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

Abstract

We consider $\Sigma$ 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 $\Sigma$ is congruent to a critical rotational annulus.