A gap theorem for free boundary minimal surfaces in the three-ball

TL;DR

This paper characterizes the flat equatorial disk and the critical catenoid as unique free boundary minimal surfaces in the unit ball satisfying a specific pinching condition on their second fundamental form.

Contribution

It establishes a new gap theorem that uniquely identifies these minimal surfaces based on curvature pinching conditions.

Findings

The equatorial disk and critical catenoid are characterized by a curvature pinching condition.

These surfaces are uniquely determined among free boundary minimal surfaces in the unit ball.

The result provides a new geometric characterization of these classical minimal surfaces.

Abstract

We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.