Free Boundary Plateau Model Cones in $\mathbb{B}^n$ are Rigid under Conformal Minimal Immersions

Abstract

The classical theorem of Nitsche asserts that every free-boundary minimal disk in the unit ball $\mathbb{B}^3$ is an equatorial flat disk. Fraser and Schoen later generalized this rigidity theorem to arbitrary dimensions and ambient spaces of constant sectional curvature. In previous work, the author established an analogous rigidity result for the singular $Y$-cone: any conformal free-boundary minimal immersion of the flat $Y$-cone into $\mathbb{B}^n$ is congruent to the flat $Y$-cone. In this paper we treat the remaining classical two-dimensional Plateau singularity model, namely the tetrahedral $T$-cone. We prove that every conformal free-boundary minimal immersion of the flat $T$-cone into $\mathbb{B}^n$ is congruent, up to an orthogonal transformation, to the flat $T$-cone itself. As a consequence, combining this result with the Nitsche--Fraser--Schoen theorem and the previously established $Y$-cone rigidity theorem, we obtain a unified rigidity theorem for the classical Plateau model domains: any free-boundary minimal Plateau surface in $\mathbb{B}^n$ conformal to a plane disk, a $Y$-cone, or a $T$-cone must be congruent to the corresponding model.