Existence and Classification of $$\pmb {\mathbb {S}}^1$$S1 -Invariant Free Boundary Minimal Annuli and Möbius Bands in $$\pmb {\mathbb {B}}^n$$Bn

We explicitly classify all $$\mathbb {S}^1$$-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 $$\mathbb {S}^1$$-invariant metrics on the annulus and Mobius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all $$\mathbb {S}^1$$-invariant metrics on the Mobius band for each $$k \ge 1$$, 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 kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $$\mathbb {S}^1$$-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 $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$.