Co-Dimension One Area-Minimizing Currents with $C^{1,\alpha}$ Tangentially Immersed Boundary

TL;DR

This paper investigates co-dimension one area-minimizing currents with complex boundary structures, establishing regularity results near boundary points and exploring currents with mean curvature boundary conditions.

Contribution

It introduces a class of area-minimizing currents with $C^{1,eta}$ tangentially immersed boundaries and analyzes their regularity and boundary behavior.

Findings

Supports are smooth hypersurfaces near boundary points with hyperplane tangent cones.

Boundary currents can have generalized mean curvature with specified normal components.

Abstract

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,\alpha}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a smooth hypersurface near any point on the support of $\partial T$ where $T$ has tangent cone which is a hyperplane with constant orientation but non-constant multiplicity. We also introduce and study co-dimensional one area-minimizing locally rectifiable currents $T$ with boundary having co-oriented mean curvature: $\partial T$ has generalized mean curvature $H_{\partial T} = h \nu_{T}$ with $h$ a real-valued function and $\nu_{T}$ the generalized outward pointing unit normal of $\partial T$ with respect to $T.$