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Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of $ {\bf C}\sp{n}$

Author: Al Boggess
Journal: Trans. Amer. Math. Soc. 272 (1982), 351-374
MSC: Primary 32F20; Secondary 32F25, 35N15
MathSciNet review: 656494
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Abstract: The relationship between the Levi geometry of a submanifold of $ {{\mathbf{C}}^n}$ and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of $ {{\mathbf{C}}^n}$, 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 $ {{\mathbf{C}}^n}$. 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 $ {{\mathbf{C}}^n}$. In fact, we show that if $ S$ is a real analytic, generic, submanifold of $ {{\mathbf{C}}^n}$ (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 $ {{\mathbf{C}}^n}$.

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