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Asymptotics of high-codimensional area-minimizing currents in hyperbolic space

Xumin Jiang, Jiongduo Xie

TL;DR

This work studies high-codimensional area-minimizing locally rectifiable currents $T$ in hyperbolic space $\mathbb{H}^ rak m$ that are asymptotic to a boundary $\Gamma$ at infinity, addressing Lin's open problem on the smoothness of $\operatorname{supp}(T)\cup \Gamma$ in Euclidean metrics. It derives a minimal-surface system for graphical representations near the asymptotic boundary, proves local mass bounds and $C^{1,\alpha}$ regularity up to the boundary, and establishes higher-order analytic regularity with explicit asymptotic expansions in powers of the height coordinate $y^n$ and logarithmic corrections, including convergence results under analytic $\Gamma$. The results generalize boundary regularity at infinity to high codimension, reveal higher-order obstruction structures tied to the geometry of $\Gamma$, and provide precise asymptotics that underpin gluing and asymptotic theory for minimal surfaces in hyperbolic space. The combination of mass-projection techniques, Schauder-type estimates, and Fuchsian-system analysis yields both analytic and convergent expansions, offering a rigorous framework for the asymptotic study of minimal currents near infinity in hyperbolic geometry.

Abstract

We investigate the asymptotic behavior of high-codimensional area-minimizing locally rectifiable currents in hyperbolic space, addressing a problem posed by F.H. Lin and establishing ``boundary regularity at infinity" results for such currents near their asymptotic boundaries under the standard Euclidean metric. Intrinsic obstructions to high-order regularity arise for odd-dimensional minimal surfaces, revealing a constraint dependent on the geometry of the asymptotic boundary. Our work advances the asymptotic theory of high-codimensional minimal surfaces in hyperbolic space.

Asymptotics of high-codimensional area-minimizing currents in hyperbolic space

TL;DR

This work studies high-codimensional area-minimizing locally rectifiable currents in hyperbolic space that are asymptotic to a boundary at infinity, addressing Lin's open problem on the smoothness of in Euclidean metrics. It derives a minimal-surface system for graphical representations near the asymptotic boundary, proves local mass bounds and regularity up to the boundary, and establishes higher-order analytic regularity with explicit asymptotic expansions in powers of the height coordinate and logarithmic corrections, including convergence results under analytic . The results generalize boundary regularity at infinity to high codimension, reveal higher-order obstruction structures tied to the geometry of , and provide precise asymptotics that underpin gluing and asymptotic theory for minimal surfaces in hyperbolic space. The combination of mass-projection techniques, Schauder-type estimates, and Fuchsian-system analysis yields both analytic and convergent expansions, offering a rigorous framework for the asymptotic study of minimal currents near infinity in hyperbolic geometry.

Abstract

We investigate the asymptotic behavior of high-codimensional area-minimizing locally rectifiable currents in hyperbolic space, addressing a problem posed by F.H. Lin and establishing ``boundary regularity at infinity" results for such currents near their asymptotic boundaries under the standard Euclidean metric. Intrinsic obstructions to high-order regularity arise for odd-dimensional minimal surfaces, revealing a constraint dependent on the geometry of the asymptotic boundary. Our work advances the asymptotic theory of high-codimensional minimal surfaces in hyperbolic space.
Paper Structure (6 sections, 21 theorems, 240 equations)

This paper contains 6 sections, 21 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. From Local Rectifiability to $C^{1,\alpha}$ Regularity
  4. System of Minimal Surface Equations
  5. Higher Order Regularity Theorems
  6. Convergence theorems

Key Result

Theorem 1.2

Let $T$ be an area-minimizing locally rectifiable $n$-current in $\mathbb{H}^\mathfrak{m}$, and let $\Gamma$ be a closed $C^{1,\alpha}$ submanifold of $\mathbb{R}^{\mathfrak{m}-1} \times \{0\}$ of dimension $n-1$, for some $0<\alpha\leqslant 1$. Assume that $T$ is a normal current with respect to th admits a representation as an $n$-dimensional $C^{1,\alpha}$ submanifold of $\mathbb{R}^\mathfrak{m

Theorems & Definitions (35)

  • Theorem 1.2: From locally rectifiable to $C^{1,\alpha}$ regularity
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6: From $C^{1,\alpha}$ to $C^{n,\alpha}$
  • Theorem 1.7: Boundary Regularity Theorem I
  • Theorem 1.8: Boundary Regularity Theorem II
  • Theorem 1.9: Convergence Theorem
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 25 more