On the Greenfield-Wallach and Katok conjectures in dimension three

Cohomologically rigid vector fields: the Katok conjecture in dimension 3

“…We would like to remark that, while this work was in progress, Forni [5] and Matsumoto [15] independently proved the same result.…”

“…Our techniques are completely different to those of Forni [5]. The main novelty of ours consists in using the integrability condition in a very indirect way, and this lets us believe that most of our proof could be reusable to solve the "contact case" independently of Taube's proof of Weinstein conjecture [27].…”

Linearization of cohomology-free vector fields

“…In fact, linear flows on d-dimensional tori provide the classical and only known examples with one-dimensional reduced flow cohomology. It is well known that such a flow is C ∞ -stable if and only if the direction numbers α ∈ R d of the vector field X is a Diophantine vector, that is, if there exist positive constants C and τ such that Chen and Chi [CC00] have essentially proved that Katok's conjecture is equivalent to a conjecture formulated by Greenfield and Wallach in [GW73] and which states that a globally hypoelliptic vector field is a linear diophantine flow on the torus (see [For08] for a review of the relation betwwen the two conjectures).…”

“…In [LdS98] the analogous problem for diffeomorphisms is solved for tori of dimension four or less. Recently independent work of G. Forni [For08], A. Kocsard [Koc09] and S. Matsumoto [Mat09], have proved the Katok-Greenfield-Wallach conjecture when dim M ≤ 3, using Taubes' proof of Weinstein's Conjecture [Tau07].…”

On manifolds supporting distributionally uniquely ergodic diffeomorphisms

“…In 2008, Forni conjectured in [For08] that tori are the only closed manifolds supporting DUE diffeomorphisms.…”