Free boundary minimal surfaces with connected boundary

All the previously constructed free boundary minimal surfaces either have genus 0 or high topological complexity (large genus or many boundary components). Developing methods to generate examples with low genus and few boundary components turned out to be hard. Especially the question whether an embedded free boundary minimal surface with genus 1 and connected boundary existed in the 3-dimensional Euclidean unit ball was rather elusive until it was answered by the following result: The Euclidean unit ball contains embedded free boundary minimal surfaces with connected boundary, arbitrary genus and dihedral symmetry.