Free boundary minimal surfaces of genus zero
The critical catenoid is conjectured to be the unique embedded free boundary minimal surface with genus zero and two boundary components in the three-dimensional Euclidean unit ball. Even more challenging would be a rigorous classification of all embedded free boundary minimal surface with genus zero and more than two boundary components.
References
- M. Karpukhin and D. Stern, From Steklov to Laplace: free boundary minimal surfaces with many boundary components, Duke Math. J. 173 (2024).
- 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 (to appear).
- G. Franz and M. B. Schulz, Topological control for min-max free boundary minimal surfaces, preprint (arXiv:2307.00941).
- M. Karpukhin, R. Kusner, P. McGrath and D. Stern, Embedded minimal surfaces in 𝕊3 and 𝔹3 via equivariant eigenvalue optimization, preprint (arXiv:2402.13121).
Free boundary minimal surfaces of genus zero with 3, 2 and 1 boundary components
free boundary minimal Trinoid
area ≈ 1.91176 π ≈ 6.0
area ≈ 1.91176 π ≈ 6.0
Critical Catenoid
area ≈ 1.66711 π
area ≈ 1.66711 π
Equatorial Disc
area = π
area = π
Free boundary minimal surfaces of genus zero with 4 boundary components
area ≈ 1.9783 π
area ≈ 2.1752 π
Free boundary minimal surfaces of genus zero with 5 boundary components
area ≈ 1.9944 π
area ≈ 2.2968 π
area ≈ 2.3027 π
Free boundary minimal surfaces of genus zero with 6 boundary components
area ≈ 1.9986 π
area ≈ 2.35858 π
area ≈ 2.4074 π
area ≈ 2.45495 π
Free boundary minimal surfaces of genus zero with 7 boundary components
area ≈ 1.9999 π
area ≈ 2.3940 π
area ≈ 2.4591 π
area ≈ 2.4605 π
area ≈ 2.5330 π
area ≈ 2.53346 π
area ≈ 2.5389 π
Free boundary minimal surfaces of genus zero with 8 boundary components
area ≈ 1.9999 π
area ≈ 2.4165 π
area ≈ 2.49604 π
area ≈ 2.59049 π
area ≈ 2.59195 π
area ≈ 2.59265 π
area ≈ 2.6141 π
area ≈ 2.62454 π
area ≈ 2.62536 π
Free boundary minimal surfaces of genus zero with many boundary components
genus 0 with 18 boundary components
genus 0 with 12 boundary components
genus 0 with 20 boundary components
genus 0 with 32 boundary components
genus 0 with 64 boundary components
Conjecture (M. B. Schulz, 2021).
There exists a fbms with 61 boundaries
and trivial symmetry group.
There exists a fbms with 61 boundaries
and trivial symmetry group.
