A Construction of Homologically Area Minimizing Hypersurfaces with Higher Dimensional Singular Sets

Abstract:

We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if $N$ is smooth, compact $n+1$ dimensional manifold, $n\geq 7$, and if $S$ is an embedded, orientable submanifold of dimension $n$, then we construct metrics on $N$ such that the homologically area minimizing hypersurface $M$, homologous to $S$, has a singular set equal to a prescribed number of spheres and tori of codimension less than $n-7$. Near each component $\Sigma$ of the singular set, $M$ is isometric to a product $C\times \Sigma$, where $C$ is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.