On the Greenfield-wallach and Katok Conjectures in Dimension Three

that is, the problem of finding a function u on M whose derivative along the flow is equal to a given function f on M . Roughly speaking, the Ccohomology of the flow is the space of non-trivial obstructions to the existence of a C solution u of (1.1) for any given function f ∈ C(M). This notion is well-defined if the range of the Lie derivative operator on the space C(M) is closed. In this case the vector field is called C-stable. In the 80’s A.Katok [Hur85], [Kat01], [Kat03] proposed the following conjecture. A vector field X on a closed, connected orientable manifold is called cohomology free (CF) or rigid if it is stable and the space of obstructions to the existence of solutions of the cohomological equation (1.1) for smooth data is 1-dimensional. A classical example of (CF) vector field, well-known from KAM theory, is given by constant Diophantine vector fields on tori. Katok conjectured that these are the only examples up to smooth conjugacies. A related conjecture has been proposed earlier in 1973 by S. J. Greenfield and N. R. Wallach [GW73]. They introduced and studied [GW73], [GW] the notion of a globally hypoelliptic (GH) vector field and conjectured that the only such vector fields (up to smooth conjugacies) are constant Diophantine vector fields on tori. A (GH) vector field X is characterized by the property that if XU is smooth for some distribution U ∈ D(M) then U is smooth. This notion is modeled on the definition of a hypoelliptic differential operator in the theory of partial differential equations.