The Fraser-Li conjecture states that the critical catenoid is the only embedded free-boundary minimal annulus in \(\mathbb{B}^3\), up to rigid motions [A. Fraser and M. M. C. Li, J. Differ. Geom. 96, No. 2, 183–200 (2014; Zbl 1295.53062)]. Fraser and Li work with Steklov eigenvalues while this paper suggests an approach based on the Weierstrass representation and the Gauss map. The authors give a sufficient condition for a curve in \(\mathbb{R}^3\) to be an orthogonal intersection of a surface with a sphere; this allows for an expression of the boundary condition on a free-boundary minimal surface in \(\mathbb{B}^3\) in terms of the Weierstrass data without integration. Then they show that the Gauss map of a free-boundary embedded minimal annulus in \(\mathbb{B}^3\) is one-to-one. The Fraser-Li conjecture thus becomes the problem of determining the Gauss map.
The last section describes minimal annuli orthogonal to spheres in terms of solutions of a Liouville-type boundary problem in an annulus. To quote the authors, this “gives some new insights into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture”.