On the l(infinity) regularity of CR mappings of positive codimension Academic Article uri icon

abstract

  • 2018 Elsevier Inc. The present paper tackles the C regularity problem for CR maps h:MM between C-smooth CR submanifolds M,M embedded in complex spaces of possibly different dimensions. For real hypersurfaces MCn+1 and MCn+1 with n>n1 and M strongly pseudoconvex, we prove that every CR transversal map of class Cnn+1 that is nowhere C on some non-empty open subset of M must send this open subset to the set of D'Angelo infinite points of M. As a corollary, we obtain that every CR transversal map h:MM of class Cnn+1 must be C-smooth on a dense open subset of M when M is of D'Angelo finite type. Another consequence establishes the following boundary regularity result for proper holomorphic maps in positive codimension: given Cn+1 and Cn+1 pseudoconvex domains with smooth boundaries and both of D'Angelo finite type, n>n1, any proper holomorphic map h: that extends Cnn+1-smoothly up to must be C-smooth on a dense open subset of . More generally, for CR submanifolds M and M of higher codimensions, our main result describes the impact of the existence of a nowhere smooth CR map h:MM on the CR geometry of M, 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.

published proceedings

  • ADVANCES IN MATHEMATICS

author list (cited authors)

  • Lamel, B., & Mir, N.

citation count

  • 12

complete list of authors

  • Lamel, Bernhard||Mir, Nordine

publication date

  • January 2018