Free boundary minimal surfaces in the unit 3-ball

Abstract

A. Fraser and R. Schoen proved the existence of free boundary minimal surfaces $$\Sigma _n$$ in $$B^3$$ which have genus 0 and n boundary components, for all $$ n \ge 3$$ . For large n, we give an independent construction of $$\Sigma _n$$ and prove the existence of free boundary minimal surfaces $${{\tilde{\Sigma }}}_n$$ in $$B^3$$ which have genus 1 and n boundary components. As n tends to infinity, the sequence $$\Sigma _n$$ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of $$B^3$$ while the sequence $${{\tilde{\Sigma }}}_n$$ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of $$B^3 \setminus \{0\}$$ .