Constant Harmonic Mean Curvature Foliation in Asymptotic Schwarzschild Spaces-II

TL;DR

This paper generalizes the existence and uniqueness of constant harmonic mean curvature foliations from 3-dimensional to higher-dimensional asymptotically Schwarzschild manifolds, expanding understanding of geometric structures in these spaces.

Contribution

It extends previous results by constructing such foliations in arbitrary dimensions and proving local uniqueness in the 3D case under stronger decay conditions.

Findings

Existence of constant harmonic mean curvature foliations in higher dimensions.

Local uniqueness of the foliation in 3D under stronger decay conditions.

Generalization of geometric foliation results to broader asymptotic settings.

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