Properly immersed minimal disks bounded by straight lines

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, $Y=Y_{\theta},\theta \in [0,\pi[$ .