Genus-zero minimizers with k boundary components in the 3-ball
Determine whether, for each integer k ≥ 2, the area-minimizing free boundary minimal surface in the Euclidean 3-ball among genus-zero surfaces with k boundary components is the Z2×Dk-symmetric “k-noid” constructed in the paper, and whether for k > 2 its area exceeds that of the critical catenoid.
References
(1) For each k≥2, the minimum area A(0, k) is realized by the Z_2× D_k-symmetric “k-noid” from Theorem \ref{thm:group_Sexistence}. (2) For each k>2, the “k-noid” from Theorem \ref{thm:group_Sexistence} has area greater than that of the critical catenoid.
— Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization
(2402.13121 - Karpukhin et al., 20 Feb 2024) in Section 1.6 Discussion and open questions