Pluriharmonic interpolation and hulls of \(C^ 1\) curves in the unit sphere (Q1907046)

The authors study polynomial hulls \(\widehat {\Gamma}\) of simple closed \(C^1\) curves \(\Gamma\) lying in the unit sphere of \(\mathbb{C}^n\). If \(\Gamma\) is not polynomially convex then the one-dimensional analytic variety \(\widehat {\Gamma} \setminus \Gamma\) is shown to be locally a graph of a holomorphic function, the index of transversality \(T(s)\) of \(\Gamma\) has constant sign and \[ \int {{ds} \over {|T(s) |^p}}< \infty \qquad \forall\;p>0. \] Finally, if a curve \(\Gamma\) has \(T(s)\) of constant sign and if \(\int {{ds} \over {T(s)}}\) converges, then the space of pluriharmonic functions in the unit ball forms a closed subspace of \(C(\Gamma)\) of finite dimension (that was proved earlier by Berndtsson and Bruna for \(C^3\) curves \(\Gamma\)).