta>0$, the authors prove local uniqueness of the CHMC foliation in large exterior regions, using spectral-gap arguments and perturbative analysis of the height function over a reference CHMC surface. Together, these results extend CHMC foliation theory to higher dimensions, link CHMC centers to ADM mass, and set the stage for elliptic methods in future work.

Abstract

This paper extends the results of [GLS24], where the existence of a constant harmonic mean curvature foliation was established in the setting of a 3-dimensional asymptotically Schwarzschild manifold. Here, we generalize this construction to higher dimensions, proving the existence of foliations by constant harmonic mean curvature hypersurfaces in an asymptotically Schwarzschild manifold of arbitrary dimension. Furthermore, in 3 dimensional case, we demonstrate the local uniqueness of this foliation under a stronger decay conditions on the asymptotically Schwarzschild metric

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4: Lemma 1.4 in Huisken1996DefinitionOC
• Proposition 2.5
• proof
• Proposition 2.6
• proof