Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds (Q1884542)

The authors start out by observing that the classical Schwarz reflection principle may be stated in the following way. If a continuous map \(f\) between real analytic curves \(M\) and \(M^{\prime}\) in the complex plane locally extends holomorphically to one side of \(M\), then it extends holomorphically to a neighborhood of \(M\). They then prove a generalization of this principle to CR-maps between CR-manifolds in \(\mathbb{C}^{n}\). Denote by \(E^{\prime}\) the set of all points \(p^{\prime} \in M^{\prime}\) through which there exist irreducible complex-analytic subvarieties of \(M^{\prime}\) of positive dimension. Assume that \(M\) is real-analytic and \(M^{\prime}\) is real-algebraic, and that \(M\) is minimal (in the sense of Tumanov) at a point \(p\). Then for any \(C^{\infty}\)-smooth CR-map \(f: M \rightarrow M^{\prime}\) at least one of the following conditions hold: (i) \(f\) extends holomorphically to a neighborhood of \(p\) in \(\mathbb{C}^{n}\) (ii) \(f\) sends a neighborhood of \(p\) in \(M\) into \(E^{\prime}\). Note that if \(M\) is minimal, then saying that \(f\) is CR is the same as saying that \(f\) extends into a wedge with edge \(M\), so that the theorem indeed fits the paradigm of the Schwarz reflection principle as stated above. The authors also obtain various interesting corollaries.