  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. This
result makes it possible to express the boundary condition entirely in terms of
the Weierstrass data without integration when dealing with free boundary
minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map
of an embedded free boundary minimal annulus is one to one. By using this, the
Fraser-Li conjecture can be translated into the problem of determining the
Gauss map. On the other hand, we show that the Liouville type boundary value
problem in an annulus gives some new insight into the structure of immersed
minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic
approach to the Fraser-Li conjecture.
