C ∞ regularity of CR mappings of positive codimension]]>

摘要

The present paper tackles theC∞regularity problem for CR mapsh:M→M′betweenC∞-smooth CR submanifoldsM,M′embedded in complex spaces of possibly different dimensions. For real hypersurfacesM?Cn+1andM′?Cn′+1withn′>n≥1andMstrongly pseudoconvex, we prove that every CR transversal map of classCn′?n+1that is nowhereC∞on some non-empty open subset ofMmust send this open subset to the set of D'Angelo infinite points ofM′. As a corollary, we obtain that every CR transversal maph:M→M′of classCn′?n+1must beC∞-smooth on a dense open subset ofMwhenM′is of D'Angelo finite type. Another consequence establishes the following boundary regularity result for proper holomorphic maps in positive codimension: givenΩ?Cn+1andΩ′?Cn′+1pseudoconvex domains with smooth boundaries ?Ω and?Ω′both of D'Angelo finite type,n′>n≥1, any proper holomorphic maph:Ω→Ω′that extendsCn′?n+1-smoothly up to ?Ω must beC∞-smooth on a dense open subset of ?Ω. More generally, for CR submanifoldsMandM′of higher codimensions, our main result describes the impact of the existence of a nowhere smooth CR maph:M→M′on the CR geometry ofM′, allowing to extend the previously mentioned results in the hypersurface case to any codimension, as well as deriving a number of regularity results for CR maps with D'Angelo infinite type targets.