Free boundary minimal surfaces of unbounded genus

TL;DR

This paper constructs new free boundary minimal surfaces in the 3-ball with unbounded genus and specific symmetries, proving their existence via variational methods and analyzing their convergence and regularity properties.

Contribution

It introduces the first known examples of free boundary minimal surfaces with genus greater than one, constructed using variational methods and symmetry considerations.

Findings

Surfaces $oldsymbol{ ext{Σ}_g}$ have genus $g$ and three boundary components for large $g$.

As $g o abla$, $oldsymbol{ ext{Σ}_g}$ converge to a union of a disk and a catenoid.

Existence of free boundary minimal surfaces with cube, tetrahedron, and dodecahedron symmetries.

Abstract

For each integer $g\geq 1$ 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 $g\rightarrow\infty$ 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 $B$.