Full generic regularity I for area-minimizing hypersurfaces

Let $M^{n+1}$ be a closed smooth manifold. A Riemannian metric $g$ is called Baire $C^\infty$-generic if it belongs to a Baire generic subset of the space of smooth Riemannian metrics on $M$. A $g$-area homology minimizer is a hypersurface minimizing area in its homology class. Full generic regularity I. For a closed smooth manifold $M^{n+1}$ endowed with a Baire $C^\infty$-generic Riemannian metric $g$, every $g$-area homology minimizer is smooth. This is presented as a more optimistic form of Yau's generic regularity question. It concerns regularity of area minimizers for generic metrics, especially in dimensions where minimizing hypersurfaces can have singularities.