Construction of Homologically Area Minimizing Hypersurfaces with HigherDimensional Singular Sets

摘要

The authors show that a large variety of singular sets can occur forhomologically 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 > or = 7, and if S is an embedded, orientable submanifold of dimension n, then the authors construct metrics on N such that the homologically area minimizing surface M, homologous to S, has singular set equal to a prescribed number of speres and tori of codimension less than n-7. Near each component sigma of the singular set, M looks like a product C x sigma, where C is any prescribed, strictly stable, strictly minimizing cone. In Thereom B, other singular examples are constructed.