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

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.03238
 Bibcode:
 2019arXiv191003238F
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory;
 53A10;
 35P15
 EPrint:
 19 pages