Local interpolation for minimal surfaces

Let $a: I\to \mathbb{R}^3$ be a real analytic curve satisfying some conditions. In this article, we show that for any real analytic curve $l:I\to \mathbb R^3$ close to $a$ (in a sense which is precisely defined in the paper) there exists a translation of $l$, and a minimal surface which contains both $a$ and the translated $l$.