The Y-Cone type Free Boundary Minimal Surfaces in the unit ball

Abstract: J. C. C. Nitsche proved that any minimal disk satisfying the free boundary condition in the unit ball B3 must be an equatorial flat disk. In this short note, we extend Nitsche's result to Y-singular minimal surfaces that satisfy the free boundary condition in B3. Specifically, we show that any conformal and minimal immersion of the standard compact flat Y-cone that meets B3 orthogonally must itself be a flat Y-cone. Furthermore, we compute the Morse index of the free boundary flat Y-cone, showing that it is two, and we prove that Y-cones are the only free boundary minimal surfaces of index two in B3.