On the C regularity of CR mappings of positive codimension

Bernhard Lamel, Nordine Mir

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The present paper tackles the C regularity problem for CR maps h:M→M between C-smooth CR submanifolds M,M embedded in complex spaces of possibly different dimensions. For real hypersurfaces M⊂Cn+1 and M⊂Cn+1 with n>n≥1 and M strongly pseudoconvex, we prove that every CR transversal map of class Cn−n+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:M→M of class Cn−n+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 Ω⊂Cn+1 and Ω⊂Cn+1 pseudoconvex domains with smooth boundaries ∂Ω and ∂Ω both of D'Angelo finite type, n>n≥1, any proper holomorphic map h:Ω→Ω that extends Cn−n+1-smoothly up to ∂Ω must be C-smooth on a dense open subset of ∂Ω. 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:M→M 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.

Original languageEnglish
Pages (from-to)696-734
Number of pages39
JournalAdvances in Mathematics
Volume335
DOIs
Publication statusPublished - 7 Sep 2018

Fingerprint

CR Mappings
Codimension
Regularity
CR-submanifold
Proper Map
Holomorphic Maps
Subset
Finite Type
Boundary Regularity
Real Hypersurfaces
Pseudoconvex Domain
Pseudoconvex
Hypersurface
Corollary
Target

Keywords

  • C regularity
  • CR map
  • Points of finite type

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the C regularity of CR mappings of positive codimension. / Lamel, Bernhard; Mir, Nordine.

In: Advances in Mathematics, Vol. 335, 07.09.2018, p. 696-734.

Research output: Contribution to journalArticle

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abstract = "The present paper tackles the C∞ regularity problem for CR maps h:M→M′ between C∞-smooth CR submanifolds M,M′ embedded in complex spaces of possibly different dimensions. For real hypersurfaces M⊂Cn+1 and M′⊂Cn′+1 with n′>n≥1 and M strongly pseudoconvex, we prove that every CR transversal map of class Cn′−n+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:M→M′ of class Cn′−n+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 Ω⊂Cn+1 and Ω′⊂Cn′+1 pseudoconvex domains with smooth boundaries ∂Ω and ∂Ω′ both of D'Angelo finite type, n′>n≥1, any proper holomorphic map h:Ω→Ω′ that extends Cn′−n+1-smoothly up to ∂Ω must be C∞-smooth on a dense open subset of ∂Ω. 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:M→M′ 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.",
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