A free boundary minimal surface via a 6-sweepout

TL;DR

This paper establishes a new upper bound for the 6-width of the unit 3-ball and constructs a free boundary minimal surface with specific topological and geometric properties, expanding understanding of minimal surfaces in 3-balls.

Contribution

It proves the 6-width of the unit 3-ball is less than 2π and constructs a new free boundary minimal surface with genus at most 1, index at most 5, and area less than 2π.

Findings

6-width of the unit 3-ball is less than 2π

Existence of a free boundary minimal surface with genus ≤ 1 and area < 2π

Surface is not the equatorial disk or critical catenoid

Abstract

We prove that the Almgren-Pitts 6-width of the unit 3-ball is less than $2\pi$. We also prove that there exists a free boundary minimal surface in the unit 3-ball that has genus at most 1, index at most 5, area less than $2\pi$, and is not the equatorial disk or the critical catenoid.