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

Shihang He, Yuguang Shi, Haobin Yu

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})

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

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 (cf. Theorem \ref{thm: 8dim Schoen conj})
Paper Structure (12 sections, 16 theorems, 90 equations, 2 figures)

This paper contains 12 sections, 16 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Foliations of Area-minimizing hypersurfaces in General AF manifolds
  3. Outline of Proofs
  4. Organization of The Paper
  5. Proof of Theorem \ref{['thm: foliation']}
  6. Plateau problem in the cylinder $C_a$
  7. Density estimate for area-minimizing hypersurfaces in asymptotically flat manifolds
  8. Curvature estimate for area-minimizing hypersurfaces in asymptotically flat manifolds
  9. Smoothness of $\Sigma_{a,t}$ outside a compact set
  10. Curvature estimate of $\Sigma_{a,t}$ outside a compact set
  11. Foliation of area-minimizing hypersurfaces in higher dimension
  12. Proof of Theorem \ref{['thm: 8dim Schoen conj']}

Key Result

Theorem 1.1

Let $(M^{n+1},g)$ be an AF manifold of dimension $n\geq 3$ with an AF end $E$ of order $\tau>\frac{n}{2}$ and some arbitrary ends. Suppose $(M^{n+1},g)$ is geometric bounded, i.e. its sectional curvature is bounded, injective radius has a positive uniform lower bound. Then (1) For each $t\in\mathbf{ for each $t\in\mathbf{R}$, where $\mathbf{D}$ is a compact set of $\mathbf{R}^{n}$ and $u_t\in C^\i

Figures (2)

  • Figure 1: An illustration of Theorem \ref{['thm: 8dim Schoen conj']}: Vertical pairs of lines represent coordinate cylinders, while horizontal curves represent free boundary minimizing hypersurfaces.
  • Figure 2: This figure demonstrates the free boundary problem with inner obstacle defined by \ref{['eq: 79']}. The left rectangle denotes part of the cylinder $C_R$; the gray area denotes $\Omega$; the red line denotes $\Sigma$, and $\partial V_{R_i}$ is the inner obstacle lying in arbitrary ends.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 25 more