Rigidity of free boundary minimal $Y$-surfaces in the unit ball

Abstract

We study rigidity for free boundary minimal $Y$-surfaces in the unit ball $\B^n$, $n\ge 3$, whose three disk faces meet at $120^{\circ }$ along an arc. The face immersions are $C^{2,\alpha }$ and remain of rank two at the two junction endpoints. For a conformal minimal immersion of the standard model of three half-disks, matching parametrizations on the common diameter force the image to be the flat $Y$-cone. We then give geometric criteria that do not require this matching. If $c_j=\ip{\II_j(T,T)}{\II_j(T,τ_j)}$ along the junction, rigidity holds when one $c_j$ changes sign at most twice, or when all three do so at most three times. In $\B^3$, a single sign change of the common geodesic torsion is still enough. For the area problem in which the junction may move and the outer boundary remains on the sphere, every such surface has Morse index at least $n-1$, and equality characterizes the flat $Y$-cone. Its normal nullity is $3(n-2)$, and the null fields are precisely the normal parts of ambient rotations. The arguments combine holomorphic differentials and boundary moment identities with a coupled Steklov calculation and the second variation of area.