Compact complete minimal immersions in $\mathbb{R}^3$

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\mathbb{R}^3$ which can be extended to a continuous map $X:\overline{M}\to\mathbb{R}^3,$ such that $X_{\vert\partial M}$ is an embedding and the Hausdorff dimension of $X(\partial M)$ is $1.$

We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $\mathbb{R}^3$, endowed with the topology of the Hausdorff distance.