Free boundary minimal annuli immersed in the unit ball

TL;DR

This paper constructs new examples of free boundary minimal annuli in the unit ball, solving longstanding problems about their existence and properties, including non-embedded and embedded cases, with implications for classical minimal surface theory.

Contribution

It introduces the first non-critical catenoid free boundary minimal annuli in the ball and constructs embedded capillary minimal annuli, addressing open problems from 1985 and 1995.

Findings

Constructed non-critical catenoid free boundary minimal annuli.

Proved the critical catenoid is the only embedded annulus with spherical curvature lines.

Established existence of embedded capillary minimal annuli in the ball.

Abstract

We construct a family of compact free boundary minimal annuli immersed in the unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in $\mathbb{B}^3$ foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in $\mathbb{B}^3$. Their existence solves in the negative a problem proposed by Wente in 1995.