Parametric argument principle and its applications to CR functions and manifolds

It follows from the classical argument principle that if a holomorphic mapping f   from the unit disc Δ to CC or, more generally, to CnCn, smooth in the closed disc, is homologically trivial on the unit circle (i.e. H1(γ)=0H1(γ)=0, γ=f(S1)γ=f(S1), which is equivalent to either γ   being a point or ∂γ≠∅∂γ≠∅), then f=constf=const, i.e. the image of the unit disc degenerates to a point. We establish a parametric version of this fact, for a variety of holomorphic mappings from Δ to CnCn in place of a single mapping. We find conditions for a holomorphic mapping of the unit disc, depending on additional real parameters, under which homological triviality of the boundary image implies collapse of the dimension of the image of the interior. As an application, we obtain estimates of dimensions of complex tangent bundles of real submanifolds in CnCn, in terms of zero moment conditions on families of closed curves covering the manifold. Applying this result to the graphs of functions, we obtain solution of several known problems about characterization of holomorphic CR functions in terms of moment conditions on families of curves.