The exterior Plateau problem (Q2640150)

For each \(r>0\), let \(D_ r\) denote the closed disc of radius r in \({\mathbb{R}}^ 2\). Let \(\Gamma \subset {\mathbb{R}}^ 3\) be a closed rectifiable Jordan curve. Let e be a unit vector in \({\mathbb{R}}^ 3\) and \(\pi_ 0\) the plane normal to e. The main result of this paper is that there exists a conformal minimal immersion u: \(\overset \circ D_ 1\setminus \{0\} {\mathbb{R}}^ 3\) with the following properties: (1) u extends continuously to \(D_ 1\setminus \{0\}\) and \(u|_{\partial D_ 1}\) yields a topological parametrization of \(\Gamma\), (2) u has least area (in an appropriately defined sense), (3) \(u|_{D_ 1\setminus D_{\epsilon}}\) has finite area for each \(\epsilon >0\), (4) \(\lim_{z\to 0} | u(z)| =+\infty\), (5) \(\lim_{z\to 0} n(z)=e\) with n denoting a continuous unit normal of u, (6) \(u|_{D_{\epsilon_ 0}\setminus \{0\}}\) is an embedding and \(u(D_{\epsilon_ 0}\setminus \{0\})\) is a graph over \(\pi_ 0\) for some \(\epsilon_ 0\in (0,1)\), (7) the total Gaussian curvature of u over \(D_{\epsilon_ 0}\setminus \{0\}\) is finite. The conformal map is obtained by extracting a convergent subsequence from a sequence of least area annuli bounded by \(\Gamma \cup \Gamma_ R\) with \(\Gamma_ R\) the round circle of radius R about the origin in \(\pi_ 0\). The behavior at infinity is studied with the aid of curvature estimates due to \textit{R. Schoen} [Semin. on minimal submanifolds, Ann. Math. Stud. 103, 111-126 (1983; Zbl 0532.53042)]. In the final section embedded solutions are obtained in certain cases and a similar higher dimensional result is obtained.