Abstract.
Let \(\pi_1\) and \(\pi_2\) be two distinct parallel planes in \({\mathbb R}^3\). Let \(o_1 \in \pi_1\) and \(o_2 \in \pi_2\) denote two points such that the segment \(l_0=[o_1,o_2]\) meets \(\pi_1\) and \(\pi_2\) orthogonally. Let \(l_1 \subset \pi_1\) be a straight line containing \(o_1\), and denote \(\cal{L}\) as the set of straight lines in \(\pi_2\) containing \(o_2\). Then there exists an analytic family \(\{Y_{\theta}:D_{\theta} \rightarrow {\mathbb R}^3 \;:\;\theta \in [0,\pi[ \}\) of proper pairwise non congruent minimal immersions satisfying:
1. \(D_{\theta}\) is homeomorphic to \(\overline{D(0,1)}-\{P_1,Q_1\}\), where \(\{P_1,Q_1\} \subset \s^1=\partial \overline{D(0,1)}.\)
2. \(Y_{\theta} (\partial D_{\theta})=l_1 \cup l_0 \cup l_2\), where \(l_2 \in \cal{L}\).
3. \(Y_{\theta}(D_{\theta})\) is contained in the slab determined by \(\pi_1\) and \(\pi_2\).
4. If \(c_1\) and \(c_2\) are the two connected components of \(\partial D_{\theta}\), then \(Y_{\theta}|_{D_{\theta}-c_i}\) is injective, \(i=1,2\).
5. The parameter \(\theta\) is an analytic determination of the angle that the orthogonal projection of \(l_1\) on \(\pi_2\) makes with \(l_2,\) and \(Y_\theta (D_\theta)\) is invariant under the reflection around a straight line not contained in the surface.
6. If \(Y:D \rightarrow {\mathbb R}^3\) is a proper minimal immersion satisfying 1, 2, 3 and4, then, up to a rigid motion, \(Y=Y_{\theta},\theta \in [0,\pi[\).
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Received December 28, 1997 / Revised November 20, 1999 / Published online October 11, 2000
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López, F., Wei, F. Properly immersed minimal disks bounded by straight lines. Math Ann 318, 667–706 (2000). https://doi.org/10.1007/s002080000134
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DOI: https://doi.org/10.1007/s002080000134