On Steklov eigenspaces for free boundary minimal surfaces in the unit ball

We develop new methods to compare the span (Σ) of the coordinate functions on a free boundary minimal submanifold Σ embedded in the unit n-ball ⁿ with its first Steklov eigenspace (Σ). Using these methods, we show that (A)=(A) for any embedded free boundary minimal annulus A in ³ invariant under the antipodal map, and thus prove that A is congruent to the critical catenoid. We also confirm that = for any free boundary minimal surface embedded in ³ with the symmetries of many known or expected examples, including: examples of any positive genus from stacking at least three disks; two infinite families of genus $0$ examples with dihedral symmetry, as well as a finite family with the various Platonic symmetries; and examples of any genus by desingularizing several disks that meet at equal angles along a diameter of the ball.