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

## TL;DR

This paper establishes partial boundary regularity for co-dimension one area-minimizing currents at points with tangential boundary intersections of $C^{1,eta}$ submanifolds, ensuring tangent cone uniqueness under certain conditions.

## Contribution

It extends boundary regularity results to cases with tangential boundary intersections, providing conditions for tangent cone uniqueness in area-minimizing currents.

## Key findings

- Proves partial boundary regularity at tangential boundary points.
- Shows tangent cone is unique under specified conditions.
- Builds on and closely follows Hardt and Simon's boundary regularity methods.

## 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 closely follows that giving the boundary regularity result of Hardt and Simon.