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

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 $T.$