Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture

In this paper, we demonstrate that any asymptotically flat manifold (Mⁿ, g) with 4 n 7 can be foliated by a family of area-minimizing hypersurfaces, each of which is asymptotic to Cartesian coordinate hyperplanes defined at an end of (Mⁿ, g). As an application of this foliation, we show that for any asymptotically flat manifold (Mⁿ, g) with 4 n 7, nonnegative scalar curvature and positive mass, the solution of free boundary problem for area-minimizing hypersurface in coordinate cylinder Cₑ㶁 in (Mⁿ, g) either does not exist or drifts to infinity of (Mⁿ, g) as Rᵢ tends to infinity. Additionally, we introduce a concept of globally minimizing hypersurface in (Mⁿ, g), and verify a version of the Schoen Conjecture.