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

Abstract

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

39pages, 8 figures. Comments are welcome!