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Asymptotic behavior of area-minimizing currents in hyperbolic space. (English) Zbl 0688.49042

The paper contains a proof of the following theorem. Let \(\Gamma\) be a compact (n-1)-dimensional \(C^{1,\alpha}\) smooth submanifold of \({\mathbb{R}}^{n+k-1}\times \{0\}\) with \(0\leq \alpha \leq 1\). Then there exists a complete, area-minimizing locally rectifiable n-dimensional current T in \({\mathbb{H}}^{n+k}\) asymptotic to \(\Gamma\) at infinity. Moreover, the set spt(T)\(\cup \Gamma\), in the ordinary Euclidean metric, is, near \(\Gamma\), a \(C^{1,\alpha}\) submanifold with boundary \(\Gamma\) which meets \({\mathbb{R}}^{n+k-1}\times \{0\}\) orthogonally at \(\Gamma\).
In a forthcoming paper the author is going to show that spt(T)\(\cup \Gamma\) is smooth near \(\Gamma\) if \(\Gamma\) is smooth.
At the end of the paper five open problems are listed concerning the interior mass bound, local regularity, boundary regularity and higher multiplicity boundary.
Reviewer: V.Zoller

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
58E99 Variational problems in infinite-dimensional spaces
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