Free boundary minimal surfaces in the unit 3-ball

TL;DR

This paper constructs new free boundary minimal surfaces in the 3-ball with specific genus and boundary components, analyzing their convergence behavior as the number of boundary components increases.

Contribution

It provides an independent construction of free boundary minimal surfaces with genus one and multiple boundary components, extending previous genus zero results.

Findings

Sequences converge to double disks or punctured disks

Construction of surfaces with genus one and many boundaries

Analysis of convergence behavior as boundary count increases

Abstract

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $\Sigma\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $ n \geq 3$. For large $n$, we give an independent construction of $\Sigma\_n$ and prove the existence of free boundary minimal surfaces $\tilde \Sigma\_n$ in $B^3$ which have genus $1$ and $n$ boundary components. As $n$ tends to infinity, the sequence $\Sigma\_n$ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of $B^3$ while the sequence $\tilde \Sigma\_n$ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of $B^3-\{0\}$.