Hypoellipticity of nonsingular closed 1-forms on compact manifolds.

*(English)*Zbl 1017.58014The paper offers a characterization of hypoelliptic non-singular closed 1-forms on closed manifolds in terms of generators of the group of periods of the form (a “dophantine condition”).

In order to present the main results of the paper let us recall that a \(C^\infty\), nonsingular and closed 1-form \(\omega\) on a smooth closed manifold \(M\) is said to be hypoelliptic if any distribution \(u\in D'(M)\) such that \(\omega\wedge du \in\Lambda^2 (C^\infty (M))\) is actually a smooth function. The form \(\omega\) is rational if its group of periods is cyclic and is said to be non-Liouville if generators of its groups of periods form a non-Liouville vector (a technical notion extending the notion of the non-Liouville real number, proved by the author to be equivalent to the existence of a sequence of “good” approximations of \(\omega\) by forms \(\omega_i/q_i\), where \(\omega_i\) are integral closed forms and \(q_i\) are integers \(\geq 2)\).

It is well known that a closed manifold \(M\) that admits a non-singular closed 1-form is diffeomorphic to the mapping torus of a diffeomorphism \(\varphi: K\to K\), where \(K\) is a leaf of the foliation associated to a rational form \(\omega_\varepsilon\) which is sufficiently close to \(\omega\), and therefore \(M\) fibers over the circle \(S^1\). In particular \(M\) is diffeomorphic to \(S^1\times K\) if \(\varphi\) is isotopic to the identity. In this last case a theorem of Bergamasco, Cordaro and Malagutti (BCM theorem) states that the 1-form \(d\theta+ \alpha\) \((\theta\) is the natural parameter on \(S^1\) and \(\alpha\) is a closed 1-form on \(K)\) is hypoelliptic if and only if \(\alpha\) is neither rational nor Liouville.

The main results of the paper generalize the BCM theorem and state that, for the first, a non-singular closed 1-form \(\omega\) on a closed manifold, which is rational or Liouville is not hypoelliptic, for the second, such a form \(\omega\) on \(M\) diffeomorphic to the mapping torus of a periodic \(\varphi\) is hypoelliptic provided that \(\omega\) is non-Liouville, and, for the third, \(\omega\) is hypoelliptic provided that the fundamental group of \(M\) is abelian and \(\omega\) is strongly non-Liouville (i.e. one of the periods of an auxiliary 1-form on the leaf \(K\) is a non-Liouville number; \(\varphi\) not necessarily isotopic to a periodic diffeomorphism).

The proof of the first of these statements consists in constructing explicitly an appropriate non-smooth \(u\) using the above-mentioned result of the author on approximation of non-Liouville \(\omega\) by the forms \(\omega_i/q_i\). The method of the proof of the last two theorems consists in reducing them to the case of BCM theorem. Namely, che first of the statements is proved by applying BCM theorem to the cover of \(M\) of the form \(S^1 \times K\) (which exists thanks to the periodicity assumption on \(\varphi)\), and the second one is proved by constructing an open subset in \(M\) of the form \(S^1 \times S^1 \times D\) \((D\) is a disc), on which \(\omega\) is non-Liouville (here the assumption on the fundamental group of \(M\) to be Abelian plays an essential role).

The paper is reasonably self-contained and very nice to read.

In order to present the main results of the paper let us recall that a \(C^\infty\), nonsingular and closed 1-form \(\omega\) on a smooth closed manifold \(M\) is said to be hypoelliptic if any distribution \(u\in D'(M)\) such that \(\omega\wedge du \in\Lambda^2 (C^\infty (M))\) is actually a smooth function. The form \(\omega\) is rational if its group of periods is cyclic and is said to be non-Liouville if generators of its groups of periods form a non-Liouville vector (a technical notion extending the notion of the non-Liouville real number, proved by the author to be equivalent to the existence of a sequence of “good” approximations of \(\omega\) by forms \(\omega_i/q_i\), where \(\omega_i\) are integral closed forms and \(q_i\) are integers \(\geq 2)\).

It is well known that a closed manifold \(M\) that admits a non-singular closed 1-form is diffeomorphic to the mapping torus of a diffeomorphism \(\varphi: K\to K\), where \(K\) is a leaf of the foliation associated to a rational form \(\omega_\varepsilon\) which is sufficiently close to \(\omega\), and therefore \(M\) fibers over the circle \(S^1\). In particular \(M\) is diffeomorphic to \(S^1\times K\) if \(\varphi\) is isotopic to the identity. In this last case a theorem of Bergamasco, Cordaro and Malagutti (BCM theorem) states that the 1-form \(d\theta+ \alpha\) \((\theta\) is the natural parameter on \(S^1\) and \(\alpha\) is a closed 1-form on \(K)\) is hypoelliptic if and only if \(\alpha\) is neither rational nor Liouville.

The main results of the paper generalize the BCM theorem and state that, for the first, a non-singular closed 1-form \(\omega\) on a closed manifold, which is rational or Liouville is not hypoelliptic, for the second, such a form \(\omega\) on \(M\) diffeomorphic to the mapping torus of a periodic \(\varphi\) is hypoelliptic provided that \(\omega\) is non-Liouville, and, for the third, \(\omega\) is hypoelliptic provided that the fundamental group of \(M\) is abelian and \(\omega\) is strongly non-Liouville (i.e. one of the periods of an auxiliary 1-form on the leaf \(K\) is a non-Liouville number; \(\varphi\) not necessarily isotopic to a periodic diffeomorphism).

The proof of the first of these statements consists in constructing explicitly an appropriate non-smooth \(u\) using the above-mentioned result of the author on approximation of non-Liouville \(\omega\) by the forms \(\omega_i/q_i\). The method of the proof of the last two theorems consists in reducing them to the case of BCM theorem. Namely, che first of the statements is proved by applying BCM theorem to the cover of \(M\) of the form \(S^1 \times K\) (which exists thanks to the periodicity assumption on \(\varphi)\), and the second one is proved by constructing an open subset in \(M\) of the form \(S^1 \times S^1 \times D\) \((D\) is a disc), on which \(\omega\) is non-Liouville (here the assumption on the fundamental group of \(M\) to be Abelian plays an essential role).

The paper is reasonably self-contained and very nice to read.

Reviewer: Wieslaw Oledzki (Bialystok)