Existence and Classification of S1-Invariant Free Boundary Minimal Annuli and Möbius Bands in Bn.

We explicitly classify all S 1 -invariant free boundary minimal annuli and Möbius bands in 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 Möbius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all S 1 -invariant metrics on the Möbius band for each k ≥ 1 , and show that it is achieved by the induced metric from a free boundary minimal embedding of the Möbius band into B 4 by kth 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 Möbius band are the induced metrics on explicit free boundary minimal annuli and Möbius bands in B 3 and B 4 , including some new families of free boundary minimal annuli and Möbius bands in B 4 . Finally, we prove that these are the only S 1 -invariant free boundary minimal annuli and Möbius bands in B n . [ABSTRACT FROM AUTHOR]