Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space
Yaoting Gui, Yuqiao Li, Jun Sun
TL;DR
This work analyzes the stability of volume- and area-preserving mean curvature flows in Schwarzschild and asymptotically Schwarzschild spaces. It develops a unified approach combining isoperimetric control with perturbation/center-manifold methods to prove global existence and smooth convergence to constant mean curvature surfaces when starting near coordinate spheres, and extends these results to asymptotically Schwarzschild geometries with exponential convergence toward isoperimetric CMC hypersurfaces. The findings yield existence results for CMC hypersurfaces in asymptotically flat spaces and reinforce the geometric-isoperimetric framework underpinning Huisken–Yau-type foliations in these manifolds. The use of isoperimetric ratios, center-manifold analysis, and known isoperimetric foliations (Huisken–Yau, Eichmair–Metzger) provides robust stability conclusions for both flows in these curved backgrounds.
Abstract
In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.