Free boundary minimal disc doublings with arbitrary topology

Realisation problem with area bound.
Any compact, orientable surface with boundary can be realised as an embedded free boundary minimal surface with area strictly below $2\pi$ in the three-dimensional Euclidean unit ball.

Given any pair $g$, $b$ of nonnegative integers such that $g+b\ge 3$, we can arrange $g$ necks and $b$ half-necks arbitrarily near the corners of a regular $(g+b)$-gon in order to glue two parallel discs. The resulting surface has genus $g$ and $b$ boundary components whenever $b>0$; in the case $b=0$ the surface has instead genus $g-1$ and $2$ boundary components. The following simulations indicate that any such configuration can be deformed into a free boundary minimal surface in the unit ball. Together with the disc, the critical catenoid and the surface with genus one and connected boundary, we obtain examples for every orientable topological type. The respective symmetry group may vary, but each surface in this family has at least one plane of reflection. Being essentially a gluing of two discs, each surface in this family has area strictly below $2\pi$. Unless the genus is equal to $1$, a multitude of free boundary minimal surfaces with larger area are known to exist in the unit ball, as illustrated in the other sections.

References

• M. Karpukhin, R. Kusner, P. McGrath and D. Stern, Embedded minimal surfaces in 𝕊³ and 𝔹³ via equivariant eigenvalue optimization. Preprint (arXiv:2402.13121).