Existence and classification of $S^1$-invariant free boundary annuli and M\"obius bands in $\mathbb{B}^n$.

We explicitly classify all $S^1$-invariant free boundary minimal annuli and Mobius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Mobius band. First, we determine the supremum of the $k$-th normalized Steklov eigenvalue among all $S^1$-invariant metrics on the Mobius band for each $k \geq 1$, and show that it is achieved by the induced metric from a free boundary minimal embedding of the Mobius band into $\mathbb{B}^4$ by $k$-th Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $S^1$-invariant metrics on the annulus and Mobius band are the induced metrics on explicit free boundary minimal annuli and Mobius bands in $\mathbb{B}^3$ and $\mathbb{B}^4$, including some new families of free boundary minimal annuli and Mobius bands in $\mathbb{B}^4$. Finally, we prove that these are the only $S^1$-invariant free boundary minimal annuli and Mobius bands in $\mathbb{B}^n$.