Area-minimizing hypersurfaces defined by homotopy classes of mappings of 1-essential manifolds (Q923362)

Let M be a smooth, compact, n-dimensional, 1-essential manifold with the additional property that if any loop in some connected open set \(U\subset M\) is contractible in M then U is contained in a coordinate neighborhood of M. Let furthermore N be a smooth, compact, \((n+1)\)-dimensional Riemannian manifold and \(f_ 0: M\to N\) a continuous mapping which induces an injective mapping of the fundamental groups. The author considers te problem of minimizing the area among all \(C^ 1\)-mappings from M to N homotopic to \(f_ 0\) and he proves the existence of a generalized solution to this problem, i.e. an n-dimensional varifold \(V_ 0\) in N which belongs to the closure (with respect to varifold convergence) of the homotopy class of \(f_ 0\) and whose mass equals the inf of the area over this homotopy class. Moreover, it is shown that the singular support of the weight \(\| V_ 0\|\) has k-dimensional Hausdorff measure zero for \(k>n-7\).