seminario 24/2 – Carlo Mazza

Il giorno giovedi' 24 Febbraio
ore 14:30 Aula 5

Il dott. Alexandre Kirilov
(Dipartimento di Matematica ed Informatica, Università di Cagliari)

terrà un seminario su

"Global regularity and global solvability for smooth vector fields on S3"

Abstract:
The main goal of this work is to address global properties for classes of smooth (non)singular vector fields on $S3$ . It is known that on $S3$ no $C^infty$ globally hypoelliptic (GH) and no $C^infty$ globally solvable (cohomologically free (CF) in the sense of Katok) vector fields exist.

First we consider classes of vector fields with one or two cycles which are attractors for all other integral curves. We show that the cohomological equation $Lu=f$ , $fin C^infty (S3)$ has at least one $C^infty$ solution defined on $S3$ except the attractors and we can extend the soluion as a weak $L1$ function near the attractors. Moreover, we describe completely the propagation of singularities of all solutions $u$ . As a particular case, we exhibit explicit vector fields whose integral curves coincied with the foliations obtained by transversal intersection of linear holomorphic flows in $C2$ with $S3$ under nondegeneracy conditions.

Secondly, for a class of nonsingular vector fields which are invariant on the fibers (two dimensional tori) in “generalized solid tori" foliations of $S3$ , we derive necessary and sufficient conditions for partial $C^infty$ GH and partial $C^infty$ GS with respect to the fibers. We point out that our construction of solid tori yields novel family of $2D$ tori shrinking to cycles at the end points, different from the Clifford and Lwoson tori.

Moreover, the integral curves of the vector field correspond to foliations obtained by intersection of linear holomorphic flows with
degeneracies in the Poincare' domain.

We provide also results on the global solvability for more general classes of smooth vector fields (admitting also singular points) associated in a natural way to intersection of linear $C$ holomorphic flows and linear $R2$ actions on $S3$ . In particular, we are able to classify completely the global properties of the cohomological equation $Lu =f$ on $S3$ , provided $L$ is obtained by foliations of linear holomorphic $C$ or linear $R2$ actions.

[Joint work with A.Bergamasco and T. Gramchev]

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