Density Theorems for Complete Minimal Surfaces in ${\mathbb{R}}^{3}$

Abstract.

In this paper we have proved several approximation theorems for the family of minimal surfaces in ${\mathbb{R}}^{3}$ that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C ^k convergence on compact sets, for any $k \in {\mathbb{N}}$ .

As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in ${\mathbb{R}}^{3}$ with uncountably many ends.