Co-dimension one area-minimizing currents with $C^{1,\alpha}$ tangentially immersed boundary having Lipschitz co-oriented mean curvature

TL;DR

This paper investigates the regularity of area-minimizing currents with boundaries that are tangentially immersed, Lipschitz mean curvature, and meet tangentially, providing partial boundary regularity results under these conditions.

Contribution

It introduces a partial boundary regularity theorem for area-minimizing currents with complex boundary structures involving $C^{1,eta}$ tangential immersions and Lipschitz mean curvature.

Findings

Support of $T$ near boundary points is either highly irregular or composed of finitely many $C^{1,eta}$ hypersurfaces with shared boundary.

Boundary $oundary T$ consists of $C^{1,eta}$ orientable submanifolds meeting tangentially.

Partial regularity result clarifies structure of currents near boundary points.

Abstract

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,\alpha}$ orientable submanifolds which only meet tangentially and with same orientation, for some $\alpha \in (0,1]$; $\partial T$ has mean curvature $=h \nu_{T}$ where $h$ is a Lipschitz scalar-valued function and $\nu_{T}$ is the generalized outward pointing normal of $\partial T$ with respect to $T.$ We give a partial boundary regularity result for such currents $T.$ We show that near any point $x$ in the support of $\partial T,$ either the support of $T$ has very uncontrolled structure, or the support of $T$ near $x$ is the finite union of orientable $C^{1,\alpha}$ hypersurfaces-with-boundary with disjoint interiors and common boundary points only along the support of $\partial T.$