Free boundary minimal surfaces of genus zero

equatorial disc

critical catenoid

3 boundary components

4 boundary components

5 boundary components

6 boundary components

4 boundaries around the vertices of a
tetrahedron

6 boundaries around the vertices of an
octahedron

8 boundaries around the vertices of a
hexahedron

12 boundaries around the vertices of an
icosahedron

20 boundaries around the vertices of a
dodecahedron

32 boundaries around the vertices of a
pentakisdodecahedron (Conjecture)

References

• A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016), 823–890.
• D. Ketover, Free boundary minimal surfaces of unbounded genus, preprint (arXiv:1612.08691).