Regularity of minimal hypersurfaces with a common free boundary

Let be a Riemannian manifold and consider a stationary union of three or more hypersurfaces-with-boundary with a common boundary . We show that if is smooth, then is smooth and each is smooth up to (real analytic in the case is real analytic). Consequently we strengthen a result of Wickramasekera for stable codimension 1 integral varifolds regularity to conclude that under the stronger hypothesis that is a stationary, stable, integral -varifold in an -dimensional, smooth (real analytic) Riemannian manifold such that the support of is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of is empty if , is discrete if , and has Hausdorff dimension at most if n >= 8.