A Construction of Outer Functions along Curves on the Sphere inCn

For a simple smooth curveΓon the unit sphere inCntransverse at all its points, and a non-negative continuous functionϕonΓsatisfying ∫Γ log ϕ ds<−∞ together with a finite number of necessary compatibility conditions, an holomorphic functionfon the unit ball is constructed such that |f|=ϕonΓ. The case of a general, not necessarily transverse, curve is also investigated.