Free boundary minimal annuli in convex balls

Abstract

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $\omega_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.