Partial boundary regularity for co-dimension one area-minimizing currents at immersed $C^{1,\alpha}$ tangential boundary points

TL;DR

This paper establishes partial boundary regularity for co-dimension one area-minimizing currents at points with tangential boundary submanifolds, ensuring tangent cone uniqueness under specific geometric conditions.

Contribution

It extends boundary regularity results to currents with boundaries composed of $C^{1,eta}$ submanifolds meeting tangentially, with a focus on tangent cone uniqueness.

Findings

Partial boundary regularity at tangential boundary points

Uniqueness of tangent cones under specified conditions

Extension of Hardt and Simon's boundary regularity results

Abstract

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; this partial regularity is such that we can conclude the tangent cone is unique. The proof follows closely the boundary regularity result given by Hardt and Simon in [9].