Local polynomial convexity of certain graphs in \(C^{3}\)
The aim of the authors is to prove the following theorem: Let \(f\) be a \(C^{2}\) smooth function on a neighborhood of \(0\in\mathbb{C}\). Assume that there exist a holomorphic function \(g\) defined near \(0\in\mathbb{C}\) and a constant \(\lambda \in (0,1)\) satisfying the following conditions: (i) \(|f|^{2} \leq Re(fg)\); (ii) \(g(0)=0\); (iii) \(|g\triangle \overline{f}|<\lambda |\frac{\partial f}{\partial \overline{z}}|^{2}+Re(f\triangle \overline{f}\) for every \(z\neq 0\). Then \(\Gamma_{f}\) is locally polynomially convex at the origin in \(0\in\mathbb{C}^{2}\). Furthermore, there exists an \(r>0\) small enough such that a continuous function on \(X_{r}:=\Gamma_{f}\cap \overline{B}(0,r)\) can be approximated uniformly by polynomials. Few corollaries of this theorem show that the local polynomial convexity of totally real graphs is well-behaved under non-degenerate holomorphic mappings.
scientific article; zbMATH DE number 5648488
scientific article