## 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, the topology does not necessarily determine a surface in this family uniquely, as illustrated by the examples with four and five boundary components. A free boundary minimal surface 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.

genus 2 with 2 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 2 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 3 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 4 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 4 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 5 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 5 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 5 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 6 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 7 boundary components

and pyramidal symmetry

and pyramidal symmetry

genus 2 with 8 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 9 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 12 boundary components

and prismatic symmetry

and prismatic symmetry

genus 2 with 14 boundary components

and prismatic symmetry

and prismatic symmetry