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

In this paper, we demonstrate that any asymptotically flat manifold ( M n + 1 , g ) with 3 ⩽ n ⩽ 6 can be foliated by a family of area-minimizing hypersurfaces, each of which is asymptotic to Cartesian coordinate hyperplanes defined at an end of ( M n + 1 , g ) . As an application of this foliation, we show that for any asymptotically flat manifold ( M n + 1 , g ) with 3 ⩽ n ⩽ 6 , nonnegative scalar curvature and positive mass, the solution of free boundary problem for area-minimizing hypersurface in coordinate cylinder C R i in ( M n + 1 , g ) either does not exist or drifts to infinity of ( M n + 1 , g ) as R i tends to infinity. Additionally, we introduce a concept of globally minimizing hypersurface in ( M n + 1 , g ) , and verify a version of the Schoen’s Conjecture.