Differential operators associated with holomorphic mappings
Two differential operators, \(\Sigma\) and \(L\), are studied. They act on holomorphic mappings from domains in \(\mathbb{C}^n\) to \(\mathbb{C} P^n\). \(\Sigma\) is of second order and plays a role in several complex variables similar to that of the Schwarzian (which is of third order) in one variable. The newly introduced operator \(L\) is of third order and is analogous to the Schwarzian considered as a curvature. It is shown that for \(g : B^n \to \mathbb{C} P^n\) locally injective holomorphic, \(n > 1\), \(g\) is projective linear iff \(\Sigma \equiv 0\), and if \(n = 1\) then \(\Sigma\equiv 0\) for all such \(g\). Finally, approximation by projective linear mappings is described.
scientific article; zbMATH DE number 729758