A Method of Prolongation of Tangential Cauchy–Riemann Equations

Abstract

In this paper we present a method of prolongation of tangential Cauchy-Riemann equations. The technique is, roughly speaking, separating the holomorphic derivatives of CR functions from their complex conjugates and applying the tangential Cauchy-Riemann operators to the holomorphic part. Using this method we show that under generic assumptions mappings of a CR manifold into a CR manifold of higher dimension satisfy a certain Pfaffi.an system in the jet space, which implies the rigidity and the regularity of CR mappings.Let M be a differentiable manifold of dimension 2m + 1. A CR structure on M is a subbundle V of the complexified tangent bundle Tc,M having the following properties: i) each fiber is of complex dimension m,It is well known that if ( M, V) is real analytic (cw) M is locally embeddable into crn+l as a real hypersurface. In this paper we are concerned with CR mappings of M into a cw real hypersurface N of cn+l, n 2: m. Let N be a cw real hypersurface of nondegenerate Levi form in cn+l defined by r(z, z) = 0, where z = (z1, ... 'Zn+1)-Let A and B be (n+ 1)-tuple of nonnegative integers and let zA = zf 1 • • • z~+Y if A= (a1 , • • •, an+1). After a holomorphic change of coordinates r(z, z) can be written as n