Free Boundary Minimal Annuli Immersed in the Unit Ball

Abstract

We construct a family of compact free boundary minimal annuli immersed in the unit ball B3 of R3, 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 embeddedinB3 foliatedbyspherical 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 B3. Their existence solves in the negative a problem proposed by Wente in 1995.