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Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

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Abstract

Let M be a generic CR manifold in \(\Bbb{C}^{m+d}\) of codimension d, locally given as the common zero set of real-valued functions r 1,…,r d. Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r 1,…,r d of third order.

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Authors and Affiliations

  1. Department of Mathematics, POSTECH, Pohang, 790-784, Korea

    Heungju Ahn

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea

    Chong-Kyu Han

Authors
  1. Heungju Ahn
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  2. Chong-Kyu Han
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Corresponding author

Correspondence to Heungju Ahn.

Additional information

Communicated by Alexander Isaev.

H. Ahn was partially supported by NRF-Korea 2010-0024633.

C.-K. Han was partially supported by NRF-Korea 2009-0070971.

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Ahn, H., Han, CK. Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds. J Geom Anal 22, 561–582 (2012). https://doi.org/10.1007/s12220-010-9208-2

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  • DOI: https://doi.org/10.1007/s12220-010-9208-2

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