Area-minimizing hypersurfaces defined by homotopy classes of mappings of 1-essential manifolds. (English) Zbl 0711.49063
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 \(k>n-7\).
Reviewer: F.Tomi
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 49Q05 | Minimal surfaces and optimization |
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