The complex Frobenius Theorem and uniqueness of solutions to the tangential Cauchy-Riemann equations

Abstract

Suppose M is a C ∞ real k-dimensional CR-submanifold of C n , n > 1, and suppose that ∂ ̄ t6 M is the tangential Cauchy-Riemann operator on M. Let S be a C 1 real ( k − 1)-dimensional submanifold of M which is noncharacteristic for ∂̄t6 M at p ϵ S. Conditions are found so that a C ∞ solution f of ∂ ̄ t6 Mf = 0 which vanishes on one side of S in M must vanish in a neighborhood of p in M. If M is a real hypersurface, it is known that such unique continuation always exists. If the codimension of M in C n is greater than 1, and if the excess dimension of the Levi algebra on M is constant, then it is proved that CR-functions on M which vanish on one side of S must vanish in a full neighborhood of p. The assumption on the dimension of the Levi algebra allows us to use the Complex Frobenius Theorem. Other methods to prove such unique continuation results are also developed.