  We develop new methods to compare the span $\mathcal{C}(\Sigma)$ of the
coordinate functions on a free boundary minimal submanifold $\Sigma$ embedded
in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace
$\mathcal{E}_{\sigma_1}(\Sigma)$. Using these methods, we show that
$\mathcal{C}(A)=\mathcal{E}_{\sigma_1}(A)$ for any embedded free boundary
minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and
thus prove that $A$ is congruent to the critical catenoid. We also confirm that
$\mathcal{C}=\mathcal{E}_{\sigma_1}$ for any free boundary minimal surface
embedded in $\mathbb{B}^3$ 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.
