Free boundary minimal surfaces with 3 nested boundary components

The symmetry and topology of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely. There exist infinitely many pairs of non-isometric free boundary minimal surfaces having the same genus $g$, three boundary components and antiprismatic symmetry group of order $4(g+1)$.

References

• A. Carlotto, M. B. Schulz and D. Wiygul, Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group, Memoirs of the AMS 313-1591 (2025).
• N. Kapouleas and M. Li, Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc, J. Reine Angew. Math. 776 (2021), 201–254.

Free boundary minimal surfaces with 4 nested boundary components

A doubling of the the critical catenoid can be glued to an embedded free boundary minimal surface with sufficiently large, odd genus and four boundary components.

Conjecturally, the Euclidean unit ball also contains free boundary minimal surfaces which look like the higher order Costa–Wohlgemuth surfaces, i.e. they have sufficiently large, even genus and four nested boundary components.

References

• N. Kapouleas and P. McGrath, Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers. Camb. J. Math. 11 (2023).

Free boundary minimal surfaces with 5 nested boundary components

Free boundary minimal surfaces with 7 nested boundary components

Free boundary minimal surfaces with 9 nested boundary components