Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk

TL;DR

This paper constructs new high genus free boundary minimal surfaces in the unit 3-ball by desingularizing a critical catenoid and equatorial disk intersection, expanding the known family of such surfaces.

Contribution

It introduces a novel desingularization method to produce high genus free boundary minimal surfaces with three boundary components in the 3-ball.

Findings

Constructed a new family of high genus free boundary minimal surfaces.

Surfaces have three boundary components and are analogues of known minimal surfaces.

Potential equivalence with surfaces obtained via min-max methods.

Abstract

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa-Hoffman-Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover using the min-max method.