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 Dirichlet-to-Neumann 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 $k\ge 1$, 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 ${\mathbb{B}}^n$.

19 pages

References in corpus (3)

• Laplacian eigenvalues functionals and metric deformations on compact manifolds
• Free boundary minimal surfaces of unbounded genus
• Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk