Foliation of area-minimizing hypersurfaces in asymptotically flat manifolds of higher dimension

TL;DR

This paper proves the existence and describes the behavior of foliations by area-minimizing hypersurfaces in higher-dimensional asymptotically flat manifolds, extending previous results to arbitrary ends and dimensions.

Contribution

It establishes the existence of such foliations in higher dimensions and arbitrary ends, and analyzes their behavior near infinity and the singular set.

Findings

Existence of foliations by area-minimizing hypersurfaces in AF manifolds of arbitrary dimension.

Behavior of hypersurfaces near infinity of AF ends.

Singular set of hypersurfaces lies outside AF ends.

Abstract

We prove the existence of foliations by area-minimizing hypersurfaces in asymptotically flat (AF) manifolds with arbitrary dimension and arbitrary ends. Also we provide behaviors of those hypersurfaces near the infinity of AF ends and demonstrate that the singular set of those area-minimizing hypersurfaces is outside AF ends (cf Theorem \ref{thm: foliation}). Building on the positive mass theorem for AF manifolds with arbitrary ends, we establish a global behavior for free-boundary area-minimizing hypersurfaces inside coordinate cylinders in AF manifolds of dimension less than or equal to $8$ (cf. Theorem \ref{thm: 8dim Schoen conj})