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

TL;DR

This paper proves that asymptotically flat manifolds with dimensions 4 to 7 can be foliated by area-minimizing hypersurfaces and verifies a version of Schoen's conjecture related to these structures.

Contribution

It establishes a foliation by area-minimizing hypersurfaces in asymptotically flat manifolds and verifies a related version of Schoen's conjecture.

Findings

Existence of foliation by area-minimizing hypersurfaces in specified manifolds.

Behavior of solutions to free boundary problems in these manifolds.

Verification of a version of Schoen's conjecture.

Abstract

In this paper, we demonstrate that any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$ 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 $4\leq n\leq 7$, 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.