Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces
Yaoting Gui, Yuqiao Li, Jun Sun
TL;DR
The paper studies the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces, showing that starting from a large coordinate sphere yields a smooth solution for all time that converges exponentially to a surface of constant harmonic mean curvature, $F= rac{H^2-|A|^2}{2H}$. The resulting family of constant-$F$ surfaces forms a proper foliation of the exterior region, enabling a geometric center of mass, $C_{HM}$, defined from the foliation to be identified with the ADM center of mass. The authors extend the Huisken-Yau foliation program to the nonlinear setting of HM curvature by developing new a priori estimates, spectral analysis of the associated linearized operator, and a careful treatment of the lapse that guarantees foliation. The work provides intrinsic infinity structure and a robust notion of center of mass for isolated gravitating systems, with potential extensions to other asymptotic geometries. Key techniques include exploiting the nondivergence elliptic operator ${ L}$ and its adjoint, establishing exponential convergence, and proving the HM foliation aligns with the ADM mass framework.
Abstract
This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.