## Free boundary minimal surfaces of genus two

Conjecturally the Euclidean unit ball contains a family of free boundary minimal surfaces with genus 2, any number of boundary components and either pyramidal symmetry group of order 6 or prismatic symmetry group of order 12. Curiously, it seems that if the number of boundary components is 2 or 5 this family of surfaces has two different members: one with pyramidal and one with prismatic symmetry. A free boundary minimal surfaces with genus 2 and connected boundary and two more conjectural examples with genus 2 and three boundary components are visualised in the linked sections but the examples there have a slightly different symmetry group: the antiprismatic group of order 12.

2 boundary components, genus = 2, prismatic symmetry

2 boundary components, genus = 2, pyramidal symmetry

3 boundary components, genus = 2, prismatic symmetry

4 boundary components, genus = 2, pyramidal symmetry

5 boundary components, genus = 2, pyramidal symmetry

5 boundary components, genus = 2, prismatic symmetry

6 boundary components, genus = 2, prismatic symmetry

7 boundary components, genus = 2, pyramidal symmetry

8 boundary components, genus = 2, prismatic symmetry

9 boundary components, genus = 2, prismatic symmetry

12 boundary components, genus = 2, prismatic symmetry

14 boundary components, genus = 2, prismatic symmetry