Free boundary minimal surfaces in the unit 3-ball

A. Fraser and R. Schoen proved the existence of free boundary minimal surfaces Σ _n in B³ which have genus 0 and n boundary components, for all n≥ 3. For large n, we give an independent construction of Σ _n and prove the existence of free boundary minimal surfaces Σ ~ _n in B³ which have genus 1 and n boundary components. As n tends to infinity, the sequence Σ _n converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of B³ while the sequence Σ ~ _n converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of B³\ { 0 }.