Rigidity and index of free boundary minimal Y-cones in the unit ball

J. C. C. Nitsche proved that any minimal disk satisfying the free boundary condition in the unit ball $B^3$ must be an equatorial flat disk. Later, Fraser and Schoen extended this rigidity theorem to higher dimensions and to ambient spaces of constant curvature. In this paper, we establish an analogue of the Nitsche-Fraser-Schoen theorem for singular free boundary minimal surfaces of Y-type in $B^n$. Specifically, we prove that any conformal and minimal immersion of the standard compact flat Y-cone meeting $\mathrm{\partial }{B}^n$ orthogonally must itself be a flat Y-cone. In addition, we compute the Morse index of the free boundary flat Y-cone and show that it equals $2(n-2)$. Furthermore, we prove that Y-cones are the only free boundary minimal surfaces in $B^n$ with Morse index $2(n-2)$.