Compact complete null curves in Complex 3-space

Abstract: We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset\mathcal{M}$ and a continuous map $X:\bar{M}\to\mathbb{C}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain no relatively compact components in $\mathcal{M},$ $X|M$ is a complete null holomorphic curve, $X|{\bar{M}-M}:\bar{M}-M\to\mathbb{C}^3$ is an embedding and the Hausdorff dimension of $X(\bar{M}-M)$ is $1.$ Moreover, for any $\epsilon>0$ and compact null holomorphic curve $Y:N\to\mathbb{C}^3$ with non-empty boundary $Y(\partial N),$ there exist Riemann surfaces $M$ and $\mathcal{M}$ homeomorphic to $N^\circ$ and a map $X:\bar{M}\to\mathbb{C}^3$ in the above conditions such that $\delta^H(Y(\partial N),X(\bar{M}-M))<\epsilon,$ where $\delta^H(\cdot,\cdot)$ means Hausdorff distance in $\mathbb{C}^3.$