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In this paper, we provide a sufficient condition for a curve on a surface in$\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. Thisresult makes it possible to express the boundary condition entirely in terms ofthe Weierstrass data without integration when dealing with free boundaryminimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss mapof an embedded free boundary minimal annulus is one to one. By using this, theFraser-Li conjecture can be translated into the problem of determining theGauss map. On the other hand, we show that the Liouville type boundary valueproblem in an annulus gives some new insight into the structure of immersedminimal annuli orthogonal to spheres. It also suggests a new PDE theoreticapproach to the Fraser-Li conjecture.

A new approach to the Fraser-Li conjecture with the Weierstrass representation formula