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. The images below are a first attempt at visualising all such surfaces with up to 8 boundary components.
Free boundary minimal surfaces of genus zero with 3, 2 and 1 boundary components
free boundary Trinoid
Critical Catenoid
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 ≈ 2.5330 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 ≈ 2.6141 π
area ≈ 2.62536 π
Free boundary minimal surfaces of genus zero with many boundary components
18 boundaries around the vertices of a bipyramid
12 boundaries around the vertices of an
icosahedron
icosahedron
20 boundaries around the vertices of a
dodecahedron
dodecahedron
32 boundaries around the vertices of a
pentakisdodecahedron
pentakisdodecahedron
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.
