Free boundary minimal surfaces of any topological type in Euclidean balls via shape optimization

Abstract

For any compact surface $\Sigma$ with smooth, non-empty boundary, we construct a free boundary minimal immersion into a Euclidean Ball ${\mathbb{B}}^N$ where $N$ is controlled in terms of the topology of $\Sigma$. We obtain these as maximizing metrics for the isoperimetric problem for the first non-trivial Steklov eigenvalue. Our main technical result concerns asymptotic control on eigenvalues in a delicate glueing construction which allows us to prove the remaining spectral gap conditions to complete the program by Fraser--Schoen and the second named author to obtain such mazimizing metrics. Our construction draws motivation from earlier work by the first named author with Siffert on the corresponding problem in the closed case.

39 pages, all comments welcome; v2: minor changes, fewer typos

References in corpus (4)

• 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
• Equivariant min-max theory
• Handle attachment and the normalized first eigenvalue

Cited by in corpus (5)

• Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
• Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli
• A remark on the rigidity of the first conformal Steklov eigenvalue
• From Steklov to Laplace: free boundary minimal surfaces with many boundary components
• Higher dimensional surgery and Steklov eigenvalues