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 for generating 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 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.

References

• A. Carlotto, G. Franz and M. B. Schulz, Free boundary minimal surfaces with connected boundary and arbitrary genus, Cambridge Journal of Mathematics 10-4 (2022).

A two-parameter family with connected boundary

Conjecturally, the family of surfaces presented above can be extended to a two-parameter family of free boundary minimal surfaces with connected boundary. The first parameter (the "stacking number") represents the odd number of intersections with the vertical axis. The second parameter is determined by the order of the symmetry group.

Exotic examples with connected boundary

Conjecturally, the Euclidean unit ball contains a multitude of free boundary minimal surfaces with connected boundary, beyond the ones already described in the two-parameter family above. Notably, certain surfaces amongst them may have the same topology and symmetry groups without being congruent. For instance, this page showcases six examples with genus 6 and connected boundaries, three of which have the same symmetry group.