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

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 ${{\mathbf {C}}^n}$.