# Co-dimension One Area-Minimizing Currents with \(C^{1,\alpha }\) Tangentially Immersed Boundary

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## 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 \(C^{1,\alpha }\) 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*.

## Keywords

Currents Area-minimizing Boundary regularity## Mathematics Subject Classification

28A75 49Q05 49Q15## Notes

### Acknowledgements

This work was partly supported by the associate membership program of the Korea Institute for Advanced Study. We wish to thank KIAS, and in particular Dr. Hojoo Lee, for their continued support.

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