Free boundary minimal surfaces of unbounded genus

Abstract

For each integer $g\ge 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface ${\Sigma }_g$ of symmetry group ${\mathbb{D}}_{g+1}$. For $g$ large, ${\Sigma }_g$ has three boundary components and genus $g$. As $g\to \infty$ the surfaces ${\Sigma }_g$ converge as varifolds to the union of the disk and critical catenoid. These examples are the first with genus greater than $1$ and were conjectured to exist by Fraser-Schoen. We also construct several new free boundary minimal surfaces in $B$ with the symmetry groups of the cube, tetrahedron and dodecahedron. Finally, we prove that free boundary minimal surfaces isotopic to those of Fraser-Schoen can be constructed variationally using an equivariant min-max procedure. We also prove an $\epsilon$-regularity theorem for free boundary minimal surfaces in $B$.

References in corpus (1)

• Equivariant min-max theory

Cited by in corpus (6)

• Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk
• Free boundary minimal surfaces of any topological type in Euclidean balls via shape optimization
• Maximal metrics for the first Steklov eigenvalue on surfaces
• Free boundary minimal hypersurfaces with least area
• Existence and classification of $S^1$-invariant free boundary annuli and Möbius bands in ${\mathbb{B}}^n$
• Gap results for free boundary CMC surfaces in conformally Euclidean three-balls