### Abstract

The relationship between the Levi geometry of a submanifold of C” and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of C”, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in C". This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of C". In fact, we show that if S is a real analytic, generic, submanifold of C" (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C".

Original language | English (US) |
---|---|

Pages (from-to) | 351-374 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 272 |

Issue number | 1 |

DOIs | |

State | Published - 1982 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Levi geometry.** / Boggess, Albert.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 272, no. 1, pp. 351-374. https://doi.org/10.1090/S0002-9947-1982-0656494-3

}

TY - JOUR

T1 - Levi geometry

AU - Boggess, Albert

PY - 1982

Y1 - 1982

N2 - The relationship between the Levi geometry of a submanifold of C” and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of C”, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in C". This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of C". In fact, we show that if S is a real analytic, generic, submanifold of C" (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C".

AB - The relationship between the Levi geometry of a submanifold of C” and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of C”, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in C". This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of C". In fact, we show that if S is a real analytic, generic, submanifold of C" (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C".

UR - http://www.scopus.com/inward/record.url?scp=84968483641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968483641&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1982-0656494-3

DO - 10.1090/S0002-9947-1982-0656494-3

M3 - Article

AN - SCOPUS:84968483641

VL - 272

SP - 351

EP - 374

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -