Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems,
1) fix a homology class $\alpha \in H_n(M, \mathbb{Z})$ and minimize $n-1$ volume among all integral cycles supported in $M$ and homologous to $\alpha$ ;
2) fix an integral cycle supported in the boundary, and solve (oriented or not) Plateau problem supported in $M$,
3) find min-max solutions in both settings (prescribe boundary or not) above
Can we find hypersurface with promising regularity (e.g. smooth outside a codimension 7 subset in the interior of $M$ and $C^{1,1}$ when hit the boundary), as solutions to above problems in $M$ ?
