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

AbstractSuppose M is a C∞ real k-dimensional CR-submanifold of Cn, n > 1, and suppose that ∂̄t6M is the tangential Cauchy-Riemann operator on M. Let S be a C1 real (k − 1)-dimensional submanifold of M which is noncharacteristic for ∂̄t6M at p ϵ S. Conditions are found so that a C∞ solution f of ∂̄t6Mf = 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 Cn 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