Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary

TL;DR

This paper constructs new free boundary minimal surfaces in the 3D unit ball that are close to the boundary, have genus zero, and feature many boundary components, using PDE gluing techniques.

Contribution

It introduces a novel PDE gluing method to create free boundary minimal surfaces with prescribed boundary complexity and proximity to the boundary.

Findings

Surfaces are embedded in the unit ball and close to the boundary sphere.

Constructed surfaces have genus zero and multiple boundary components.

Method involves desingularization of unions of catenoidal annuli and flat discs.

Abstract

We construct free boundary minimal surfaces (FBMS) embedded in the unit ball in the Euclidean three-space which are compact, lie arbitrarily close to the boundary unit sphere, are of genus zero, and their boundary has an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations of unions of many catenoidal annuli and two flat discs. The union of the boundaries of the catenoidal annuli and discs is the union of a large finite number of parallel circles contained in the unit sphere, with each parallel circle contained in the boundary of exactly two of the catenoidal annuli and discs.