On the \(\mathcal{C}^\infty\) regularity of CR mappings of positive codimension
The authors study the smooth (\(C^\infty\)) regularity problem for CR maps between smooth embedded CR submanifolds (embedded in possibly different dimensions). An embedded submanifold \(M \subset {\mathbb{C}}^n\) is CR if the Cauchy-Riemann equations induce a vector bundle on \(M\), the so-called CR structure, for example if \(M\) is a real hypersurface. A map \(h : M \to M'\) is CR if it preserves the CR structure. The question is: Under what conditions is a \(C^k\) CR map \(h\) actually smooth, at least on a dense open subset. The basic motivating question is the boundary regularity of holomorphic maps of domains with smooth boundaries. Suppose \(M \subset {\mathbb{C}}^{n+1}\), \(M' \subset {\mathbb{C}}^{n'+1}\), and \(n' > n \geq 1\), and \(M\) is strongly pseudoconvex. The authors prove that if \(h : M \to M'\) is a CR transversal CR map of class \(C^{n'-n+1}\) that is nowhere smooth on an open subset of \(M\), then \(h\) must send this set to the set of D'Angelo infinite type points of \(M'\) (points where the order of contact of \(M'\) with possibly singular complex curves is unbounded). A map \(h\) is CR transversal if the push-forward of the tangent space of \(M\) together with CR and anti-CR vector fields span the tangent space of \(M'\). A first corollary is that, if \(M'\) is of D'Angelo finite type, any such map is in fact smooth on a dense open subset. A second corollary is a boundary regularity result. Suppose \(\Omega \subset {\mathbb{C}}^{n+1}\) and \(\Omega' \subset {\mathbb{C}}^{n'+1}\) are pseudoconvex domains with smooth boundary, \(n' > n\), and the boundaries are of D'Angelo finite type. Any proper holomorphic map that extends to be of class \(C^{n'-n+1}\) up to the boundary is in fact smooth on an open dense subset of \(\partial \Omega\). The results are also extended to the higher-codimension case.
scientific article; zbMATH DE number 6929163
scientific article