On uniqueness of free boundary minimal annuli in geodesic balls of ${\mathbb{S}}_{+}^3$ and ${\mathbb{H}}^3$

We consider $\mathrm{\Sigma }$ an embedded free boundary minimal annulus in a geodesic ball in the round hemisphere ${\mathbb{S}}_{+}^3$ or in the hyperbolic space ${\mathbb{H}}^3$. Under the hypothesis of invariance due to an antipodal map on the geodesic ball and using the fact that this surface satisfies the Steklov problem with frequency, we prove that $\mathrm{\Sigma }$ is congruent to a critical rotational annulus.