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On the unique continuation problem for CR mappings into nonminimal hypersurfaces

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Abstract

We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ℂn. If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp 0 εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′0 =H(p 0 (i.e.,M′ is nonminimal atp′ 0), andH(M) ⊄ Σ′ (forcingM to be nonminimal atp 0), then the transversal component ofH is not flat atp 0. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ⊂ ℂ2, real analytic and of infinite type atp 0 andp′0 respectively (without being Levi flat), such thatH(M) ⊄ Σ′ but the transversal component ofH is flat atp 0 (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ⊂ Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.

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  1. Peter Ebenfelt

    Present address: Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

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  1. Department of Mathematics, University of California at San Diego, 92093, La Jolla, CA

    Peter Ebenfelt

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  1. Peter Ebenfelt
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Correspondence to Peter Ebenfelt.

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The author was supported by a grant from the Swedish Natural Science Research Council.

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Ebenfelt, P. On the unique continuation problem for CR mappings into nonminimal hypersurfaces. J Geom Anal 6, 385–405 (1996). https://doi.org/10.1007/BF02921657

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  • DOI: https://doi.org/10.1007/BF02921657

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