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

TL;DR

This paper studies the behavior of high-codimensional area-minimizing currents in hyperbolic space, revealing boundary regularity results and geometric obstructions that advance the understanding of minimal surfaces in this setting.

Contribution

It establishes boundary regularity at infinity for high-codimensional currents and uncovers intrinsic obstructions to higher regularity related to the asymptotic boundary geometry.

Findings

Boundary regularity at infinity established for certain currents

Obstructions to high-order regularity in odd dimensions identified

Advances in the asymptotic theory of minimal surfaces in hyperbolic space

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.