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Mathematical Research Letters

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11 pages
Mathematical Research Letters 10, 447–457 (2003) ANALYTIC REGULARITY OF CR MAPS INTO SPHERES Nordine Mir Abstract. LetM ? CN be a connected real-analytic hypersurface and S2N??1 ? CN ? the unit real sphere, N ? > N ≥ 2. Assume that M does not contain any complex-analytic hypersurface of CN and that there exists at least one strongly pseudoconvex point on M . We show that any CR map f : M ? S2N??1 of class CN??N+1 extends holomorphically to a neighborhood of M in CN . 1. Introduction In this paper we are interested in the analytic regularity of CR mappings from real-analytic hypersurfaces into higher dimensional unit spheres in com- plex space. While there is a wide literature deciding when CR maps, of a given smoothness, between two real-analytic hypersurfaces in the same complex space must be real-analytic (see e.g. [BN90, Fo93, BER99, Hu01] for complete ref- erences up to 1999), very little is known about the analyticity of such maps when the hypersurfaces lie in complex spaces of di?erent dimension. The case of CR maps with target unit spheres, arising e.g. from the embedding problem for pseudoconvex domains into balls (see e.g. [Fo86, EHZ02]), has attracted the at- tention of many authors.

  • s2n ??1 ?

  • no complex-analytic

  • dense open

  • then any

  • strongly pseudoconvex

  • analytic hypersurface

  • constant cr

  • points forms

  • cr map


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Mathematical Research Letters10, 447–457 (2003)
ANALYTIC REGULARITY OF
CR MAPS INTO SPHERES
Nordine Mir
N2N1 Abstract.LetMCbe a connected real-analytic hypersurface andSNCthe unit real sphere,NN > 2. Assume thatMdoes not contain any N complex-analytic hypersurface ofCand that there exists at least one strongly 2N1 pseudoconvex point onM. We show that any CR mapf:MSof class NN+1N Cextends holomorphically to a neighborhood ofMinC.
1. Introduction In this paper we are interested in the analytic regularity of CR mappings from real-analytic hypersurfaces into higher dimensional unit spheres in com-plex space. While there is a wide literature deciding when CR maps, of a given smoothness, between two real-analytic hypersurfaces in the same complex space must be real-analytic (see e.g. [BN90, Fo93, BER99, Hu01] for complete ref-erences up to 1999), very little is known about the analyticity of such maps when the hypersurfaces lie in complex spaces of different dimension. The case of CR maps with target unit spheres, arising e.g. from the embedding problem for pseudoconvex domains into balls (see e.g. [Fo86, EHZ02]), has attracted the at-tention of many authors. The first regularity result in such a situation was given 3 by Webster [W79] who showed that any CR map of classCfrom a real-analytic N2N+1N+1 strongly pseudoconvex hypersurface inCinto the unit sphereSCis real-analyticonadenseopensubsetofthesourcehypersurface.LaterForstnericˇ [Fo89] generalized Webster’s result by showing that the same conclusion holds   NN+1 2N1N for any CR map of classCwith an arbitrary unit sphereSC as a target,N > NHe also asked in the same paper2 (see also [Hu94]). whether the real-analyticity, or equivalently, the holomorphic extendability of such maps holds at every point. A partial positive answer in codimension one (i.e. forNN= 1) was given by Baouendi, Huang and Rothschild [BHR96] for the case of CR maps from a real-algebraic hypersurface of D’Angelo finite type N (i.e. not containing any positive dimensional complex-analytic subvariety) inC 2N+1N+1 into the unit sphereSC[BHR96], the required smoothness for. (In the maps depends on the so-called D’Angelo type of the reference point, which is always greater or equal to two.) In this paper we prove the following theorem,
Received February 24, 2003. Revised version received March 20, 2003. 2000Mathematics Subject Classification.32H02, 32H04, 32V20, 32V30, 32V40.
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