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

abstract: We develop new methods to compare the span $\Coord(\Sigma)$ of the coordinate functions on a free boundary minimal submanifold $\Sigma$ embedded in the unit $n$-ball $\B^n$ with its first Steklov eigenspace $\firsteigen(\Sigma)$. Using these methods, we show that $\Coord(A)=\firsteigen(A)$ for any embedded free boundary minimal annulus $A$ in $\B^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\Coord=\firsteigen$ for any free boundary minimal surface embedded in $\B^3$ 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.