Free Boundary Minimal Annuli Immersed in the Unit Ball

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.