Density theorems for complete minimal surfaces in R^3

TL;DR

This paper proves that complete minimal surfaces are dense among all minimal surfaces in R^3 and constructs a novel example of a complete proper minimal surface with uncountably many ends.

Contribution

It introduces approximation theorems showing the density of complete minimal surfaces and constructs the first example with uncountably many ends.

Findings

Complete minimal surfaces are dense in the space of all minimal surfaces.

Constructed the first complete proper minimal surface in R^3 with uncountably many ends.

Established new approximation theorems for minimal surfaces.

Abstract

In this paper we have proved several approximation theorems for the family of minimal surfaces in 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. As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in R^3 with uncountably many ends.